Theorem bdayfinbndlem1 28444

Description: Lemma for bdayfinbnd 28446. Show the first half of the inductive step. (Contributed by Scott Fenton, 26-Feb-2026.)

Hypotheses

Ref	Expression
bdayfinbndlem.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
bdayfinbndlem.2	⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))

Assertion

Ref	Expression
bdayfinbndlem1	⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))

Distinct variable groups:   𝜑,𝑎,𝑏,𝑝,𝑞,𝑤,𝑥,𝑦   𝑁,𝑎,𝑏,𝑝,𝑞,𝑤,𝑥,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem bdayfinbndlem1

Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayfinbndlem.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
2		bdayn0p1 28346	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0s → ( bday ‘(𝑁 +s 1s )) = suc ( bday ‘𝑁))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘(𝑁 +s 1s )) = suc ( bday ‘𝑁))
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ( bday ‘(𝑁 +s 1s )) = suc ( bday ‘𝑁))
5	4	sseq2d 3965	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ↔ ( bday ‘𝑤) ⊆ suc ( bday ‘𝑁)))
6		bdayelon 27750	. . . . . . 7 ⊢ ( bday ‘𝑤) ∈ On
7		bdayelon 27750	. . . . . . . 8 ⊢ ( bday ‘𝑁) ∈ On
8	7	onsuci 7781	. . . . . . 7 ⊢ suc ( bday ‘𝑁) ∈ On
9		onsseleq 6357	. . . . . . 7 ⊢ ((( bday ‘𝑤) ∈ On ∧ suc ( bday ‘𝑁) ∈ On) → (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) ↔ (( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁))))
10	6, 8, 9	mp2an 693	. . . . . 6 ⊢ (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) ↔ (( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
11		onsssuc 6408	. . . . . . . . 9 ⊢ ((( bday ‘𝑤) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ∈ suc ( bday ‘𝑁)))
12	6, 7, 11	mp2an 693	. . . . . . . 8 ⊢ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ∈ suc ( bday ‘𝑁))
13	12	orbi1i 914	. . . . . . 7 ⊢ ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)) ↔ (( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
14	13	bicomi 224	. . . . . 6 ⊢ ((( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
15	10, 14	bitri 275	. . . . 5 ⊢ (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
16		bdayfinbndlem.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
17		fveq2 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → ( bday ‘𝑧) = ( bday ‘𝑤))
18	17	sseq1d 3964	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ⊆ ( bday ‘𝑁)))
19		breq2 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤))
20	18, 19	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤)))
21		eqeq1 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝑁 ↔ 𝑤 = 𝑁))
22		eqeq1 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
23	22	3anbi1d 1443	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
24	23	rexbidv 3159	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
25	24	2rexbidv 3200	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
26	21, 25	orbi12d 919	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) ↔ (𝑤 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
27	20, 26	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))))
28	27	rspccva 3574	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
29	16, 28	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
30	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑁 ∈ ℕ0s)
31		0n0s 28308	. . . . . . . . . . . . . . 15 ⊢ 0s ∈ ℕ0s
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 0s ∈ ℕ0s)
33		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑤 = 𝑁)
34	1	n0snod 28304	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ No )
35	34	addsridd 27945	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 +s 0s ) = 𝑁)
36	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → (𝑁 +s 0s ) = 𝑁)
37	33, 36	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑤 = (𝑁 +s 0s ))
38		0slt1s 27808	. . . . . . . . . . . . . . 15 ⊢ 0s <s 1s
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 0s <s 1s )
40	34	sltp1d 27995	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 <s (𝑁 +s 1s ))
41	35, 40	eqbrtrd 5119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 +s 0s ) <s (𝑁 +s 1s ))
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → (𝑁 +s 0s ) <s (𝑁 +s 1s ))
43		oveq1 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑁 +s (𝑏 /su (2s↑s𝑞))))
44	43	eqeq2d 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞)))))
45		oveq1 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → (𝑎 +s 𝑞) = (𝑁 +s 𝑞))
46	45	breq1d 5107	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑁 +s 𝑞) <s (𝑁 +s 1s )))
47	44, 46	3anbi13d 1441	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ((𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑁 +s 𝑞) <s (𝑁 +s 1s ))))
48		oveq1 7365	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0s → (𝑏 /su (2s↑s𝑞)) = ( 0s /su (2s↑s𝑞)))
49	48	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 0s → (𝑁 +s (𝑏 /su (2s↑s𝑞))) = (𝑁 +s ( 0s /su (2s↑s𝑞))))
50	49	eqeq2d 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0s → (𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞)))))
51		breq1 5100	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0s → (𝑏 <s (2s↑s𝑞) ↔ 0s <s (2s↑s𝑞)))
52	50, 51	3anbi12d 1440	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 0s → ((𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑁 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞))) ∧ 0s <s (2s↑s𝑞) ∧ (𝑁 +s 𝑞) <s (𝑁 +s 1s ))))
53		oveq2 7366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = 0s → (2s↑s𝑞) = (2s↑s 0s ))
54		2sno 28396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2s ∈ No
55		exps0 28404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
56	54, 55	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2s↑s 0s ) = 1s
57	53, 56	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 0s → (2s↑s𝑞) = 1s )
58	57	oveq2d 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 0s → ( 0s /su (2s↑s𝑞)) = ( 0s /su 1s ))
59		0sno 27805	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0s ∈ No
60		divs1 28184	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
61	59, 60	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 0s /su 1s ) = 0s
62	58, 61	eqtrdi 2786	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 0s → ( 0s /su (2s↑s𝑞)) = 0s )
63	62	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 0s → (𝑁 +s ( 0s /su (2s↑s𝑞))) = (𝑁 +s 0s ))
64	63	eqeq2d 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → (𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s 0s )))
65	57	breq2d 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → ( 0s <s (2s↑s𝑞) ↔ 0s <s 1s ))
66		oveq2 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 0s → (𝑁 +s 𝑞) = (𝑁 +s 0s ))
67	66	breq1d 5107	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → ((𝑁 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑁 +s 0s ) <s (𝑁 +s 1s )))
68	64, 65, 67	3anbi123d 1439	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 0s → ((𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞))) ∧ 0s <s (2s↑s𝑞) ∧ (𝑁 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑁 +s 0s ) ∧ 0s <s 1s ∧ (𝑁 +s 0s ) <s (𝑁 +s 1s ))))
69	47, 52, 68	rspc3ev 3592	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s) ∧ (𝑤 = (𝑁 +s 0s ) ∧ 0s <s 1s ∧ (𝑁 +s 0s ) <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
70	30, 32, 32, 37, 39, 42, 69	syl33anc 1388	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
71	70	expr 456	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (𝑤 = 𝑁 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
72		idd 24	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
73		idd 24	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑦 <s (2s↑s𝑝) → 𝑦 <s (2s↑s𝑝)))
74		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ ℕ0s)
75	74	n0snod 28304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ No )
76	75	3adant2 1132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ No )
77	76	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑥 +s 𝑝) ∈ No )
78	77	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) ∈ No )
79	34	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → 𝑁 ∈ No )
80	79	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → 𝑁 ∈ No )
81		peano2no 27964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ No → (𝑁 +s 1s ) ∈ No )
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑁 +s 1s ) ∈ No )
83		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) <s 𝑁)
84	79	sltp1d 27995	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → 𝑁 <s (𝑁 +s 1s ))
85	84	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → 𝑁 <s (𝑁 +s 1s ))
86	78, 80, 82, 83, 85	slttrd 27733	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) <s (𝑁 +s 1s ))
87	86	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → ((𝑥 +s 𝑝) <s 𝑁 → (𝑥 +s 𝑝) <s (𝑁 +s 1s )))
88	72, 73, 87	3anim123d 1446	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑁 +s 1s ))))
89		oveq1 7365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑥 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑥 +s (𝑏 /su (2s↑s𝑞))))
90	89	eqeq2d 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞)))))
91		oveq1 7365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑥 → (𝑎 +s 𝑞) = (𝑥 +s 𝑞))
92	91	breq1d 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑥 +s 𝑞) <s (𝑁 +s 1s )))
93	90, 92	3anbi13d 1441	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑥 → ((𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑥 +s 𝑞) <s (𝑁 +s 1s ))))
94		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑦 → (𝑏 /su (2s↑s𝑞)) = (𝑦 /su (2s↑s𝑞)))
95	94	oveq2d 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑦 → (𝑥 +s (𝑏 /su (2s↑s𝑞))) = (𝑥 +s (𝑦 /su (2s↑s𝑞))))
96	95	eqeq2d 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞)))))
97		breq1 5100	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏 <s (2s↑s𝑞) ↔ 𝑦 <s (2s↑s𝑞)))
98	96, 97	3anbi12d 1440	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑥 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞))) ∧ 𝑦 <s (2s↑s𝑞) ∧ (𝑥 +s 𝑞) <s (𝑁 +s 1s ))))
99		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑞 = 𝑝 → (2s↑s𝑞) = (2s↑s𝑝))
100	99	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = 𝑝 → (𝑦 /su (2s↑s𝑞)) = (𝑦 /su (2s↑s𝑝)))
101	100	oveq2d 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝑝 → (𝑥 +s (𝑦 /su (2s↑s𝑞))) = (𝑥 +s (𝑦 /su (2s↑s𝑝))))
102	101	eqeq2d 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
103	99	breq2d 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → (𝑦 <s (2s↑s𝑞) ↔ 𝑦 <s (2s↑s𝑝)))
104		oveq2 7366	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝑝 → (𝑥 +s 𝑞) = (𝑥 +s 𝑝))
105	104	breq1d 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → ((𝑥 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑥 +s 𝑝) <s (𝑁 +s 1s )))
106	102, 103, 105	3anbi123d 1439	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑝 → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞))) ∧ 𝑦 <s (2s↑s𝑞) ∧ (𝑥 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑁 +s 1s ))))
107	93, 98, 106	rspc3ev 3592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) ∧ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
108	107	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑁 +s 1s )) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
109	108	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑁 +s 1s )) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
110	88, 109	syld 47	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
111	110	rexlimdvvva 3193	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
112	111	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
113	71, 112	jaod 860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((𝑤 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
114	29, 113	syld 47	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
115	114	impr 454	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
116	115	olcd 875	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤))) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
117	116	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
118	117	expd 415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ⊆ ( bday ‘𝑁) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
119	4	eqeq2d 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ↔ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
120		df-ne 2932	. . . . . . . . . . 11 ⊢ (𝑤 ≠ (𝑁 +s 1s ) ↔ ¬ 𝑤 = (𝑁 +s 1s ))
121		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑤 ∈ No )
122		sleloe 27724	. . . . . . . . . . . . . . . . . 18 ⊢ (( 0s ∈ No ∧ 𝑤 ∈ No ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤)))
123	59, 121, 122	sylancr 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤)))
124		simplrl 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑤 ∈ No )
125		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 0s <s 𝑤)
126	124, 125	0elleft 27891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 0s ∈ ( L ‘𝑤))
127	126	ne0d 4293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( L ‘𝑤) ≠ ∅)
128		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
129	128	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
130		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0s → (𝑁 +s 1s ) ∈ ℕ0s)
131	1, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ ℕ0s)
132		n0sbday 28330	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 +s 1s ) ∈ ℕ0s → ( bday ‘(𝑁 +s 1s )) ∈ ω)
133	131, 132	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ( bday ‘(𝑁 +s 1s )) ∈ ω)
134	133	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( bday ‘(𝑁 +s 1s )) ∈ ω)
135	134	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘(𝑁 +s 1s )) ∈ ω)
136	129, 135	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘𝑤) ∈ ω)
137		oldfi 27894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( bday ‘𝑤) ∈ ω → ( O ‘( bday ‘𝑤)) ∈ Fin)
138	136, 137	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( O ‘( bday ‘𝑤)) ∈ Fin)
139		leftssold 27859	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( L ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))
140		ssfi 9099	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( O ‘( bday ‘𝑤)) ∈ Fin ∧ ( L ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))) → ( L ‘𝑤) ∈ Fin)
141	138, 139, 140	sylancl 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( L ‘𝑤) ∈ Fin)
142		leftssno 27861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( L ‘𝑤) ⊆ No
143		sltso 27646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ <s Or No
144		soss 5551	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (( L ‘𝑤) ⊆ No → ( <s Or No → <s Or ( L ‘𝑤)))
145	142, 143, 144	mp2 9	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ <s Or ( L ‘𝑤)
146		fimax2g 9188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( <s Or ( L ‘𝑤) ∧ ( L ‘𝑤) ∈ Fin ∧ ( L ‘𝑤) ≠ ∅) → ∃𝑐 ∈ ( L ‘𝑤)∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒)
147	145, 146	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( L ‘𝑤) ∈ Fin ∧ ( L ‘𝑤) ≠ ∅) → ∃𝑐 ∈ ( L ‘𝑤)∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒)
148	142	sseli 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑒 ∈ ( L ‘𝑤) → 𝑒 ∈ No )
149	142	sseli 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ ( L ‘𝑤) → 𝑐 ∈ No )
150		slenlt 27722	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑒 ∈ No ∧ 𝑐 ∈ No ) → (𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒))
151	148, 149, 150	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑒 ∈ ( L ‘𝑤) ∧ 𝑐 ∈ ( L ‘𝑤)) → (𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒))
152	151	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑐 ∈ ( L ‘𝑤) ∧ 𝑒 ∈ ( L ‘𝑤)) → (𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒))
153	152	ralbidva 3156	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ ( L ‘𝑤) → (∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐 ↔ ∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒))
154	153	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ 𝑐 ∈ ( L ‘𝑤)) → (∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐 ↔ ∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒))
155		rightssold 27860	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( R ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))
156		ssfi 9099	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((( O ‘( bday ‘𝑤)) ∈ Fin ∧ ( R ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))) → ( R ‘𝑤) ∈ Fin)
157	138, 155, 156	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( R ‘𝑤) ∈ Fin)
158	157	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ( R ‘𝑤) ∈ Fin)
159	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → 𝑤 ∈ No )
160		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑤 ≠ (𝑁 +s 1s ))
161	160	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑤 ≠ (𝑁 +s 1s ))
162	161	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → 𝑤 ≠ (𝑁 +s 1s ))
163	162	neneqd 2936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ¬ 𝑤 = (𝑁 +s 1s ))
164		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑤 ∈ Ons) → 𝑤 ∈ Ons)
165	131	ad4antr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑤 ∈ Ons) → (𝑁 +s 1s ) ∈ ℕ0s)
166		n0ons 28314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 +s 1s ) ∈ ℕ0s → (𝑁 +s 1s ) ∈ Ons)
167	165, 166	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑤 ∈ Ons) → (𝑁 +s 1s ) ∈ Ons)
168	129	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
169	168	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑤 ∈ Ons) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
170		bday11on 28244	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑤 ∈ Ons ∧ (𝑁 +s 1s ) ∈ Ons ∧ ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s ))) → 𝑤 = (𝑁 +s 1s ))
171	164, 167, 169, 170	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑤 ∈ Ons) → 𝑤 = (𝑁 +s 1s ))
172	163, 171	mtand 816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ¬ 𝑤 ∈ Ons)
173		elons 28232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 ∈ Ons ↔ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
174	173	notbii 320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ 𝑤 ∈ Ons ↔ ¬ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
175		imnan 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅) ↔ ¬ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
176	174, 175	bitr4i 278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ 𝑤 ∈ Ons ↔ (𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅))
177	172, 176	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → (𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅))
178	159, 177	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ¬ ( R ‘𝑤) = ∅)
179	178	neqned 2938	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ( R ‘𝑤) ≠ ∅)
180		rightssno 27862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ( R ‘𝑤) ⊆ No
181		soss 5551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (( R ‘𝑤) ⊆ No → ( <s Or No → <s Or ( R ‘𝑤)))
182	180, 143, 181	mp2 9	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ <s Or ( R ‘𝑤)
183		fimin2g 9404	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (( <s Or ( R ‘𝑤) ∧ ( R ‘𝑤) ∈ Fin ∧ ( R ‘𝑤) ≠ ∅) → ∃𝑑 ∈ ( R ‘𝑤)∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑)
184	182, 183	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((( R ‘𝑤) ∈ Fin ∧ ( R ‘𝑤) ≠ ∅) → ∃𝑑 ∈ ( R ‘𝑤)∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑)
185	180	sseli 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑑 ∈ ( R ‘𝑤) → 𝑑 ∈ No )
186	180	sseli 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ ( R ‘𝑤) → 𝑓 ∈ No )
187		slenlt 27722	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑑 ∈ No ∧ 𝑓 ∈ No ) → (𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑))
188	185, 186, 187	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑 ∈ ( R ‘𝑤) ∧ 𝑓 ∈ ( R ‘𝑤)) → (𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑))
189	188	ralbidva 3156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑑 ∈ ( R ‘𝑤) → (∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑))
190	189	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑑 ∈ ( R ‘𝑤)) → (∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑))
191		simp2l 1201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑐 ∈ ( L ‘𝑤))
192		simp2r 1202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)
193		simp3l 1203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑑 ∈ ( R ‘𝑤))
194		simp3r 1204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)
195	191, 192, 193, 194	cutminmax 27916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑤 = ({𝑐} |s {𝑑}))
196		simpl2l 1228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑐 ∈ ( L ‘𝑤))
197	139, 196	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑐 ∈ ( O ‘( bday ‘𝑤)))
198	142, 196	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑐 ∈ No )
199		oldbday 27881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((( bday ‘𝑤) ∈ On ∧ 𝑐 ∈ No ) → (𝑐 ∈ ( O ‘( bday ‘𝑤)) ↔ ( bday ‘𝑐) ∈ ( bday ‘𝑤)))
200	6, 198, 199	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → (𝑐 ∈ ( O ‘( bday ‘𝑤)) ↔ ( bday ‘𝑐) ∈ ( bday ‘𝑤)))
201	197, 200	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑐) ∈ ( bday ‘𝑤))
202	129	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
203	202	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
204	1	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑁 ∈ ℕ0s)
205	204	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑁 ∈ ℕ0s)
206	205	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑁 ∈ ℕ0s)
207	206	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑁 ∈ ℕ0s)
208	207, 2	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘(𝑁 +s 1s )) = suc ( bday ‘𝑁))
209	203, 208	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑤) = suc ( bday ‘𝑁))
210	201, 209	eleqtrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑐) ∈ suc ( bday ‘𝑁))
211		bdayelon 27750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ( bday ‘𝑐) ∈ On
212		onsssuc 6408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((( bday ‘𝑐) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ∈ suc ( bday ‘𝑁)))
213	211, 7, 212	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ∈ suc ( bday ‘𝑁))
214	210, 213	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑐) ⊆ ( bday ‘𝑁))
215		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 0s → (𝑒 ≤s 𝑐 ↔ 0s ≤s 𝑐))
216		simpl2r 1229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)
217		simpl1 1193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤))
218	217, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 0s ∈ ( L ‘𝑤))
219	215, 216, 218	rspcdva 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 0s ≤s 𝑐)
220		fveq2 6833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑧 = 𝑐 → ( bday ‘𝑧) = ( bday ‘𝑐))
221	220	sseq1d 3964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 = 𝑐 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ⊆ ( bday ‘𝑁)))
222		breq2 5101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 = 𝑐 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑐))
223	221, 222	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑧 = 𝑐 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐)))
224		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 = 𝑐 → (𝑧 = 𝑁 ↔ 𝑐 = 𝑁))
225		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
226	225	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑧 = 𝑐 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
227	226	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑐 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
228	227	2rexbidv 3200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑧 = 𝑐 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
229		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑥 = 𝑔 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑔 +s (𝑦 /su (2s↑s𝑝))))
230	229	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑥 = 𝑔 → (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝)))))
231		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑥 = 𝑔 → (𝑥 +s 𝑝) = (𝑔 +s 𝑝))
232	231	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑥 = 𝑔 → ((𝑥 +s 𝑝) <s 𝑁 ↔ (𝑔 +s 𝑝) <s 𝑁))
233	230, 232	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑥 = 𝑔 → ((𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁)))
234	233	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑥 = 𝑔 → (∃𝑝 ∈ ℕ0s (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁)))
235		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑦 = ℎ → (𝑦 /su (2s↑s𝑝)) = (ℎ /su (2s↑s𝑝)))
236	235	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑦 = ℎ → (𝑔 +s (𝑦 /su (2s↑s𝑝))) = (𝑔 +s (ℎ /su (2s↑s𝑝))))
237	236	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑦 = ℎ → (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝)))))
238		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑦 = ℎ → (𝑦 <s (2s↑s𝑝) ↔ ℎ <s (2s↑s𝑝)))
239	237, 238	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 = ℎ → ((𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁) ↔ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ∧ ℎ <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁)))
240	239	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑦 = ℎ → (∃𝑝 ∈ ℕ0s (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ∧ ℎ <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁)))
241		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑝 = 𝑖 → (2s↑s𝑝) = (2s↑s𝑖))
242	241	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑝 = 𝑖 → (ℎ /su (2s↑s𝑝)) = (ℎ /su (2s↑s𝑖)))
243	242	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑝 = 𝑖 → (𝑔 +s (ℎ /su (2s↑s𝑝))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
244	243	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑝 = 𝑖 → (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖)))))
245	241	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑝 = 𝑖 → (ℎ <s (2s↑s𝑝) ↔ ℎ <s (2s↑s𝑖)))
246		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑝 = 𝑖 → (𝑔 +s 𝑝) = (𝑔 +s 𝑖))
247	246	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑝 = 𝑖 → ((𝑔 +s 𝑝) <s 𝑁 ↔ (𝑔 +s 𝑖) <s 𝑁))
248	244, 245, 247	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑝 = 𝑖 → ((𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ∧ ℎ <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁) ↔ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
249	248	cbvrexvw 3214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∃𝑝 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ∧ ℎ <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁) ↔ ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))
250	240, 249	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 = ℎ → (∃𝑝 ∈ ℕ0s (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑔 +s 𝑝) <s 𝑁) ↔ ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
251	234, 250	cbvrex2vw 3218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))
252	228, 251	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 = 𝑐 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
253	224, 252	orbi12d 919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑧 = 𝑐 → ((𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) ↔ (𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))))
254	223, 253	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑐 → (((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ ((( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐) → (𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))))
255	16	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
256	255	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
257	256	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
258	257	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
259	254, 258, 198	rspcdva 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ((( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐) → (𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))))
260		simp1ll 1238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝜑)
261	260	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝜑)
262		n0ons 28314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
263	1, 262	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝜑 → 𝑁 ∈ Ons)
264	261, 263	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑁 ∈ Ons)
265		simpl3l 1230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑑 ∈ ( R ‘𝑤))
266	155, 265	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑑 ∈ ( O ‘( bday ‘𝑤)))
267		oldbdayim 27869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑑 ∈ ( O ‘( bday ‘𝑤)) → ( bday ‘𝑑) ∈ ( bday ‘𝑤))
268	266, 267	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑑) ∈ ( bday ‘𝑤))
269	268, 209	eleqtrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑑) ∈ suc ( bday ‘𝑁))
270		bdayelon 27750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ( bday ‘𝑑) ∈ On
271		onsssuc 6408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((( bday ‘𝑑) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ∈ suc ( bday ‘𝑁)))
272	270, 7, 271	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ∈ suc ( bday ‘𝑁))
273	269, 272	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ( bday ‘𝑑) ⊆ ( bday ‘𝑁))
274	180, 265	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑑 ∈ No )
275		madebday 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((( bday ‘𝑁) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( M ‘( bday ‘𝑁)) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
276	7, 274, 275	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → (𝑑 ∈ ( M ‘( bday ‘𝑁)) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
277	273, 276	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑑 ∈ ( M ‘( bday ‘𝑁)))
278		onsbnd 28260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑁 ∈ Ons ∧ 𝑑 ∈ ( M ‘( bday ‘𝑁))) → 𝑑 ≤s 𝑁)
279	264, 277, 278	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑑 ≤s 𝑁)
280	207	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑁 ∈ No )
281	274, 280	slenltd 27726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → (𝑑 ≤s 𝑁 ↔ ¬ 𝑁 <s 𝑑))
282	279, 281	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ¬ 𝑁 <s 𝑑)
283		lltropt 27852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ( L ‘𝑤) <<s ( R ‘𝑤)
284	283	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ( L ‘𝑤) <<s ( R ‘𝑤))
285	284, 191, 193	ssltsepcd 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑐 <s 𝑑)
286	285	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → 𝑐 <s 𝑑)
287		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑐 = 𝑁 → (𝑐 <s 𝑑 ↔ 𝑁 <s 𝑑))
288	287	bicomd 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑐 = 𝑁 → (𝑁 <s 𝑑 ↔ 𝑐 <s 𝑑))
289	286, 288	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → (𝑐 = 𝑁 → 𝑁 <s 𝑑))
290	282, 289	mtod 198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ¬ 𝑐 = 𝑁)
291		orel1 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑐 = 𝑁 → ((𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)) → ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
292	290, 291	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ((𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)) → ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
293	259, 292	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ((( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐) → ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁)))
294		simp3l1 1280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑔 ∈ ℕ0s)
295	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑔 ∈ ℕ0s)
296	295	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑔 ∈ No )
297		simp3l3 1282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑖 ∈ ℕ0s)
298	297	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑖 ∈ ℕ0s)
299		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (𝑔 +s 𝑖) ∈ ℕ0s)
300	295, 298, 299	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 𝑖) ∈ ℕ0s)
301	300	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 𝑖) ∈ No )
302	260	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝜑)
303	302	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝜑)
304	303, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑁 ∈ No )
305		n0sge0 28316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑖 ∈ ℕ0s → 0s ≤s 𝑖)
306	298, 305	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 0s ≤s 𝑖)
307	298	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑖 ∈ No )
308	296, 307	addsge01d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ( 0s ≤s 𝑖 ↔ 𝑔 ≤s (𝑔 +s 𝑖)))
309	306, 308	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑔 ≤s (𝑔 +s 𝑖))
310		simp3r3 1285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (𝑔 +s 𝑖) <s 𝑁)
311	310	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 𝑖) <s 𝑁)
312	296, 301, 304, 309, 311	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑔 <s 𝑁)
313	303, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑁 ∈ ℕ0s)
314		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑔 <s 𝑁 ↔ (𝑔 +s 1s ) ≤s 𝑁))
315	295, 313, 314	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 <s 𝑁 ↔ (𝑔 +s 1s ) ≤s 𝑁))
316	312, 315	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 1s ) ≤s 𝑁)
317		sltirr 27716	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑁 ∈ No → ¬ 𝑁 <s 𝑁)
318	304, 317	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ¬ 𝑁 <s 𝑁)
319		1n0s 28326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ 1s ∈ ℕ0s
320		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝑔 ∈ ℕ0s ∧ 1s ∈ ℕ0s) → (𝑔 +s 1s ) ∈ ℕ0s)
321	295, 319, 320	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 1s ) ∈ ℕ0s)
322	321	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 1s ) ∈ No )
323	322, 304	sltnled 27727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) <s 𝑁 ↔ ¬ 𝑁 ≤s (𝑔 +s 1s )))
324	302	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝜑)
325	131	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ No )
326	324, 325	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ∈ No )
327	324, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑁 ∈ No )
328	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 2s ∈ No )
329	327, 328	subscld 28043	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 -s 2s) ∈ No )
330		1sno 27806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ 1s ∈ No
331	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s ∈ No )
332	329, 331, 331	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (((𝑁 -s 2s) +s 1s ) +s 1s ) = ((𝑁 -s 2s) +s ( 1s +s 1s )))
333		1p1e2s 28393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ( 1s +s 1s ) = 2s
334	333	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((𝑁 -s 2s) +s ( 1s +s 1s )) = ((𝑁 -s 2s) +s 2s)
335		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((𝑁 ∈ No ∧ 2s ∈ No ) → ((𝑁 -s 2s) +s 2s) = 𝑁)
336	327, 54, 335	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 2s) = 𝑁)
337	334, 336	eqtrid 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s ( 1s +s 1s )) = 𝑁)
338	332, 337	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (((𝑁 -s 2s) +s 1s ) +s 1s ) = 𝑁)
339	338, 327	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (((𝑁 -s 2s) +s 1s ) +s 1s ) ∈ No )
340	329, 331	addscld 27960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) ∈ No )
341	202	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
342	341	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
343		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑤 = ({𝑐} |s {𝑑}))
344	142, 191	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑐 ∈ No )
345	344	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑐 ∈ No )
346	345	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 ∈ No )
347	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑔 ∈ ℕ0s)
348		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑔 ∈ ℕ0s → (𝑔 +s 1s ) ∈ ℕ0s)
349	347, 348	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s 1s ) ∈ ℕ0s)
350	349	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s 1s ) ∈ No )
351		subscl 28042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((𝑁 ∈ No ∧ 1s ∈ No ) → (𝑁 -s 1s ) ∈ No )
352	34, 330, 351	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝜑 → (𝑁 -s 1s ) ∈ No )
353	324, 352	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 -s 1s ) ∈ No )
354		simp3r1 1283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
355	354	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
356		simp3r2 1284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ℎ <s (2s↑s𝑖))
357	356	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ℎ <s (2s↑s𝑖))
358	297	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑖 ∈ ℕ0s)
359		expscl 28408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((2s ∈ No ∧ 𝑖 ∈ ℕ0s) → (2s↑s𝑖) ∈ No )
360	54, 358, 359	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (2s↑s𝑖) ∈ No )
361	360	mulslidd 28123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( 1s ·s (2s↑s𝑖)) = (2s↑s𝑖))
362	357, 361	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ℎ <s ( 1s ·s (2s↑s𝑖)))
363		simp3l2 1281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ℎ ∈ ℕ0s)
364	363	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ℎ ∈ ℕ0s)
365	364	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ℎ ∈ No )
366	365, 331, 358	pw2sltdivmul2d 28434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ ℎ <s ( 1s ·s (2s↑s𝑖))))
367	362, 366	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (ℎ /su (2s↑s𝑖)) <s 1s )
368	365, 358	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (ℎ /su (2s↑s𝑖)) ∈ No )
369	347	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑔 ∈ No )
370	368, 331, 369	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
371	367, 370	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
372	355, 371	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 <s (𝑔 +s 1s ))
373		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((𝑔 +s 1s ) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑔 +s 1s ) <s 𝑁 ↔ ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁))
374	321, 313, 373	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) <s 𝑁 ↔ ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁))
375	374	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) <s 𝑁 → ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁))
376	375	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁)
377		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑁 ∈ No ∧ 1s ∈ No ) → ((𝑁 -s 1s ) +s 1s ) = 𝑁)
378	327, 330, 377	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 1s ) +s 1s ) = 𝑁)
379	376, 378	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑔 +s 1s ) +s 1s ) ≤s ((𝑁 -s 1s ) +s 1s ))
380	350, 353, 331	sleadd1d 27975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑔 +s 1s ) ≤s (𝑁 -s 1s ) ↔ ((𝑔 +s 1s ) +s 1s ) ≤s ((𝑁 -s 1s ) +s 1s )))
381	379, 380	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s 1s ) ≤s (𝑁 -s 1s ))
382	346, 350, 353, 372, 381	sltletrd 27734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 <s (𝑁 -s 1s ))
383	333	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑁 -s ( 1s +s 1s )) = (𝑁 -s 2s)
384	383	oveq1i 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑁 -s ( 1s +s 1s )) +s 1s ) = ((𝑁 -s 2s) +s 1s )
385	327, 331, 331	subsubs4d 28074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 1s ) -s 1s ) = (𝑁 -s ( 1s +s 1s )))
386	385	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (((𝑁 -s 1s ) -s 1s ) +s 1s ) = ((𝑁 -s ( 1s +s 1s )) +s 1s ))
387		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((𝑁 -s 1s ) ∈ No ∧ 1s ∈ No ) → (((𝑁 -s 1s ) -s 1s ) +s 1s ) = (𝑁 -s 1s ))
388	353, 330, 387	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (((𝑁 -s 1s ) -s 1s ) +s 1s ) = (𝑁 -s 1s ))
389	386, 388	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s ( 1s +s 1s )) +s 1s ) = (𝑁 -s 1s ))
390	384, 389	eqtr3id 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) = (𝑁 -s 1s ))
391	382, 390	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 <s ((𝑁 -s 2s) +s 1s ))
392	346, 340, 391	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → {𝑐} <<s {((𝑁 -s 2s) +s 1s )})
393	180, 193	sselid 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑑 ∈ No )
394	393	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑑 ∈ No )
395	394	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑑 ∈ No )
396		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑑 = 𝑁)
397	396	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑑 -s 1s ) = (𝑁 -s 1s ))
398	397	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 -s 1s ) = (𝑑 -s 1s ))
399	395	sltm1d 28082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑑 -s 1s ) <s 𝑑)
400	398, 399	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 -s 1s ) <s 𝑑)
401	390, 400	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) <s 𝑑)
402	340, 395, 401	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → {((𝑁 -s 2s) +s 1s )} <<s {𝑑})
403	343, 340, 392, 402	ssltbday 27897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( bday ‘𝑤) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s )))
404	342, 403	eqsstrrd 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s )))
405	131, 166	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ Ons)
406	324, 405	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ∈ Ons)
407	327, 331, 328	addsubsd 28062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 +s 1s ) -s 2s) = ((𝑁 -s 2s) +s 1s ))
408		n0sge0 28316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑔 ∈ ℕ0s → 0s ≤s 𝑔)
409	347, 408	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 0s ≤s 𝑔)
410	331, 369	addsge01d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( 0s ≤s 𝑔 ↔ 1s ≤s ( 1s +s 𝑔)))
411	409, 410	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s ≤s ( 1s +s 𝑔))
412	369, 331	addscomd 27947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s 1s ) = ( 1s +s 𝑔))
413	411, 412	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s ≤s (𝑔 +s 1s ))
414		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s 1s ) <s 𝑁)
415	331, 350, 327, 413, 414	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s <s 𝑁)
416	331, 327, 415	sltled 27743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s ≤s 𝑁)
417	331, 327, 331	sleadd1d 27975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( 1s ≤s 𝑁 ↔ ( 1s +s 1s ) ≤s (𝑁 +s 1s )))
418	416, 417	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( 1s +s 1s ) ≤s (𝑁 +s 1s ))
419	333, 418	eqbrtrrid 5133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 2s ≤s (𝑁 +s 1s ))
420		2nns 28395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ 2s ∈ ℕs
421		nnn0s 28306	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
422	420, 421	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ 2s ∈ ℕ0s
423	422	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 2s ∈ ℕ0s)
424	302, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (𝑁 +s 1s ) ∈ ℕ0s)
425	424	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ∈ ℕ0s)
426		n0subs 28340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ ℕ0s ∧ (𝑁 +s 1s ) ∈ ℕ0s) → (2s ≤s (𝑁 +s 1s ) ↔ ((𝑁 +s 1s ) -s 2s) ∈ ℕ0s))
427	423, 425, 426	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (2s ≤s (𝑁 +s 1s ) ↔ ((𝑁 +s 1s ) -s 2s) ∈ ℕ0s))
428	419, 427	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 +s 1s ) -s 2s) ∈ ℕ0s)
429	407, 428	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) ∈ ℕ0s)
430		n0ons 28314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((𝑁 -s 2s) +s 1s ) ∈ ℕ0s → ((𝑁 -s 2s) +s 1s ) ∈ Ons)
431	429, 430	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) ∈ Ons)
432	406, 431	onsled 28249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 +s 1s ) ≤s ((𝑁 -s 2s) +s 1s ) ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s ))))
433	404, 432	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ≤s ((𝑁 -s 2s) +s 1s ))
434	340	sltp1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ((𝑁 -s 2s) +s 1s ) <s (((𝑁 -s 2s) +s 1s ) +s 1s ))
435	326, 340, 339, 433, 434	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) <s (((𝑁 -s 2s) +s 1s ) +s 1s ))
436	326, 339, 435	sltled 27743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ≤s (((𝑁 -s 2s) +s 1s ) +s 1s ))
437	436, 338	breqtrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ≤s 𝑁)
438	324, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑁 ∈ ℕ0s)
439		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 <s 𝑁 ↔ (𝑁 +s 1s ) ≤s 𝑁))
440	438, 438, 439	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 <s 𝑁 ↔ (𝑁 +s 1s ) ≤s 𝑁))
441	437, 440	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑁 <s 𝑁)
442	441	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) <s 𝑁 → 𝑁 <s 𝑁))
443	323, 442	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (¬ 𝑁 ≤s (𝑔 +s 1s ) → 𝑁 <s 𝑁))
444	318, 443	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → 𝑁 ≤s (𝑔 +s 1s ))
445	444	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) ≤s 𝑁 → 𝑁 ≤s (𝑔 +s 1s )))
446	316, 445	jcai 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s (𝑔 +s 1s )))
447	322, 304	sletri3d 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) = 𝑁 ↔ ((𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s (𝑔 +s 1s ))))
448	446, 447	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → (𝑔 +s 1s ) = 𝑁)
449	310	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑔 +s 𝑖) <s 𝑁)
450		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑔 +s 1s ) = 𝑁)
451	449, 450	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑔 +s 𝑖) <s (𝑔 +s 1s ))
452	297	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑖 ∈ ℕ0s)
453	452	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑖 ∈ No )
454	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 1s ∈ No )
455	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑔 ∈ ℕ0s)
456	455	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑔 ∈ No )
457	453, 454, 456	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑖 <s 1s ↔ (𝑔 +s 𝑖) <s (𝑔 +s 1s )))
458	451, 457	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑖 <s 1s )
459		n0slt1e0 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑖 ∈ ℕ0s → (𝑖 <s 1s ↔ 𝑖 = 0s ))
460	452, 459	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑖 <s 1s ↔ 𝑖 = 0s ))
461	458, 460	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → 𝑖 = 0s )
462	354	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
463	356	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ <s (2s↑s𝑖))
464		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑖 = 0s → (2s↑s𝑖) = (2s↑s 0s ))
465	464	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) → (2s↑s𝑖) = (2s↑s 0s ))
466	465	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (2s↑s𝑖) = (2s↑s 0s ))
467	466, 56	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (2s↑s𝑖) = 1s )
468	463, 467	breqtrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ <s 1s )
469	363	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ ∈ ℕ0s)
470		n0slt1e0 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (ℎ ∈ ℕ0s → (ℎ <s 1s ↔ ℎ = 0s ))
471	469, 470	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (ℎ <s 1s ↔ ℎ = 0s ))
472	468, 471	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ = 0s )
473	472, 467	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (ℎ /su (2s↑s𝑖)) = ( 0s /su 1s ))
474	473, 61	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (ℎ /su (2s↑s𝑖)) = 0s )
475	474	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (𝑔 +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s 0s ))
476	294	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑔 ∈ No )
477	476	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → 𝑔 ∈ No )
478	477	addsridd 27945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (𝑔 +s 0s ) = 𝑔)
479	462, 475, 478	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → 𝑐 = 𝑔)
480	56	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑔 /su (2s↑s 0s )) = (𝑔 /su 1s )
481	476	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑔 ∈ No )
482		divs1 28184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑔 ∈ No → (𝑔 /su 1s ) = 𝑔)
483	481, 482	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 /su 1s ) = 𝑔)
484		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑐 = 𝑔)
485	483, 484	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 /su 1s ) = 𝑐)
486	480, 485	eqtrid 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 /su (2s↑s 0s )) = 𝑐)
487	486	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → {(𝑔 /su (2s↑s 0s ))} = {𝑐})
488	56	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((𝑔 +s 1s ) /su (2s↑s 0s )) = ((𝑔 +s 1s ) /su 1s )
489	294, 348	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (𝑔 +s 1s ) ∈ ℕ0s)
490	489	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 +s 1s ) ∈ ℕ0s)
491	490	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 +s 1s ) ∈ No )
492		divs1 28184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((𝑔 +s 1s ) ∈ No → ((𝑔 +s 1s ) /su 1s ) = (𝑔 +s 1s ))
493	491, 492	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ((𝑔 +s 1s ) /su 1s ) = (𝑔 +s 1s ))
494		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) → (𝑔 +s 1s ) = 𝑁)
495	494	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 +s 1s ) = 𝑁)
496		simplll 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) → 𝑑 = 𝑁)
497	496	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑑 = 𝑁)
498	495, 497	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 +s 1s ) = 𝑑)
499	493, 498	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ((𝑔 +s 1s ) /su 1s ) = 𝑑)
500	488, 499	eqtrid 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ((𝑔 +s 1s ) /su (2s↑s 0s )) = 𝑑)
501	500	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → {((𝑔 +s 1s ) /su (2s↑s 0s ))} = {𝑑})
502	487, 501	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ({(𝑔 /su (2s↑s 0s ))} |s {((𝑔 +s 1s ) /su (2s↑s 0s ))}) = ({𝑐} |s {𝑑}))
503	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑔 ∈ ℕ0s)
504	503	n0zsd 28367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑔 ∈ ℤs)
505	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 0s ∈ ℕ0s)
506	504, 505	pw2cutp1 28438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ({(𝑔 /su (2s↑s 0s ))} |s {((𝑔 +s 1s ) /su (2s↑s 0s ))}) = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))))
507	506	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) = ({(𝑔 /su (2s↑s 0s ))} |s {((𝑔 +s 1s ) /su (2s↑s 0s ))}))
508		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑤 = ({𝑐} |s {𝑑}))
509	502, 507, 508	3eqtr4rd 2781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑤 = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))))
510		mulscl 28114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((2s ∈ No ∧ 𝑔 ∈ No ) → (2s ·s 𝑔) ∈ No )
511	54, 481, 510	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (2s ·s 𝑔) ∈ No )
512	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 1s ∈ No )
513		addslid 27948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
514	330, 513	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ( 0s +s 1s ) = 1s
515	514, 319	eqeltri 2831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ( 0s +s 1s ) ∈ ℕ0s
516	515	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ( 0s +s 1s ) ∈ ℕ0s)
517	511, 512, 516	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) = (((2s ·s 𝑔) /su (2s↑s( 0s +s 1s ))) +s ( 1s /su (2s↑s( 0s +s 1s )))))
518		exps1 28405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
519	54, 518	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (2s↑s 1s ) = 2s
520	519	oveq1i 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((2s↑s 1s ) ·s 𝑔) = (2s ·s 𝑔)
521	520	oveq1i 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((2s↑s 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))) = ((2s ·s 𝑔) /su (2s↑s( 0s +s 1s )))
522	319	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 1s ∈ ℕ0s)
523	481, 505, 522	pw2divscan4d 28421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 /su (2s↑s 0s )) = (((2s↑s 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))))
524	480, 483	eqtrid 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 /su (2s↑s 0s )) = 𝑔)
525	523, 524	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (((2s↑s 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))) = 𝑔)
526	521, 525	eqtr3id 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ((2s ·s 𝑔) /su (2s↑s( 0s +s 1s ))) = 𝑔)
527	514	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (2s↑s( 0s +s 1s )) = (2s↑s 1s )
528	527, 519	eqtri 2758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (2s↑s( 0s +s 1s )) = 2s
529	528	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ( 1s /su (2s↑s( 0s +s 1s ))) = ( 1s /su 2s)
530	529	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ( 1s /su (2s↑s( 0s +s 1s ))) = ( 1s /su 2s))
531	526, 530	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (((2s ·s 𝑔) /su (2s↑s( 0s +s 1s ))) +s ( 1s /su (2s↑s( 0s +s 1s )))) = (𝑔 +s ( 1s /su 2s)))
532	517, 531	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) = (𝑔 +s ( 1s /su 2s)))
533	532	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑤 = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) ↔ 𝑤 = (𝑔 +s ( 1s /su 2s))))
534	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑔 ∈ ℕ0s)
535	319	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 1s ∈ ℕ0s)
536		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑤 = (𝑔 +s ( 1s /su 2s)))
537		sltadd1 27972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (( 0s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s )))
538	59, 330, 330, 537	mp3an 1464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ))
539	38, 538	mpbi 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ( 0s +s 1s ) <s ( 1s +s 1s )
540	539, 514, 333	3brtr3i 5126	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ 1s <s 2s
541	540	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 1s <s 2s)
542		simp-4r 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s))) → (𝑔 +s 1s ) = 𝑁)
543	542	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → (𝑔 +s 1s ) = 𝑁)
544	302	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝜑)
545	544, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑁 ∈ No )
546	545	sltp1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑁 <s (𝑁 +s 1s ))
547	543, 546	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → (𝑔 +s 1s ) <s (𝑁 +s 1s ))
548		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑎 = 𝑔 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (𝑏 /su (2s↑s𝑞))))
549	548	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑎 = 𝑔 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞)))))
550		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑎 = 𝑔 → (𝑎 +s 𝑞) = (𝑔 +s 𝑞))
551	550	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑎 = 𝑔 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s 𝑞) <s (𝑁 +s 1s )))
552	549, 551	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑎 = 𝑔 → ((𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s ))))
553		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑏 = 1s → (𝑏 /su (2s↑s𝑞)) = ( 1s /su (2s↑s𝑞)))
554	553	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑏 = 1s → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s ( 1s /su (2s↑s𝑞))))
555	554	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑏 = 1s → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞)))))
556		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑏 = 1s → (𝑏 <s (2s↑s𝑞) ↔ 1s <s (2s↑s𝑞)))
557	555, 556	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑏 = 1s → ((𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞))) ∧ 1s <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s ))))
558		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑞 = 1s → (2s↑s𝑞) = (2s↑s 1s ))
559	558, 519	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑞 = 1s → (2s↑s𝑞) = 2s)
560	559	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑞 = 1s → ( 1s /su (2s↑s𝑞)) = ( 1s /su 2s))
561	560	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑞 = 1s → (𝑔 +s ( 1s /su (2s↑s𝑞))) = (𝑔 +s ( 1s /su 2s)))
562	561	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑞 = 1s → (𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s ( 1s /su 2s))))
563	559	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑞 = 1s → ( 1s <s (2s↑s𝑞) ↔ 1s <s 2s))
564		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑞 = 1s → (𝑔 +s 𝑞) = (𝑔 +s 1s ))
565	564	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑞 = 1s → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s 1s ) <s (𝑁 +s 1s )))
566	562, 563, 565	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑞 = 1s → ((𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞))) ∧ 1s <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s ( 1s /su 2s)) ∧ 1s <s 2s ∧ (𝑔 +s 1s ) <s (𝑁 +s 1s ))))
567	552, 557, 566	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((𝑔 ∈ ℕ0s ∧ 1s ∈ ℕ0s ∧ 1s ∈ ℕ0s) ∧ (𝑤 = (𝑔 +s ( 1s /su 2s)) ∧ 1s <s 2s ∧ (𝑔 +s 1s ) <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
568	534, 535, 535, 536, 541, 547, 567	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
569	568	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑤 = (𝑔 +s ( 1s /su 2s)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
570	533, 569	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑤 = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
571	509, 570	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
572	571	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → (𝑐 = 𝑔 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
573	479, 572	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
574	573	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → (𝑖 = 0s → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
575	461, 574	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
576	575	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ((𝑔 +s 1s ) = 𝑁 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
577	448, 576	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ 𝑑 = 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
578	577	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (𝑑 = 𝑁 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
579		simprr1 1223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
580		simprr2 1224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑘 <s (2s↑s𝑙))
581		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑙 ∈ ℕ0s)
582		expscl 28408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((2s ∈ No ∧ 𝑙 ∈ ℕ0s) → (2s↑s𝑙) ∈ No )
583	54, 581, 582	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (2s↑s𝑙) ∈ No )
584	583	mulslidd 28123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ( 1s ·s (2s↑s𝑙)) = (2s↑s𝑙))
585	580, 584	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑘 <s ( 1s ·s (2s↑s𝑙)))
586		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑘 ∈ ℕ0s)
587	586	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑘 ∈ No )
588	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 1s ∈ No )
589	587, 588, 581	pw2sltdivmul2d 28434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ((𝑘 /su (2s↑s𝑙)) <s 1s ↔ 𝑘 <s ( 1s ·s (2s↑s𝑙))))
590	585, 589	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑘 /su (2s↑s𝑙)) <s 1s )
591	587, 581	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑘 /su (2s↑s𝑙)) ∈ No )
592		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑗 ∈ ℕ0s)
593	592	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑗 ∈ No )
594	591, 588, 593	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ((𝑘 /su (2s↑s𝑙)) <s 1s ↔ (𝑗 +s (𝑘 /su (2s↑s𝑙))) <s (𝑗 +s 1s )))
595	590, 594	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑗 +s (𝑘 /su (2s↑s𝑙))) <s (𝑗 +s 1s ))
596	579, 595	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑑 <s (𝑗 +s 1s ))
597	294	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑔 ∈ ℕ0s)
598	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑔 ∈ ℕ0s)
599	598	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑔 ∈ No )
600	599	addsridd 27945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → (𝑔 +s 0s ) = 𝑔)
601	363	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ℎ ∈ ℕ0s)
602	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ℎ ∈ ℕ0s)
603		n0sge0 28316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (ℎ ∈ ℕ0s → 0s ≤s ℎ)
604	602, 603	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 0s ≤s ℎ)
605	602	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ℎ ∈ No )
606	297	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑖 ∈ ℕ0s)
607	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑖 ∈ ℕ0s)
608	605, 607	pw2ge0divsd 28423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ( 0s ≤s ℎ ↔ 0s ≤s (ℎ /su (2s↑s𝑖))))
609	59	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 0s ∈ No )
610	605, 607	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → (ℎ /su (2s↑s𝑖)) ∈ No )
611	609, 610, 599	sleadd2d 27976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ( 0s ≤s (ℎ /su (2s↑s𝑖)) ↔ (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖)))))
612	608, 611	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ( 0s ≤s ℎ ↔ (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖)))))
613	604, 612	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖))))
614	600, 613	eqbrtrrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑔 ≤s (𝑔 +s (ℎ /su (2s↑s𝑖))))
615	354	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
616	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
617	614, 616	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑔 ≤s 𝑐)
618	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ∈ ℕ0s)
619	618	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ∈ No )
620	345	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑐 ∈ No )
621	620	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑐 ∈ No )
622	394	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑑 ∈ No )
623	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑑 ∈ No )
624		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ≤s 𝑐)
625	286	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 𝑐 <s 𝑑)
626	625	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑐 <s 𝑑)
627	626	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑐 <s 𝑑)
628	619, 621, 623, 624, 627	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 <s 𝑑)
629	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑔 ∈ ℕ0s)
630	629	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑔 ∈ No )
631	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑑 ∈ No )
632	592	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑗 ∈ ℕ0s)
633		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ ℕ0s)
634	632, 633	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → (𝑗 +s 1s ) ∈ ℕ0s)
635	634	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → (𝑗 +s 1s ) ∈ No )
636		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑔 <s 𝑑)
637		simprll 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑑 <s (𝑗 +s 1s ))
638	630, 631, 635, 636, 637	slttrd 27733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑔 <s (𝑗 +s 1s ))
639		n0sleltp1 28343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s) → (𝑔 ≤s 𝑗 ↔ 𝑔 <s (𝑗 +s 1s )))
640	629, 632, 639	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → (𝑔 ≤s 𝑗 ↔ 𝑔 <s (𝑗 +s 1s )))
641	638, 640	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑑)) → 𝑔 ≤s 𝑗)
642	641	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 <s 𝑑 → 𝑔 ≤s 𝑗))
643	628, 642	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ≤s 𝑗)
644	593	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑗 ∈ No )
645	619, 644	sleloed 27728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 ≤s 𝑗 ↔ (𝑔 <s 𝑗 ∨ 𝑔 = 𝑗)))
646	592	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑗 ∈ ℕ0s)
647		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s) → (𝑔 <s 𝑗 ↔ (𝑔 +s 1s ) ≤s 𝑗))
648	618, 646, 647	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 <s 𝑗 ↔ (𝑔 +s 1s ) ≤s 𝑗))
649	648	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 <s 𝑗 → (𝑔 +s 1s ) ≤s 𝑗))
650	649	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗)) → (𝑔 +s 1s ) ≤s 𝑗)
651	489	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑔 +s 1s ) ∈ ℕ0s)
652	651	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ∈ ℕ0s)
653	652	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ∈ No )
654	592	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑗 ∈ ℕ0s)
655	654	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑗 ∈ No )
656	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑑 ∈ No )
657		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ≤s 𝑗)
658	586	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑘 ∈ ℕ0s)
659		n0sge0 28316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑘 ∈ ℕ0s → 0s ≤s 𝑘)
660	658, 659	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 0s ≤s 𝑘)
661	658	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑘 ∈ No )
662	581	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑙 ∈ ℕ0s)
663	661, 662	pw2ge0divsd 28423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → ( 0s ≤s 𝑘 ↔ 0s ≤s (𝑘 /su (2s↑s𝑙))))
664	661, 662	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑘 /su (2s↑s𝑙)) ∈ No )
665	655, 664	addsge01d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → ( 0s ≤s (𝑘 /su (2s↑s𝑙)) ↔ 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
666	663, 665	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → ( 0s ≤s 𝑘 ↔ 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
667	660, 666	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙))))
668	579	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
669	667, 668	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑗 ≤s 𝑑)
670	653, 655, 656, 657, 669	sletrd 27736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ≤s 𝑑)
671	592	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑗 ∈ ℕ0s)
672	581	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑙 ∈ ℕ0s)
673		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑗 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (𝑗 +s 𝑙) ∈ ℕ0s)
674	671, 672, 673	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑗 +s 𝑙) ∈ ℕ0s)
675	674	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑗 +s 𝑙) ∈ No )
676	302	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝜑)
677	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝜑)
678	677, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑁 ∈ No )
679	677, 325	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑁 +s 1s ) ∈ No )
680		simprr3 1225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑗 +s 𝑙) <s 𝑁)
681	680	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑗 +s 𝑙) <s 𝑁)
682	678	sltp1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑁 <s (𝑁 +s 1s ))
683	675, 678, 679, 681, 682	slttrd 27733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑗 +s 𝑙) <s (𝑁 +s 1s ))
684	675, 679	sltnled 27727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → ((𝑗 +s 𝑙) <s (𝑁 +s 1s ) ↔ ¬ (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
685	683, 684	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → ¬ (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙))
686	651	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑔 +s 1s ) ∈ ℕ0s)
687	686	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑔 +s 1s ) ∈ No )
688	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑑 ∈ No )
689	687, 688	sltnled 27727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → ((𝑔 +s 1s ) <s 𝑑 ↔ ¬ 𝑑 ≤s (𝑔 +s 1s )))
690	679	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ∈ No )
691	593	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑗 ∈ No )
692	675	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑗 +s 𝑙) ∈ No )
693	651	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ∈ ℕ0s)
694	693	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ∈ No )
695	341	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
696	695	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
697		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑤 = ({𝑐} |s {𝑑}))
698	620	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑐 ∈ No )
699	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
700	356	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ℎ <s (2s↑s𝑖))
701	700	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ℎ <s (2s↑s𝑖))
702	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑖 ∈ ℕ0s)
703	54, 702, 359	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (2s↑s𝑖) ∈ No )
704	703	mulslidd 28123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ( 1s ·s (2s↑s𝑖)) = (2s↑s𝑖))
705	701, 704	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ℎ <s ( 1s ·s (2s↑s𝑖)))
706	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ℎ ∈ ℕ0s)
707	706	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ℎ ∈ No )
708	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 1s ∈ No )
709	707, 708, 702	pw2sltdivmul2d 28434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ ℎ <s ( 1s ·s (2s↑s𝑖))))
710	705, 709	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (ℎ /su (2s↑s𝑖)) <s 1s )
711	707, 702	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (ℎ /su (2s↑s𝑖)) ∈ No )
712	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑔 ∈ ℕ0s)
713	712	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑔 ∈ No )
714	711, 708, 713	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
715	710, 714	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
716	699, 715	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑐 <s (𝑔 +s 1s ))
717	698, 694, 716	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → {𝑐} <<s {(𝑔 +s 1s )})
718	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑑 ∈ No )
719		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) <s 𝑑)
720	694, 718, 719	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → {(𝑔 +s 1s )} <<s {𝑑})
721	697, 694, 717, 720	ssltbday 27897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s 1s )))
722	696, 721	eqsstrrd 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s 1s )))
723	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝜑)
724	723, 405	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ∈ Ons)
725		n0ons 28314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((𝑔 +s 1s ) ∈ ℕ0s → (𝑔 +s 1s ) ∈ Ons)
726	693, 725	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ∈ Ons)
727	724, 726	onsled 28249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ((𝑁 +s 1s ) ≤s (𝑔 +s 1s ) ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s 1s ))))
728	722, 727	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ≤s (𝑔 +s 1s ))
729		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑) → (𝑔 +s 1s ) ≤s 𝑗)
730	729	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ≤s 𝑗)
731	690, 694, 691, 728, 730	sletrd 27736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ≤s 𝑗)
732	581	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑙 ∈ ℕ0s)
733		n0sge0 28316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (𝑙 ∈ ℕ0s → 0s ≤s 𝑙)
734	732, 733	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 0s ≤s 𝑙)
735	732	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑙 ∈ No )
736	691, 735	addsge01d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ( 0s ≤s 𝑙 ↔ 𝑗 ≤s (𝑗 +s 𝑙)))
737	734, 736	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑗 ≤s (𝑗 +s 𝑙))
738	690, 691, 692, 731, 737	sletrd 27736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙))
739	738	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → ((𝑔 +s 1s ) <s 𝑑 → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
740	689, 739	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (¬ 𝑑 ≤s (𝑔 +s 1s ) → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
741	685, 740	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑑 ≤s (𝑔 +s 1s ))
742		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑔 +s 1s ) ≤s 𝑑)
743	688, 687	sletri3d 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑑 = (𝑔 +s 1s ) ↔ (𝑑 ≤s (𝑔 +s 1s ) ∧ (𝑔 +s 1s ) ≤s 𝑑)))
744	741, 742, 743	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑑 = (𝑔 +s 1s ))
745	700	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ℎ <s (2s↑s𝑖))
746	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ℎ ∈ ℕ0s)
747	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → 𝑖 ∈ ℕ0s)
748		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (2s↑s𝑖) ∈ ℕ0s)
749	422, 747, 748	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (2s↑s𝑖) ∈ ℕ0s)
750		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((ℎ ∈ ℕ0s ∧ (2s↑s𝑖) ∈ ℕ0s) → (ℎ <s (2s↑s𝑖) ↔ (ℎ +s 1s ) ≤s (2s↑s𝑖)))
751	746, 749, 750	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ <s (2s↑s𝑖) ↔ (ℎ +s 1s ) ≤s (2s↑s𝑖)))
752	745, 751	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ +s 1s ) ≤s (2s↑s𝑖))
753		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (ℎ ∈ ℕ0s → (ℎ +s 1s ) ∈ ℕ0s)
754	363, 753	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (ℎ +s 1s ) ∈ ℕ0s)
755	754	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (ℎ +s 1s ) ∈ ℕ0s)
756	755	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ +s 1s ) ∈ ℕ0s)
757	756	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ +s 1s ) ∈ No )
758	749	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (2s↑s𝑖) ∈ No )
759	757, 758	sleloed 27728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ((ℎ +s 1s ) ≤s (2s↑s𝑖) ↔ ((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖))))
760	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → 𝜑)
761	34, 325	sltnled 27727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝜑 → (𝑁 <s (𝑁 +s 1s ) ↔ ¬ (𝑁 +s 1s ) ≤s 𝑁))
762	40, 761	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝜑 → ¬ (𝑁 +s 1s ) ≤s 𝑁)
763	760, 762	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ¬ (𝑁 +s 1s ) ≤s 𝑁)
764	695	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
765		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑤 = ({𝑐} |s {𝑑}))
766	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑔 ∈ ℕ0s)
767	766	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑔 ∈ No )
768	755	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ +s 1s ) ∈ ℕ0s)
769	768	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ +s 1s ) ∈ No )
770	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑖 ∈ ℕ0s)
771	769, 770	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) ∈ No )
772	767, 771	addscld 27960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) ∈ No )
773	620	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑐 ∈ No )
774	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
775	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ℎ ∈ ℕ0s)
776	775	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ℎ ∈ No )
777	776	sltp1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ℎ <s (ℎ +s 1s ))
778	776, 769, 770	pw2sltdiv1d 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ <s (ℎ +s 1s ) ↔ (ℎ /su (2s↑s𝑖)) <s ((ℎ +s 1s ) /su (2s↑s𝑖))))
779	777, 778	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ /su (2s↑s𝑖)) <s ((ℎ +s 1s ) /su (2s↑s𝑖)))
780	776, 770	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ /su (2s↑s𝑖)) ∈ No )
781	780, 771, 767	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((ℎ /su (2s↑s𝑖)) <s ((ℎ +s 1s ) /su (2s↑s𝑖)) ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))))
782	779, 781	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))))
783	774, 782	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑐 <s (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))))
784	773, 772, 783	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → {𝑐} <<s {(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))})
785	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑑 ∈ No )
786		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ +s 1s ) <s (2s↑s𝑖))
787	54, 770, 359	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (2s↑s𝑖) ∈ No )
788	787	mulslidd 28123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( 1s ·s (2s↑s𝑖)) = (2s↑s𝑖))
789	786, 788	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖)))
790	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 1s ∈ No )
791	769, 790, 770	pw2sltdivmul2d 28434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (((ℎ +s 1s ) /su (2s↑s𝑖)) <s 1s ↔ (ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖))))
792	791	bicomd 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖)) ↔ ((ℎ +s 1s ) /su (2s↑s𝑖)) <s 1s ))
793	771, 790, 767	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (((ℎ +s 1s ) /su (2s↑s𝑖)) <s 1s ↔ (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
794	792, 793	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖)) ↔ (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
795	789, 794	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
796		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑑 = (𝑔 +s 1s ))
797	795, 796	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s 𝑑)
798	772, 785, 797	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → {(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))} <<s {𝑑})
799	765, 772, 784, 798	ssltbday 27897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))))
800	764, 799	eqsstrrd 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))))
801	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝜑)
802	801, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑁 ∈ ℕ0s)
803	310	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑔 +s 𝑖) <s 𝑁)
804	803	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s 𝑖) <s 𝑁)
805	802, 766, 768, 770, 786, 804	bdaypw2bnd 28442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))) ⊆ ( bday ‘𝑁))
806	800, 805	sstrd 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁))
807		onslt 28246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ ((𝑁 ∈ Ons ∧ (𝑁 +s 1s ) ∈ Ons) → (𝑁 <s (𝑁 +s 1s ) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
808	263, 405, 807	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (𝜑 → (𝑁 <s (𝑁 +s 1s ) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
809	801, 808	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑁 <s (𝑁 +s 1s ) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
810	809	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (¬ 𝑁 <s (𝑁 +s 1s ) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
811	325, 34	slenltd 27726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (𝜑 → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑁 +s 1s )))
812	801, 811	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑁 +s 1s )))
813		bdayelon 27750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ( bday ‘(𝑁 +s 1s )) ∈ On
814		ontri1 6350	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((( bday ‘(𝑁 +s 1s )) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
815	813, 7, 814	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s )))
816	815	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
817	810, 812, 816	3bitr4d 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
818	806, 817	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑁 +s 1s ) ≤s 𝑁)
819	818	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ((ℎ +s 1s ) <s (2s↑s𝑖) → (𝑁 +s 1s ) ≤s 𝑁))
820	763, 819	mtod 198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ¬ (ℎ +s 1s ) <s (2s↑s𝑖))
821		orel1 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (¬ (ℎ +s 1s ) <s (2s↑s𝑖) → (((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖)) → (ℎ +s 1s ) = (2s↑s𝑖)))
822	820, 821	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖)) → (ℎ +s 1s ) = (2s↑s𝑖)))
823	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑔 ∈ ℕ0s)
824	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ℎ ∈ ℕ0s)
825		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((2s ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → (2s ·s ℎ) ∈ ℕ0s)
826	422, 824, 825	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s ℎ) ∈ ℕ0s)
827		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((2s ·s ℎ) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s ℎ) +s 1s ) ∈ ℕ0s)
828	826, 319, 827	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) ∈ ℕ0s)
829		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑖 ∈ ℕ0s → (𝑖 +s 1s ) ∈ ℕ0s)
830	606, 829	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑖 +s 1s ) ∈ ℕ0s)
831	830	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (𝑖 +s 1s ) ∈ ℕ0s)
832		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑤 = ({𝑐} |s {𝑑}))
833	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
834	54, 606, 359	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (2s↑s𝑖) ∈ No )
835	834	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s𝑖) ∈ No )
836	823	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑔 ∈ No )
837	835, 836	mulscld 28115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) ·s 𝑔) ∈ No )
838	601	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ℎ ∈ No )
839	838	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ℎ ∈ No )
840	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑖 ∈ ℕ0s)
841	837, 839, 840	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s (ℎ /su (2s↑s𝑖))))
842	836, 840	pw2divscan3d 28418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) = 𝑔)
843	842	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
844	841, 843	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖)) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
845	833, 844	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑐 = ((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖)))
846	845	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → {𝑐} = {((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))})
847		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑑 = (𝑔 +s 1s ))
848	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 1s ∈ No )
849	837, 839, 848	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) = (((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )))
850	849	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )) /su (2s↑s𝑖)))
851		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (ℎ +s 1s ) = (2s↑s𝑖))
852	851, 835	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (ℎ +s 1s ) ∈ No )
853	837, 852, 840	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s ((ℎ +s 1s ) /su (2s↑s𝑖))))
854	851	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) = ((2s↑s𝑖) /su (2s↑s𝑖)))
855	840	pw2divsidd 28433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) /su (2s↑s𝑖)) = 1s )
856	854, 855	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) = 1s )
857	842, 856	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s ((ℎ +s 1s ) /su (2s↑s𝑖))) = (𝑔 +s 1s ))
858	853, 857	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )) /su (2s↑s𝑖)) = (𝑔 +s 1s ))
859	850, 858	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖)) = (𝑔 +s 1s ))
860	847, 859	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑑 = (((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖)))
861	860	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → {𝑑} = {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))})
862	846, 861	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ({𝑐} |s {𝑑}) = ({((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))} |s {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))}))
863	422, 840, 748	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s𝑖) ∈ ℕ0s)
864		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((2s↑s𝑖) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s)
865	863, 823, 864	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s)
866		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ ((((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → (((2s↑s𝑖) ·s 𝑔) +s ℎ) ∈ ℕ0s)
867	865, 824, 866	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((2s↑s𝑖) ·s 𝑔) +s ℎ) ∈ ℕ0s)
868	867	n0zsd 28367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((2s↑s𝑖) ·s 𝑔) +s ℎ) ∈ ℤs)
869	868, 840	pw2cutp1 28438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ({((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))} |s {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))}) = (((2s ·s (((2s↑s𝑖) ·s 𝑔) +s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
870	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 2s ∈ No )
871	870, 837, 839	addsdid 28136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (((2s↑s𝑖) ·s 𝑔) +s ℎ)) = ((2s ·s ((2s↑s𝑖) ·s 𝑔)) +s (2s ·s ℎ)))
872		expsp1 28406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((2s ∈ No ∧ 𝑖 ∈ ℕ0s) → (2s↑s(𝑖 +s 1s )) = ((2s↑s𝑖) ·s 2s))
873	54, 840, 872	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s(𝑖 +s 1s )) = ((2s↑s𝑖) ·s 2s))
874	835, 870	mulscomd 28120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) ·s 2s) = (2s ·s (2s↑s𝑖)))
875	873, 874	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s(𝑖 +s 1s )) = (2s ·s (2s↑s𝑖)))
876	875	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s(𝑖 +s 1s )) ·s 𝑔) = ((2s ·s (2s↑s𝑖)) ·s 𝑔))
877	870, 835, 836	mulsassd 28147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s (2s↑s𝑖)) ·s 𝑔) = (2s ·s ((2s↑s𝑖) ·s 𝑔)))
878	876, 877	eqtr2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s ((2s↑s𝑖) ·s 𝑔)) = ((2s↑s(𝑖 +s 1s )) ·s 𝑔))
879	878	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ((2s↑s𝑖) ·s 𝑔)) +s (2s ·s ℎ)) = (((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)))
880	871, 879	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (((2s↑s𝑖) ·s 𝑔) +s ℎ)) = (((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)))
881	880	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s (((2s↑s𝑖) ·s 𝑔) +s ℎ)) +s 1s ) = ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ))
882	881	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((2s ·s (((2s↑s𝑖) ·s 𝑔) +s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))) = (((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
883	869, 882	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ({((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))} |s {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))}) = (((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
884	832, 862, 883	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑤 = (((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
885		expscl 28408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ ((2s ∈ No ∧ (𝑖 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑖 +s 1s )) ∈ No )
886	54, 831, 885	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s(𝑖 +s 1s )) ∈ No )
887	886, 836	mulscld 28115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s(𝑖 +s 1s )) ·s 𝑔) ∈ No )
888	826	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s ℎ) ∈ No )
889	887, 888, 848	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) = (((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s ((2s ·s ℎ) +s 1s )))
890	889	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))) = ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s ((2s ·s ℎ) +s 1s )) /su (2s↑s(𝑖 +s 1s ))))
891	884, 890	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑤 = ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s ((2s ·s ℎ) +s 1s )) /su (2s↑s(𝑖 +s 1s ))))
892	828	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) ∈ No )
893	887, 892, 831	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s ((2s ·s ℎ) +s 1s )) /su (2s↑s(𝑖 +s 1s ))) = ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) /su (2s↑s(𝑖 +s 1s ))) +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
894	836, 831	pw2divscan3d 28418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (((2s↑s(𝑖 +s 1s )) ·s 𝑔) /su (2s↑s(𝑖 +s 1s ))) = 𝑔)
895	894	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) /su (2s↑s(𝑖 +s 1s ))) +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))) = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
896	893, 895	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s ((2s ·s ℎ) +s 1s )) /su (2s↑s(𝑖 +s 1s ))) = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
897	891, 896	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
898	848, 870, 888	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ( 1s <s 2s ↔ ((2s ·s ℎ) +s 1s ) <s ((2s ·s ℎ) +s 2s)))
899	540, 898	mpbii 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) <s ((2s ·s ℎ) +s 2s))
900	851	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (ℎ +s 1s )) = (2s ·s (2s↑s𝑖)))
901	870, 839, 848	addsdid 28136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (ℎ +s 1s )) = ((2s ·s ℎ) +s (2s ·s 1s )))
902		mulsrid 28093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
903	54, 902	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ (2s ·s 1s ) = 2s
904	903	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ ((2s ·s ℎ) +s (2s ·s 1s )) = ((2s ·s ℎ) +s 2s)
905	904	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s (2s ·s 1s )) = ((2s ·s ℎ) +s 2s))
906	901, 905	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (ℎ +s 1s )) = ((2s ·s ℎ) +s 2s))
907	900, 906	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s (2s↑s𝑖)) = ((2s ·s ℎ) +s 2s))
908	874, 907	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) ·s 2s) = ((2s ·s ℎ) +s 2s))
909	873, 908	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s(𝑖 +s 1s )) = ((2s ·s ℎ) +s 2s))
910	899, 909	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s )))
911	840	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑖 ∈ No )
912	836, 911, 848	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((𝑔 +s 𝑖) +s 1s ) = (𝑔 +s (𝑖 +s 1s )))
913	803	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (𝑔 +s 𝑖) <s 𝑁)
914	836, 911	addscld 27960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (𝑔 +s 𝑖) ∈ No )
915	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝜑)
916	915, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑁 ∈ No )
917	914, 916, 848	sltadd1d 27978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((𝑔 +s 𝑖) <s 𝑁 ↔ ((𝑔 +s 𝑖) +s 1s ) <s (𝑁 +s 1s )))
918	913, 917	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((𝑔 +s 𝑖) +s 1s ) <s (𝑁 +s 1s ))
919	912, 918	eqbrtrrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s ))
920		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑏 /su (2s↑s𝑞)) = (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)))
921	920	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))))
922	921	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)))))
923		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑏 <s (2s↑s𝑞) ↔ ((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞)))
924	922, 923	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → ((𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))) ∧ ((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s ))))
925		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (𝑞 = (𝑖 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑖 +s 1s )))
926	925	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (𝑞 = (𝑖 +s 1s ) → (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
927	926	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (𝑞 = (𝑖 +s 1s ) → (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
928	927	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑞 = (𝑖 +s 1s ) → (𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))))
929	925	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑞 = (𝑖 +s 1s ) → (((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞) ↔ ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s ))))
930		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (𝑞 = (𝑖 +s 1s ) → (𝑔 +s 𝑞) = (𝑔 +s (𝑖 +s 1s )))
931	930	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑞 = (𝑖 +s 1s ) → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s )))
932	928, 929, 931	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝑞 = (𝑖 +s 1s ) → ((𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))) ∧ ((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))) ∧ ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s )) ∧ (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s ))))
933	552, 924, 932	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((𝑔 ∈ ℕ0s ∧ ((2s ·s ℎ) +s 1s ) ∈ ℕ0s ∧ (𝑖 +s 1s ) ∈ ℕ0s) ∧ (𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))) ∧ ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s )) ∧ (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
934	823, 828, 831, 897, 910, 919, 933	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
935	934	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ((ℎ +s 1s ) = (2s↑s𝑖) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
936	822, 935	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
937	759, 936	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ((ℎ +s 1s ) ≤s (2s↑s𝑖) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
938	752, 937	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
939	938	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑑 = (𝑔 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
940	939	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑑 = (𝑔 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
941	744, 940	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
942	941	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → ((𝑔 +s 1s ) ≤s 𝑑 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
943	670, 942	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
944	943	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗)) → ((𝑔 +s 1s ) ≤s 𝑗 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
945	650, 944	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 <s 𝑗)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
946	945	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 <s 𝑗 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
947	626	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑐 <s 𝑑)
948	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
949	579	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
950		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑔 = 𝑗)
951	950	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (𝑔 +s (𝑘 /su (2s↑s𝑙))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
952	949, 951	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
953	947, 948, 952	3brtr3d 5128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s (𝑘 /su (2s↑s𝑙))))
954	838	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → ℎ ∈ No )
955	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑖 ∈ ℕ0s)
956	954, 955	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (ℎ /su (2s↑s𝑖)) ∈ No )
957	587	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑘 ∈ No )
958	581	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑙 ∈ ℕ0s)
959	957, 958	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (𝑘 /su (2s↑s𝑙)) ∈ No )
960	597	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑔 ∈ No )
961	960	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → 𝑔 ∈ No )
962	956, 959, 961	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → ((ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)) ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s (𝑘 /su (2s↑s𝑙)))))
963	953, 962	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
964	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ ∈ ℕ0s)
965		simpl3 1195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)) → 𝑙 ∈ ℕ0s)
966	965	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑙 ∈ ℕ0s)
967	966	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑙 ∈ ℕ0s)
968	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑖 ∈ ℕ0s)
969		n0subs 28340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑙 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (𝑙 ≤s 𝑖 ↔ (𝑖 -s 𝑙) ∈ ℕ0s))
970	967, 968, 969	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑙 ≤s 𝑖 ↔ (𝑖 -s 𝑙) ∈ ℕ0s))
971	970	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑙 ≤s 𝑖 → (𝑖 -s 𝑙) ∈ ℕ0s))
972	971	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑖 -s 𝑙) ∈ ℕ0s)
973		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ ℕ0s ∧ (𝑖 -s 𝑙) ∈ ℕ0s) → (2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s)
974	422, 972, 973	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s)
975	586	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑘 ∈ ℕ0s)
976		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ ℕ0s)
977	974, 975, 976	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ ℕ0s)
978	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑖 ∈ ℕ0s)
979		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
980	966	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑙 ∈ ℕ0s)
981	980	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑙 ∈ No )
982	972	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑖 -s 𝑙) ∈ No )
983	981, 982	addscomd 27947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑙 +s (𝑖 -s 𝑙)) = ((𝑖 -s 𝑙) +s 𝑙))
984	978	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑖 ∈ No )
985		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ ((𝑖 ∈ No ∧ 𝑙 ∈ No ) → ((𝑖 -s 𝑙) +s 𝑙) = 𝑖)
986	984, 981, 985	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((𝑖 -s 𝑙) +s 𝑙) = 𝑖)
987	983, 986	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑙 +s (𝑖 -s 𝑙)) = 𝑖)
988	987	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s(𝑙 +s (𝑖 -s 𝑙))) = (2s↑s𝑖))
989	988	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s(𝑙 +s (𝑖 -s 𝑙)))) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
990	989	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s(𝑙 +s (𝑖 -s 𝑙)))))
991	975	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑘 ∈ No )
992	991, 980, 972	pw2divscan4d 28421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑘 /su (2s↑s𝑙)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s(𝑙 +s (𝑖 -s 𝑙)))))
993	990, 992	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)) = (𝑘 /su (2s↑s𝑙)))
994	993	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑘 /su (2s↑s𝑙)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
995	979, 994	breqtrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (ℎ /su (2s↑s𝑖)) <s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
996	964	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ ∈ No )
997	977	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ No )
998	996, 997, 978	pw2sltdiv1d 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ↔ (ℎ /su (2s↑s𝑖)) <s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))))
999	995, 998	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘))
1000	700	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ <s (2s↑s𝑖))
1001	580	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑘 <s (2s↑s𝑙))
1002		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((2s ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (2s↑s𝑙) ∈ ℕ0s)
1003	422, 980, 1002	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s𝑙) ∈ ℕ0s)
1004	1003	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s𝑙) ∈ No )
1005	974	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s(𝑖 -s 𝑙)) ∈ No )
1006		nnsgt0 28317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (2s ∈ ℕs → 0s <s 2s)
1007	420, 1006	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ 0s <s 2s
1008		expsgt0 28414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((2s ∈ No ∧ (𝑖 -s 𝑙) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑖 -s 𝑙)))
1009	54, 1007, 1008	mp3an13 1455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑖 -s 𝑙) ∈ ℕ0s → 0s <s (2s↑s(𝑖 -s 𝑙)))
1010	972, 1009	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 0s <s (2s↑s(𝑖 -s 𝑙)))
1011	991, 1004, 1005, 1010	sltmul2d 28152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑘 <s (2s↑s𝑙) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙))))
1012	1001, 1011	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
1013		expadds 28412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ No ∧ (𝑖 -s 𝑙) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (2s↑s((𝑖 -s 𝑙) +s 𝑙)) = ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
1014	54, 1013	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((𝑖 -s 𝑙) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (2s↑s((𝑖 -s 𝑙) +s 𝑙)) = ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
1015	972, 980, 1014	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s((𝑖 -s 𝑙) +s 𝑙)) = ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
1016	986	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s((𝑖 -s 𝑙) +s 𝑙)) = (2s↑s𝑖))
1017	1015, 1016	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)) = (2s↑s𝑖))
1018	1012, 1017	breqtrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖))
1019	999, 1000, 1018	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑖) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖)))
1020	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
1021	579	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1022		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖) → 𝑔 = 𝑗)
1023	1022	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑔 = 𝑗)
1024	1023, 993	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1025	1021, 1024	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))))
1026	803	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑔 +s 𝑖) <s 𝑁)
1027	1020, 1025, 1026	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) ∧ (𝑔 +s 𝑖) <s 𝑁))
1028		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ℎ → (𝑚 <s 𝑛 ↔ ℎ <s 𝑛))
1029		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ℎ → (𝑚 <s (2s↑s𝑜) ↔ ℎ <s (2s↑s𝑜)))
1030	1028, 1029	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ℎ → ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ↔ (ℎ <s 𝑛 ∧ ℎ <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜))))
1031		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑚 = ℎ → (𝑚 /su (2s↑s𝑜)) = (ℎ /su (2s↑s𝑜)))
1032	1031	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑚 = ℎ → (𝑔 +s (𝑚 /su (2s↑s𝑜))) = (𝑔 +s (ℎ /su (2s↑s𝑜))))
1033	1032	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ℎ → (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜)))))
1034	1033	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ℎ → ((𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1035	1030, 1034	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑚 = ℎ → (((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((ℎ <s 𝑛 ∧ ℎ <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1036		breq2 5101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (ℎ <s 𝑛 ↔ ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘)))
1037		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑛 <s (2s↑s𝑜) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜)))
1038	1036, 1037	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → ((ℎ <s 𝑛 ∧ ℎ <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ↔ (ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑜) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜))))
1039		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑛 /su (2s↑s𝑜)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜)))
1040	1039	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑔 +s (𝑛 /su (2s↑s𝑜))) = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))))
1041	1040	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜)))))
1042	1041	3anbi2d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → ((𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1043	1038, 1042	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (((ℎ <s 𝑛 ∧ ℎ <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑜) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1044		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑖 → (2s↑s𝑜) = (2s↑s𝑖))
1045	1044	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑖 → (ℎ <s (2s↑s𝑜) ↔ ℎ <s (2s↑s𝑖)))
1046	1044	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑖 → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖)))
1047	1045, 1046	3anbi23d 1442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑖 → ((ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑜) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜)) ↔ (ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑖) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖))))
1048	1044	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑜 = 𝑖 → (ℎ /su (2s↑s𝑜)) = (ℎ /su (2s↑s𝑖)))
1049	1048	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑖 → (𝑔 +s (ℎ /su (2s↑s𝑜))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
1050	1049	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑖 → (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖)))))
1051	1044	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑜 = 𝑖 → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
1052	1051	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑖 → (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))))
1053	1052	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑖 → (𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))))
1054		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑖 → (𝑔 +s 𝑜) = (𝑔 +s 𝑖))
1055	1054	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑖 → ((𝑔 +s 𝑜) <s 𝑁 ↔ (𝑔 +s 𝑖) <s 𝑁))
1056	1050, 1053, 1055	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑖 → ((𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) ∧ (𝑔 +s 𝑖) <s 𝑁)))
1057	1047, 1056	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑖 → (((ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑜) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑖) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) ∧ (𝑔 +s 𝑖) <s 𝑁))))
1058	1035, 1043, 1057	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((ℎ ∈ ℕ0s ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ ((ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∧ ℎ <s (2s↑s𝑖) ∧ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖)) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1059	964, 977, 978, 1019, 1027, 1058	syl32anc 1381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1060	1059	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑙 ≤s 𝑖 → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1061		n0subs 28340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑖 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (𝑖 ≤s 𝑙 ↔ (𝑙 -s 𝑖) ∈ ℕ0s))
1062	968, 967, 1061	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑖 ≤s 𝑙 ↔ (𝑙 -s 𝑖) ∈ ℕ0s))
1063	1062	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑖 ≤s 𝑙 → (𝑙 -s 𝑖) ∈ ℕ0s))
1064	1063	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑙 -s 𝑖) ∈ ℕ0s)
1065		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ ℕ0s ∧ (𝑙 -s 𝑖) ∈ ℕ0s) → (2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s)
1066	422, 1064, 1065	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s)
1067	601	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ℎ ∈ ℕ0s)
1068		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ ℕ0s)
1069	1066, 1067, 1068	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ ℕ0s)
1070	586	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑘 ∈ ℕ0s)
1071	966	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑙 ∈ ℕ0s)
1072	1067	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ℎ ∈ No )
1073	606	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑖 ∈ ℕ0s)
1074	1072, 1073, 1064	pw2divscan4d 28421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (ℎ /su (2s↑s𝑖)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s(𝑖 +s (𝑙 -s 𝑖)))))
1075	1073	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑖 ∈ No )
1076	1064	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑙 -s 𝑖) ∈ No )
1077	1075, 1076	addscomd 27947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑖 +s (𝑙 -s 𝑖)) = ((𝑙 -s 𝑖) +s 𝑖))
1078	1077	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s(𝑖 +s (𝑙 -s 𝑖))) = (2s↑s((𝑙 -s 𝑖) +s 𝑖)))
1079	1078	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s(𝑖 +s (𝑙 -s 𝑖)))) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s((𝑙 -s 𝑖) +s 𝑖))))
1080	1074, 1079	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (ℎ /su (2s↑s𝑖)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s((𝑙 -s 𝑖) +s 𝑖))))
1081	1071	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑙 ∈ No )
1082		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((𝑙 ∈ No ∧ 𝑖 ∈ No ) → ((𝑙 -s 𝑖) +s 𝑖) = 𝑙)
1083	1081, 1075, 1082	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((𝑙 -s 𝑖) +s 𝑖) = 𝑙)
1084	1083	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = (2s↑s𝑙))
1085	1084	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s((𝑙 -s 𝑖) +s 𝑖))) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1086	1080, 1085	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (ℎ /su (2s↑s𝑖)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1087		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
1088	1086, 1087	eqbrtrrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)) <s (𝑘 /su (2s↑s𝑙)))
1089		expscl 28408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s) → (2s↑s(𝑙 -s 𝑖)) ∈ No )
1090	54, 1064, 1089	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s(𝑙 -s 𝑖)) ∈ No )
1091	1090, 1072	mulscld 28115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ No )
1092	1070	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑘 ∈ No )
1093	1091, 1092, 1071	pw2sltdiv1d 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ↔ (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)) <s (𝑘 /su (2s↑s𝑙))))
1094	1088, 1093	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘)
1095	700	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ℎ <s (2s↑s𝑖))
1096	54, 1073, 359	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s𝑖) ∈ No )
1097		expsgt0 28414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑙 -s 𝑖)))
1098	54, 1007, 1097	mp3an13 1455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑙 -s 𝑖) ∈ ℕ0s → 0s <s (2s↑s(𝑙 -s 𝑖)))
1099	1064, 1098	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 0s <s (2s↑s(𝑙 -s 𝑖)))
1100	1072, 1096, 1090, 1099	sltmul2d 28152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (ℎ <s (2s↑s𝑖) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖))))
1101	1095, 1100	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1102		expadds 28412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1103	54, 1102	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((𝑙 -s 𝑖) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1104	1064, 1073, 1103	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1105	1104, 1084	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)) = (2s↑s𝑙))
1106	1101, 1105	breqtrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙))
1107	580	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑘 <s (2s↑s𝑙))
1108	1094, 1106, 1107	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙) ∧ 𝑘 <s (2s↑s𝑙)))
1109	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
1110	1086	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))))
1111	1109, 1110	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))))
1112	579	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1113		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙) → 𝑔 = 𝑗)
1114	1113	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑔 = 𝑗)
1115	1114	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑔 +s (𝑘 /su (2s↑s𝑙))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1116	1112, 1115	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
1117	1114	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑔 +s 𝑙) = (𝑗 +s 𝑙))
1118	680	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑗 +s 𝑙) <s 𝑁)
1119	1117, 1118	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑔 +s 𝑙) <s 𝑁)
1120	1111, 1116, 1119	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))) ∧ (𝑔 +s 𝑙) <s 𝑁))
1121		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 <s 𝑛 ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛))
1122		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 <s (2s↑s𝑜) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜)))
1123	1121, 1122	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ↔ (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜))))
1124		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 /su (2s↑s𝑜)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜)))
1125	1124	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑔 +s (𝑚 /su (2s↑s𝑜))) = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))))
1126	1125	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜)))))
1127	1126	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → ((𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1128	1123, 1127	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1129		breq2 5101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = 𝑘 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘))
1130		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = 𝑘 → (𝑛 <s (2s↑s𝑜) ↔ 𝑘 <s (2s↑s𝑜)))
1131	1129, 1130	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = 𝑘 → ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ↔ (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑘 <s (2s↑s𝑜))))
1132		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑛 = 𝑘 → (𝑛 /su (2s↑s𝑜)) = (𝑘 /su (2s↑s𝑜)))
1133	1132	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑛 = 𝑘 → (𝑔 +s (𝑛 /su (2s↑s𝑜))) = (𝑔 +s (𝑘 /su (2s↑s𝑜))))
1134	1133	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑛 = 𝑘 → (𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜)))))
1135	1134	3anbi2d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = 𝑘 → ((𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1136	1131, 1135	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑛 = 𝑘 → (((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑘 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1137		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑙 → (2s↑s𝑜) = (2s↑s𝑙))
1138	1137	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑙 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙)))
1139	1137	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑙 → (𝑘 <s (2s↑s𝑜) ↔ 𝑘 <s (2s↑s𝑙)))
1140	1138, 1139	3anbi23d 1442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑙 → ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑘 <s (2s↑s𝑜)) ↔ (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙) ∧ 𝑘 <s (2s↑s𝑙))))
1141	1137	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑜 = 𝑙 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1142	1141	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑙 → (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))))
1143	1142	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑙 → (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))))
1144	1137	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑜 = 𝑙 → (𝑘 /su (2s↑s𝑜)) = (𝑘 /su (2s↑s𝑙)))
1145	1144	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑙 → (𝑔 +s (𝑘 /su (2s↑s𝑜))) = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
1146	1145	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑙 → (𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙)))))
1147		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑜 = 𝑙 → (𝑔 +s 𝑜) = (𝑔 +s 𝑙))
1148	1147	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑜 = 𝑙 → ((𝑔 +s 𝑜) <s 𝑁 ↔ (𝑔 +s 𝑙) <s 𝑁))
1149	1143, 1146, 1148	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑙 → ((𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁) ↔ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))) ∧ (𝑔 +s 𝑙) <s 𝑁)))
1150	1140, 1149	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑙 → (((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ∧ 𝑘 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) ↔ ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙) ∧ 𝑘 <s (2s↑s𝑙)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))) ∧ (𝑔 +s 𝑙) <s 𝑁))))
1151	1128, 1136, 1150	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ ((((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ∧ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙) ∧ 𝑘 <s (2s↑s𝑙)) ∧ (𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))) ∧ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))) ∧ (𝑔 +s 𝑙) <s 𝑁))) → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1152	1069, 1070, 1071, 1108, 1120, 1151	syl32anc 1381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1153	1152	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑖 ≤s 𝑙 → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))))
1154	967	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑙 ∈ No )
1155	968	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑖 ∈ No )
1156		sletric 27738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((𝑙 ∈ No ∧ 𝑖 ∈ No ) → (𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙))
1157	1154, 1155, 1156	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙))
1158	1060, 1153, 1157	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → ∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))
1159	597	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑔 ∈ ℕ0s)
1160	1159	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑔 ∈ ℕ0s)
1161		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑚 ∈ ℕ0s)
1162		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (2s ·s 𝑚) ∈ ℕ0s)
1163	422, 1161, 1162	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s 𝑚) ∈ ℕ0s)
1164		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((2s ·s 𝑚) ∈ ℕ0s → ((2s ·s 𝑚) +s 1s ) ∈ ℕ0s)
1165	1163, 1164	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s 𝑚) +s 1s ) ∈ ℕ0s)
1166		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑜 ∈ ℕ0s)
1167		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 ∈ ℕ0s → (𝑜 +s 1s ) ∈ ℕ0s)
1168	1166, 1167	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑜 +s 1s ) ∈ ℕ0s)
1169		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑤 = ({𝑐} |s {𝑑}))
1170	1169	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑤 = ({𝑐} |s {𝑑}))
1171	1170	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑤 = ({𝑐} |s {𝑑}))
1172		simprr1 1223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → 𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1173	1172	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1174		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((2s ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) → (2s↑s𝑜) ∈ ℕ0s)
1175	422, 1166, 1174	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s𝑜) ∈ ℕ0s)
1176		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((2s↑s𝑜) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s𝑜) ·s 𝑔) ∈ ℕ0s)
1177	1175, 1160, 1176	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s𝑜) ·s 𝑔) ∈ ℕ0s)
1178	1177	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s𝑜) ·s 𝑔) ∈ No )
1179	1161	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑚 ∈ No )
1180	1178, 1179, 1166	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜)) = ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s (𝑚 /su (2s↑s𝑜))))
1181	1160	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑔 ∈ No )
1182	1181, 1166	pw2divscan3d 28418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) = 𝑔)
1183	1182	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s (𝑚 /su (2s↑s𝑜))) = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1184	1180, 1183	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜)) = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1185	1173, 1184	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑐 = ((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜)))
1186	1185	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → {𝑐} = {((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜))})
1187		simprr2 1224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1188	1187	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1189	676	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝜑)
1190	1189	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝜑)
1191	1190, 762	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ¬ (𝑁 +s 1s ) ≤s 𝑁)
1192	330	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 1s ∈ No )
1193		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑛 ∈ ℕ0s)
1194	1193	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑛 ∈ No )
1195	1194, 1179	subscld 28043	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑛 -s 𝑚) ∈ No )
1196	1192, 1195	sltnled 27727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ( 1s <s (𝑛 -s 𝑚) ↔ ¬ (𝑛 -s 𝑚) ≤s 1s ))
1197	695	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
1198	1197	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
1199	1170	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑤 = ({𝑐} |s {𝑑}))
1200	1159	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑔 ∈ ℕ0s)
1201	1200	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑔 ∈ No )
1202	1161	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑚 ∈ ℕ0s)
1203		peano2n0s 28309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ ℕ0s)
1204	1202, 1203	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 +s 1s ) ∈ ℕ0s)
1205	1204	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 +s 1s ) ∈ No )
1206	1166	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑜 ∈ ℕ0s)
1207	1205, 1206	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ((𝑚 +s 1s ) /su (2s↑s𝑜)) ∈ No )
1208	1201, 1207	addscld 27960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) ∈ No )
1209	620	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑐 ∈ No )
1210	1209	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑐 ∈ No )
1211	1173	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1212	1179	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑚 ∈ No )
1213	1212	sltp1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑚 <s (𝑚 +s 1s ))
1214	1212, 1205, 1206	pw2sltdiv1d 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 <s (𝑚 +s 1s ) ↔ (𝑚 /su (2s↑s𝑜)) <s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1215	1212, 1206	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 /su (2s↑s𝑜)) ∈ No )
1216	1215, 1207, 1201	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ((𝑚 /su (2s↑s𝑜)) <s ((𝑚 +s 1s ) /su (2s↑s𝑜)) ↔ (𝑔 +s (𝑚 /su (2s↑s𝑜))) <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1217	1214, 1216	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 <s (𝑚 +s 1s ) ↔ (𝑔 +s (𝑚 /su (2s↑s𝑜))) <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1218	1213, 1217	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑔 +s (𝑚 /su (2s↑s𝑜))) <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1219	1211, 1218	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑐 <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1220	1210, 1208, 1219	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → {𝑐} <<s {(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))})
1221	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑑 ∈ No )
1222	1221	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑑 ∈ No )
1223	1179, 1192	addscomd 27947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 +s 1s ) = ( 1s +s 𝑚))
1224	1223	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑚 +s 1s ) <s 𝑛 ↔ ( 1s +s 𝑚) <s 𝑛))
1225	1192, 1179, 1194	sltaddsubd 28071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (( 1s +s 𝑚) <s 𝑛 ↔ 1s <s (𝑛 -s 𝑚)))
1226	1224, 1225	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑚 +s 1s ) <s 𝑛 ↔ 1s <s (𝑛 -s 𝑚)))
1227	1226	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ( 1s <s (𝑛 -s 𝑚) → (𝑚 +s 1s ) <s 𝑛))
1228	1227	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 +s 1s ) <s 𝑛)
1229	1193	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑛 ∈ ℕ0s)
1230	1229	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑛 ∈ No )
1231	1205, 1230, 1206	pw2sltdiv1d 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ((𝑚 +s 1s ) <s 𝑛 ↔ ((𝑚 +s 1s ) /su (2s↑s𝑜)) <s (𝑛 /su (2s↑s𝑜))))
1232	1230, 1206	pw2divscld 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑛 /su (2s↑s𝑜)) ∈ No )
1233	1207, 1232, 1201	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (((𝑚 +s 1s ) /su (2s↑s𝑜)) <s (𝑛 /su (2s↑s𝑜)) ↔ (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜)))))
1234	1231, 1233	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ((𝑚 +s 1s ) <s 𝑛 ↔ (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜)))))
1235	1228, 1234	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1236	1188	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1237	1235, 1236	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s 𝑑)
1238	1208, 1222, 1237	ssltsn 27768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → {(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))} <<s {𝑑})
1239	1199, 1208, 1220, 1238	ssltbday 27897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1240	1198, 1239	eqsstrrd 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1241	1189	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝜑)
1242	1241, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑁 ∈ ℕ0s)
1243		expscl 28408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((2s ∈ No ∧ 𝑜 ∈ ℕ0s) → (2s↑s𝑜) ∈ No )
1244	54, 1166, 1243	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s𝑜) ∈ No )
1245	1244	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (2s↑s𝑜) ∈ No )
1246		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → 𝑚 <s 𝑛)
1247	1246	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑚 <s 𝑛)
1248	1247	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑚 <s 𝑛)
1249		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s 𝑛))
1250	1202, 1229, 1249	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s 𝑛))
1251	1248, 1250	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 +s 1s ) ≤s 𝑛)
1252		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → 𝑛 <s (2s↑s𝑜))
1253	1252	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑛 <s (2s↑s𝑜))
1254	1253	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → 𝑛 <s (2s↑s𝑜))
1255	1205, 1230, 1245, 1251, 1254	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 +s 1s ) <s (2s↑s𝑜))
1256		simprr3 1225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → (𝑔 +s 𝑜) <s 𝑁)
1257	1256	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ⊢ ((((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚)) → (𝑔 +s 𝑜) <s 𝑁)
1258	1257	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑔 +s 𝑜) <s 𝑁)
1259	1242, 1200, 1204, 1206, 1255, 1258	bdaypw2bnd 28442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))) ⊆ ( bday ‘𝑁))
1260	1240, 1259	sstrd 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁))
1261	405, 263	onsled 28249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ (𝜑 → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
1262	1241, 1261	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
1263	1260, 1262	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑁 +s 1s ) ≤s 𝑁)
1264	1263	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ( 1s <s (𝑛 -s 𝑚) → (𝑁 +s 1s ) ≤s 𝑁))
1265	1196, 1264	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (¬ (𝑛 -s 𝑚) ≤s 1s → (𝑁 +s 1s ) ≤s 𝑁))
1266	1191, 1265	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑛 -s 𝑚) ≤s 1s )
1267	1161, 1193, 1249	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s 𝑛))
1268		npcans 28055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ⊢ ((𝑛 ∈ No ∧ 1s ∈ No ) → ((𝑛 -s 1s ) +s 1s ) = 𝑛)
1269	1194, 330, 1268	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑛 -s 1s ) +s 1s ) = 𝑛)
1270	1269	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑚 +s 1s ) ≤s ((𝑛 -s 1s ) +s 1s ) ↔ (𝑚 +s 1s ) ≤s 𝑛))
1271	1267, 1270	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s ((𝑛 -s 1s ) +s 1s )))
1272	1194, 1192	subscld 28043	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑛 -s 1s ) ∈ No )
1273	1179, 1272, 1192	sleadd1d 27975	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 ≤s (𝑛 -s 1s ) ↔ (𝑚 +s 1s ) ≤s ((𝑛 -s 1s ) +s 1s )))
1274	1271, 1273	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 <s 𝑛 ↔ 𝑚 ≤s (𝑛 -s 1s )))
1275	1179, 1194, 1192	slesubd 28076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 ≤s (𝑛 -s 1s ) ↔ 1s ≤s (𝑛 -s 𝑚)))
1276	1274, 1275	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 <s 𝑛 ↔ 1s ≤s (𝑛 -s 𝑚)))
1277	1247, 1276	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 1s ≤s (𝑛 -s 𝑚))
1278	1266, 1277	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑛 -s 𝑚) ≤s 1s ∧ 1s ≤s (𝑛 -s 𝑚)))
1279	1195, 1192	sletri3d 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑛 -s 𝑚) = 1s ↔ ((𝑛 -s 𝑚) ≤s 1s ∧ 1s ≤s (𝑛 -s 𝑚))))
1280	1278, 1279	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑛 -s 𝑚) = 1s )
1281	1194, 1179, 1192	subaddsd 28051	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑛 -s 𝑚) = 1s ↔ (𝑚 +s 1s ) = 𝑛))
1282	1280, 1281	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 +s 1s ) = 𝑛)
1283	1282	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑛 = (𝑚 +s 1s ))
1284	1283	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑛 /su (2s↑s𝑜)) = ((𝑚 +s 1s ) /su (2s↑s𝑜)))
1285	1284	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s (𝑛 /su (2s↑s𝑜))) = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1286	1188, 1285	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑑 = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1287	1182	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1288	1287	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) = ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1289	1161, 1203	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 +s 1s ) ∈ ℕ0s)
1290	1289	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 +s 1s ) ∈ No )
1291	1178, 1290, 1166	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)) = ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1292	1288, 1291	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) = ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)))
1293	1178, 1179, 1192	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) = (((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )))
1294	1293	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜)) = ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)))
1295	1294	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)) = (((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜)))
1296	1286, 1292, 1295	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑑 = (((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜)))
1297	1296	sneqd 4591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → {𝑑} = {(((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜))})
1298	1186, 1297	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ({𝑐} |s {𝑑}) = ({((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜))} |s {(((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜))}))
1299		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((2s↑s𝑜) ·s 𝑔) ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (((2s↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℕ0s)
1300	1177, 1161, 1299	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℕ0s)
1301	1300	n0zsd 28367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℤs)
1302	1301, 1166	pw2cutp1 28438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ({((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜))} |s {(((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜))}) = (((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))
1303	1171, 1298, 1302	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑤 = (((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))
1304	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 2s ∈ No )
1305	1304, 1178, 1179	addsdid 28136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) = ((2s ·s ((2s↑s𝑜) ·s 𝑔)) +s (2s ·s 𝑚)))
1306		expsp1 28406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ⊢ ((2s ∈ No ∧ 𝑜 ∈ ℕ0s) → (2s↑s(𝑜 +s 1s )) = ((2s↑s𝑜) ·s 2s))
1307	54, 1166, 1306	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s(𝑜 +s 1s )) = ((2s↑s𝑜) ·s 2s))
1308	1244, 1304	mulscomd 28120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s𝑜) ·s 2s) = (2s ·s (2s↑s𝑜)))
1309	1307, 1308	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s(𝑜 +s 1s )) = (2s ·s (2s↑s𝑜)))
1310	1309	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) = ((2s ·s (2s↑s𝑜)) ·s 𝑔))
1311	1304, 1244, 1181	mulsassd 28147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s (2s↑s𝑜)) ·s 𝑔) = (2s ·s ((2s↑s𝑜) ·s 𝑔)))
1312	1310, 1311	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) = (2s ·s ((2s↑s𝑜) ·s 𝑔)))
1313	1312	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)) = ((2s ·s ((2s↑s𝑜) ·s 𝑔)) +s (2s ·s 𝑚)))
1314	1305, 1313	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)))
1315	1314	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)) +s 1s ))
1316		n0expscl 28409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((2s ∈ ℕ0s ∧ (𝑜 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑜 +s 1s )) ∈ ℕ0s)
1317	422, 1168, 1316	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s(𝑜 +s 1s )) ∈ ℕ0s)
1318		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (((2s↑s(𝑜 +s 1s )) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) ∈ ℕ0s)
1319	1317, 1160, 1318	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) ∈ ℕ0s)
1320	1319	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) ∈ No )
1321	1163	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s 𝑚) ∈ No )
1322	1320, 1321, 1192	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)) +s 1s ) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )))
1323	1315, 1322	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )))
1324	1323	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) /su (2s↑s(𝑜 +s 1s ))) = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )) /su (2s↑s(𝑜 +s 1s ))))
1325	1303, 1324	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑤 = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )) /su (2s↑s(𝑜 +s 1s ))))
1326	1165	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s 𝑚) +s 1s ) ∈ No )
1327	1320, 1326, 1168	pw2divsdird 28425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )) /su (2s↑s(𝑜 +s 1s ))) = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) /su (2s↑s(𝑜 +s 1s ))) +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))))
1328	1181, 1168	pw2divscan3d 28418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (((2s↑s(𝑜 +s 1s )) ·s 𝑔) /su (2s↑s(𝑜 +s 1s ))) = 𝑔)
1329	1328	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) /su (2s↑s(𝑜 +s 1s ))) +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))) = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))))
1330	1325, 1327, 1329	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))))
1331		n0mulscl 28323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((2s ∈ ℕ0s ∧ (𝑚 +s 1s ) ∈ ℕ0s) → (2s ·s (𝑚 +s 1s )) ∈ ℕ0s)
1332	422, 1289, 1331	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) ∈ ℕ0s)
1333	1332	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) ∈ No )
1334	1317	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s↑s(𝑜 +s 1s )) ∈ No )
1335	1192, 1304, 1321	sltadd2d 27977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ( 1s <s 2s ↔ ((2s ·s 𝑚) +s 1s ) <s ((2s ·s 𝑚) +s 2s)))
1336	540, 1335	mpbii 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s 𝑚) +s 1s ) <s ((2s ·s 𝑚) +s 2s))
1337	1304, 1179, 1192	addsdid 28136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) = ((2s ·s 𝑚) +s (2s ·s 1s )))
1338	903	oveq2i 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((2s ·s 𝑚) +s (2s ·s 1s )) = ((2s ·s 𝑚) +s 2s)
1339	1337, 1338	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) = ((2s ·s 𝑚) +s 2s))
1340	1336, 1339	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s 𝑚) +s 1s ) <s (2s ·s (𝑚 +s 1s )))
1341		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁))) → 𝑚 <s (2s↑s𝑜))
1342	1341	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑚 <s (2s↑s𝑜))
1343		n0sltp1le 28342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑚 ∈ ℕ0s ∧ (2s↑s𝑜) ∈ ℕ0s) → (𝑚 <s (2s↑s𝑜) ↔ (𝑚 +s 1s ) ≤s (2s↑s𝑜)))
1344	1161, 1175, 1343	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 <s (2s↑s𝑜) ↔ (𝑚 +s 1s ) ≤s (2s↑s𝑜)))
1345	1342, 1344	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑚 +s 1s ) ≤s (2s↑s𝑜))
1346	1007	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 0s <s 2s)
1347	1290, 1244, 1304, 1346	slemul2d 28154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑚 +s 1s ) ≤s (2s↑s𝑜) ↔ (2s ·s (𝑚 +s 1s )) ≤s (2s ·s (2s↑s𝑜))))
1348	1345, 1347	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) ≤s (2s ·s (2s↑s𝑜)))
1349	1348, 1309	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (2s ·s (𝑚 +s 1s )) ≤s (2s↑s(𝑜 +s 1s )))
1350	1326, 1333, 1334, 1340, 1349	sltletrd 27734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((2s ·s 𝑚) +s 1s ) <s (2s↑s(𝑜 +s 1s )))
1351	1166	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑜 ∈ No )
1352	1181, 1351, 1192	addsassd 27986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑔 +s 𝑜) +s 1s ) = (𝑔 +s (𝑜 +s 1s )))
1353	1256	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s 𝑜) <s 𝑁)
1354		n0addscl 28322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) → (𝑔 +s 𝑜) ∈ ℕ0s)
1355	1160, 1166, 1354	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s 𝑜) ∈ ℕ0s)
1356	1355	n0snod 28304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s 𝑜) ∈ No )
1357	1190, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → 𝑁 ∈ No )
1358	1356, 1357, 1192	sltadd1d 27978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑔 +s 𝑜) <s 𝑁 ↔ ((𝑔 +s 𝑜) +s 1s ) <s (𝑁 +s 1s )))
1359	1353, 1358	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ((𝑔 +s 𝑜) +s 1s ) <s (𝑁 +s 1s ))
1360	1352, 1359	eqbrtrrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s ))
1361		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑏 /su (2s↑s𝑞)) = (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)))
1362	1361	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))))
1363	1362	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)))))
1364		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑏 <s (2s↑s𝑞) ↔ ((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞)))
1365	1363, 1364	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → ((𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))) ∧ ((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s ))))
1366		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑞 = (𝑜 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑜 +s 1s )))
1367	1366	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑞 = (𝑜 +s 1s ) → (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))
1368	1367	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑞 = (𝑜 +s 1s ) → (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))))
1369	1368	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑞 = (𝑜 +s 1s ) → (𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))))
1370	1366	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑞 = (𝑜 +s 1s ) → (((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞) ↔ ((2s ·s 𝑚) +s 1s ) <s (2s↑s(𝑜 +s 1s ))))
1371		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑞 = (𝑜 +s 1s ) → (𝑔 +s 𝑞) = (𝑔 +s (𝑜 +s 1s )))
1372	1371	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑞 = (𝑜 +s 1s ) → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s )))
1373	1369, 1370, 1372	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑞 = (𝑜 +s 1s ) → ((𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))) ∧ ((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞) ∧ (𝑔 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))) ∧ ((2s ·s 𝑚) +s 1s ) <s (2s↑s(𝑜 +s 1s )) ∧ (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s ))))
1374	552, 1365, 1373	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((𝑔 ∈ ℕ0s ∧ ((2s ·s 𝑚) +s 1s ) ∈ ℕ0s ∧ (𝑜 +s 1s ) ∈ ℕ0s) ∧ (𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))) ∧ ((2s ·s 𝑚) +s 1s ) <s (2s↑s(𝑜 +s 1s )) ∧ (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1375	1160, 1165, 1168, 1330, 1350, 1360, 1374	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) ∧ ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1376	1375	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) ∧ (𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s)) → (((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1377	1376	rexlimdvvva 3193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → (∃𝑚 ∈ ℕ0s ∃𝑛 ∈ ℕ0s ∃𝑜 ∈ ℕ0s ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ∧ (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ∧ 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ∧ (𝑔 +s 𝑜) <s 𝑁)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1378	1158, 1377	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1379	1378	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → ((ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1380	963, 1379	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1381	1380	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 = 𝑗 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1382	946, 1381	jaod 860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → ((𝑔 <s 𝑗 ∨ 𝑔 = 𝑗) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1383	645, 1382	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → (𝑔 ≤s 𝑗 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1384	643, 1383	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1385	1384	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → (𝑔 ≤s 𝑐 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1386	617, 1385	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ 𝑑 <s (𝑗 +s 1s )) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1387	1386	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → (𝑑 <s (𝑗 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1388	596, 1387	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1389	1388	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) ∧ (𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s)) → ((𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1390	1389	rexlimdvvva 3193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1391	273	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ( bday ‘𝑑) ⊆ ( bday ‘𝑁))
1392	59	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s ∈ No )
1393	124	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 𝑤 ∈ No )
1394		simp1r 1200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s <s 𝑤)
1395	1393, 1394	0elleft 27891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s ∈ ( L ‘𝑤))
1396	215, 192, 1395	rspcdva 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s ≤s 𝑐)
1397	1392, 344, 393, 1396, 285	slelttrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s <s 𝑑)
1398	1392, 393, 1397	sltled 27743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → 0s ≤s 𝑑)
1399	1398	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → 0s ≤s 𝑑)
1400		fveq2 6833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑧 = 𝑑 → ( bday ‘𝑧) = ( bday ‘𝑑))
1401	1400	sseq1d 3964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑧 = 𝑑 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
1402		breq2 5101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑧 = 𝑑 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑑))
1403	1401, 1402	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑧 = 𝑑 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑑)))
1404		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑧 = 𝑑 → (𝑧 = 𝑁 ↔ 𝑑 = 𝑁))
1405		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑧 = 𝑑 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
1406	1405	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑧 = 𝑑 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
1407	1406	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑧 = 𝑑 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
1408	1407	2rexbidv 3200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑧 = 𝑑 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
1409		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑥 = 𝑗 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑗 +s (𝑦 /su (2s↑s𝑝))))
1410	1409	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑥 = 𝑗 → (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝)))))
1411		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑥 = 𝑗 → (𝑥 +s 𝑝) = (𝑗 +s 𝑝))
1412	1411	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑥 = 𝑗 → ((𝑥 +s 𝑝) <s 𝑁 ↔ (𝑗 +s 𝑝) <s 𝑁))
1413	1410, 1412	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑥 = 𝑗 → ((𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁)))
1414	1413	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑥 = 𝑗 → (∃𝑝 ∈ ℕ0s (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁)))
1415		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑦 = 𝑘 → (𝑦 /su (2s↑s𝑝)) = (𝑘 /su (2s↑s𝑝)))
1416	1415	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑦 = 𝑘 → (𝑗 +s (𝑦 /su (2s↑s𝑝))) = (𝑗 +s (𝑘 /su (2s↑s𝑝))))
1417	1416	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = 𝑘 → (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝)))))
1418		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = 𝑘 → (𝑦 <s (2s↑s𝑝) ↔ 𝑘 <s (2s↑s𝑝)))
1419	1417, 1418	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑦 = 𝑘 → ((𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁) ↔ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ∧ 𝑘 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁)))
1420	1419	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑦 = 𝑘 → (∃𝑝 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ∧ 𝑘 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁)))
1421		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑝 = 𝑙 → (2s↑s𝑝) = (2s↑s𝑙))
1422	1421	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑝 = 𝑙 → (𝑘 /su (2s↑s𝑝)) = (𝑘 /su (2s↑s𝑙)))
1423	1422	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑝 = 𝑙 → (𝑗 +s (𝑘 /su (2s↑s𝑝))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1424	1423	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑝 = 𝑙 → (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
1425	1421	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑝 = 𝑙 → (𝑘 <s (2s↑s𝑝) ↔ 𝑘 <s (2s↑s𝑙)))
1426		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑝 = 𝑙 → (𝑗 +s 𝑝) = (𝑗 +s 𝑙))
1427	1426	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑝 = 𝑙 → ((𝑗 +s 𝑝) <s 𝑁 ↔ (𝑗 +s 𝑙) <s 𝑁))
1428	1424, 1425, 1427	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑝 = 𝑙 → ((𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ∧ 𝑘 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁) ↔ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))
1429	1428	cbvrexvw 3214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (∃𝑝 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ∧ 𝑘 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁) ↔ ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))
1430	1420, 1429	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑦 = 𝑘 → (∃𝑝 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑗 +s 𝑝) <s 𝑁) ↔ ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))
1431	1414, 1430	cbvrex2vw 3218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))
1432	1408, 1431	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑧 = 𝑑 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))
1433	1404, 1432	orbi12d 919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑧 = 𝑑 → ((𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) ↔ (𝑑 = 𝑁 ∨ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))))
1434	1403, 1433	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑑 → (((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ ((( bday ‘𝑑) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑑) → (𝑑 = 𝑁 ∨ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))))
1435	302, 16	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
1436	1434, 1435, 394	rspcdva 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ((( bday ‘𝑑) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑑) → (𝑑 = 𝑁 ∨ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))))
1437	1391, 1399, 1436	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → (𝑑 = 𝑁 ∨ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))
1438	578, 1390, 1437	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑}) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1439	1438	3expa 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) ∧ ((𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) ∧ (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1440	1439	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) ∧ (𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s)) → ((𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1441	1440	rexlimdvvva 3193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → (∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1442	293, 1441	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ((( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1443	214, 219, 1442	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) ∧ 𝑤 = ({𝑐} |s {𝑑})) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1444	1443	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → (𝑤 = ({𝑐} |s {𝑑}) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1445	195, 1444	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1446	1445	3expa 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ (𝑑 ∈ ( R ‘𝑤) ∧ ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1447	1446	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑑 ∈ ( R ‘𝑤)) → (∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1448	190, 1447	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) ∧ 𝑑 ∈ ( R ‘𝑤)) → (∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1449	1448	rexlimdva 3136	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → (∃𝑑 ∈ ( R ‘𝑤)∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1450	184, 1449	syl5 34	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ((( R ‘𝑤) ∈ Fin ∧ ( R ‘𝑤) ≠ ∅) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1451	158, 179, 1450	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ (𝑐 ∈ ( L ‘𝑤) ∧ ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1452	1451	expr 456	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ 𝑐 ∈ ( L ‘𝑤)) → (∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1453	154, 1452	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) ∧ 𝑐 ∈ ( L ‘𝑤)) → (∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1454	1453	rexlimdva 3136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → (∃𝑐 ∈ ( L ‘𝑤)∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1455	147, 1454	syl5 34	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ((( L ‘𝑤) ∈ Fin ∧ ( L ‘𝑤) ≠ ∅) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1456	141, 1455	mpand 696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → (( L ‘𝑤) ≠ ∅ → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1457	127, 1456	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1458	1457	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( 0s <s 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1459		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝜑)
1460		n0p1nns 28348	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0s → (𝑁 +s 1s ) ∈ ℕs)
1461	1, 1460	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ ℕs)
1462		nnsgt0 28317	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 +s 1s ) ∈ ℕs → 0s <s (𝑁 +s 1s ))
1463	1461, 1462	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0s <s (𝑁 +s 1s ))
1464		addslid 27948	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
1465	59, 1464	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( 0s +s 0s ) = 0s
1466	1465	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0s = ( 0s +s 0s )
1467	31, 31, 31	3pm3.2i 1341	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s)
1468		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 0s → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = ( 0s +s (𝑏 /su (2s↑s𝑞))))
1469	1468	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 0s → ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 0s = ( 0s +s (𝑏 /su (2s↑s𝑞)))))
1470		oveq1 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 0s → (𝑎 +s 𝑞) = ( 0s +s 𝑞))
1471	1470	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 0s → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ ( 0s +s 𝑞) <s (𝑁 +s 1s )))
1472	1469, 1471	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 0s → (( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ ( 0s = ( 0s +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ ( 0s +s 𝑞) <s (𝑁 +s 1s ))))
1473	48	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 = 0s → ( 0s +s (𝑏 /su (2s↑s𝑞))) = ( 0s +s ( 0s /su (2s↑s𝑞))))
1474	1473	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = 0s → ( 0s = ( 0s +s (𝑏 /su (2s↑s𝑞))) ↔ 0s = ( 0s +s ( 0s /su (2s↑s𝑞)))))
1475	1474, 51	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 0s → (( 0s = ( 0s +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ ( 0s +s 𝑞) <s (𝑁 +s 1s )) ↔ ( 0s = ( 0s +s ( 0s /su (2s↑s𝑞))) ∧ 0s <s (2s↑s𝑞) ∧ ( 0s +s 𝑞) <s (𝑁 +s 1s ))))
1476	62	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 = 0s → ( 0s +s ( 0s /su (2s↑s𝑞))) = ( 0s +s 0s ))
1477	1476	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑞 = 0s → ( 0s = ( 0s +s ( 0s /su (2s↑s𝑞))) ↔ 0s = ( 0s +s 0s )))
1478		oveq2 7366	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑞 = 0s → ( 0s +s 𝑞) = ( 0s +s 0s ))
1479	1478, 1465	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 = 0s → ( 0s +s 𝑞) = 0s )
1480	1479	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑞 = 0s → (( 0s +s 𝑞) <s (𝑁 +s 1s ) ↔ 0s <s (𝑁 +s 1s )))
1481	1477, 65, 1480	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑞 = 0s → (( 0s = ( 0s +s ( 0s /su (2s↑s𝑞))) ∧ 0s <s (2s↑s𝑞) ∧ ( 0s +s 𝑞) <s (𝑁 +s 1s )) ↔ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s (𝑁 +s 1s ))))
1482	1472, 1475, 1481	rspc3ev 3592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s) ∧ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s (𝑁 +s 1s ))) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1483	1467, 1482	mpan 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s (𝑁 +s 1s )) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1484	1466, 38, 1483	mp3an12 1454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( 0s <s (𝑁 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1485	1463, 1484	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1486	1459, 1485	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
1487		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( 0s = 𝑤 → ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞)))))
1488	1487	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( 0s = 𝑤 → (( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1489	1488	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( 0s = 𝑤 → (∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1490	1489	2rexbidv 3200	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( 0s = 𝑤 → (∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )) ↔ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1491	1486, 1490	syl5ibcom 245	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ( 0s = 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1492	1491	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( 0s = 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1493	1458, 1492	jaod 860	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → (( 0s <s 𝑤 ∨ 0s = 𝑤) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1494	123, 1493	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( 0s ≤s 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1495	1494	expr 456	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )) → ( 0s ≤s 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
1496	1495	expd 415	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) → (𝑤 ≠ (𝑁 +s 1s ) → ( 0s ≤s 𝑤 → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1497	1496	com34 91	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) → ( 0s ≤s 𝑤 → (𝑤 ≠ (𝑁 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1498	1497	impd 410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 ≠ (𝑁 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
1499	1498	impr 454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤))) → (𝑤 ≠ (𝑁 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1500	120, 1499	biimtrrid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤))) → (¬ 𝑤 = (𝑁 +s 1s ) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1501	1500	orrd 864	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤))) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))
1502	1501	expr 456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
1503	1502	expd 415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1504	119, 1503	sylbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = suc ( bday ‘𝑁) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1505	118, 1504	jaod 860	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1506	15, 1505	biimtrid 242	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1507	5, 1506	sylbid 240	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) → ( 0s ≤s 𝑤 → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s ))))))
1508	1507	impd 410	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → ((( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))
1509	1508	ralrimiva 3127	1 ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑁 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∀wral 3050  ∃wrex 3059   ⊆ wss 3900  ∅c0 4284  {csn 4579   class class class wbr 5097   Or wor 5530  Oncon0 6316  suc csuc 6318  ‘cfv 6491  (class class class)co 7358  ωcom 7808  Fincfn 8885   No csur 27609   <s cslt 27610   bday cbday 27611   ≤s csle 27714   <<s csslt 27755   |s cscut 27757   0_s c0s 27801   1_s c1s 27802   M cmade 27818   O cold 27819   L cleft 27821   R cright 27822   +_s cadds 27939   -_s csubs 28000   ·_s cmuls 28086   /_su cdivs 28167  On_scons 28230  ℕ_0scnn0s 28291  ℕ_scnns 28292  2_sc2s 28387  ↑_scexps 28389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-dc 10358  ax-ac2 10375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-fin 8889  df-card 9853  df-acn 9856  df-ac 10028  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-ons 28231  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390

This theorem is referenced by:  bdayfinbndlem2  28445