Theorem bdayfinbndlem1 28467

Description: Lemma for bdayfinbnd 28469. 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 28369	. . . . . . 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 3967	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +s 1s )) ↔ ( bday ‘𝑤) ⊆ suc ( bday ‘𝑁)))
6		bdayon 27752	. . . . . . 7 ⊢ ( bday ‘𝑤) ∈ On
7		bdayon 27752	. . . . . . . 8 ⊢ ( bday ‘𝑁) ∈ On
8	7	onsuci 7783	. . . . . . 7 ⊢ suc ( bday ‘𝑁) ∈ On
9	6, 8	onsseli 6440	. . . . . 6 ⊢ (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) ↔ (( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
10		onsssuc 6410	. . . . . . . 8 ⊢ ((( bday ‘𝑤) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ∈ suc ( bday ‘𝑁)))
11	6, 7, 10	mp2an 693	. . . . . . 7 ⊢ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ∈ suc ( bday ‘𝑁))
12	11	orbi1i 914	. . . . . 6 ⊢ ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)) ↔ (( bday ‘𝑤) ∈ suc ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
13	9, 12	bitr4i 278	. . . . 5 ⊢ (( bday ‘𝑤) ⊆ suc ( bday ‘𝑁) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∨ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
14		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))))
15		fveq2 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → ( bday ‘𝑧) = ( bday ‘𝑤))
16	15	sseq1d 3966	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑤) ⊆ ( bday ‘𝑁)))
17		breq2 5103	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤))
18	16, 17	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑤)))
19		eqeq1 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝑁 ↔ 𝑤 = 𝑁))
20		eqeq1 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
21	20	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)))
22	21	rexbidv 3161	. . . . . . . . . . . . . . . 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)))
23	22	2rexbidv 3202	. . . . . . . . . . . . . . 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)))
24	19, 23	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))))
25	18, 24	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)))))
26	25	rspccva 3576	. . . . . . . . . . . 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))))
27	14, 26	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))))
28	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑁 ∈ ℕ0s)
29		0n0s 28329	. . . . . . . . . . . . . . 15 ⊢ 0s ∈ ℕ0s
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 0s ∈ ℕ0s)
31		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑤 = 𝑁)
32	1	n0nod 28325	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ No )
33	32	addsridd 27965	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 +s 0s ) = 𝑁)
34	33	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → (𝑁 +s 0s ) = 𝑁)
35	31, 34	eqtr4d 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 𝑤 = (𝑁 +s 0s ))
36		0lt1s 27812	. . . . . . . . . . . . . . 15 ⊢ 0s <s 1s
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → 0s <s 1s )
38	32	ltsp1d 28015	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 <s (𝑁 +s 1s ))
39	33, 38	eqbrtrd 5121	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 +s 0s ) <s (𝑁 +s 1s ))
40	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ 𝑤 = 𝑁)) → (𝑁 +s 0s ) <s (𝑁 +s 1s ))
41		oveq1 7367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑁 +s (𝑏 /su (2s↑s𝑞))))
42	41	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞)))))
43		oveq1 7367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → (𝑎 +s 𝑞) = (𝑁 +s 𝑞))
44	43	breq1d 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑁 +s 𝑞) <s (𝑁 +s 1s )))
45	42, 44	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 ))))
46		oveq1 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 0s → (𝑏 /su (2s↑s𝑞)) = ( 0s /su (2s↑s𝑞)))
47	46	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 0s → (𝑁 +s (𝑏 /su (2s↑s𝑞))) = (𝑁 +s ( 0s /su (2s↑s𝑞))))
48	47	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0s → (𝑤 = (𝑁 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞)))))
49		breq1 5102	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0s → (𝑏 <s (2s↑s𝑞) ↔ 0s <s (2s↑s𝑞)))
50	48, 49	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 ))))
51		oveq2 7368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = 0s → (2s↑s𝑞) = (2s↑s 0s ))
52		2no 28419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2s ∈ No
53		exps0 28427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2s↑s 0s ) = 1s
55	51, 54	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 0s → (2s↑s𝑞) = 1s )
56	55	oveq2d 7376	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 0s → ( 0s /su (2s↑s𝑞)) = ( 0s /su 1s ))
57		0no 27809	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0s ∈ No
58		divs1 28204	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
59	57, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 0s /su 1s ) = 0s
60	56, 59	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 0s → ( 0s /su (2s↑s𝑞)) = 0s )
61	60	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 0s → (𝑁 +s ( 0s /su (2s↑s𝑞))) = (𝑁 +s 0s ))
62	61	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → (𝑤 = (𝑁 +s ( 0s /su (2s↑s𝑞))) ↔ 𝑤 = (𝑁 +s 0s )))
63	55	breq2d 5111	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → ( 0s <s (2s↑s𝑞) ↔ 0s <s 1s ))
64		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 0s → (𝑁 +s 𝑞) = (𝑁 +s 0s ))
65	64	breq1d 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 0s → ((𝑁 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑁 +s 0s ) <s (𝑁 +s 1s )))
66	62, 63, 65	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 ))))
67	45, 50, 66	rspc3ev 3594	. . . . . . . . . . . . . 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 )))
68	28, 30, 30, 35, 37, 40, 67	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 )))
69	68	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 ))))
70		idd 24	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
71		idd 24	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑦 <s (2s↑s𝑝) → 𝑦 <s (2s↑s𝑝)))
72		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ ℕ0s)
73	72	n0nod 28325	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ No )
74	73	3adant2 1132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝑥 +s 𝑝) ∈ No )
75	74	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → (𝑥 +s 𝑝) ∈ No )
76	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) ∈ No )
77	32	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → 𝑁 ∈ No )
78	77	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → 𝑁 ∈ No )
79		peano2no 27984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ No → (𝑁 +s 1s ) ∈ No )
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑁 +s 1s ) ∈ No )
81		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) <s 𝑁)
82	77	ltsp1d 28015	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → 𝑁 <s (𝑁 +s 1s ))
83	82	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → 𝑁 <s (𝑁 +s 1s ))
84	76, 78, 80, 81, 83	ltstrd 27735	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) ∧ (𝑥 +s 𝑝) <s 𝑁) → (𝑥 +s 𝑝) <s (𝑁 +s 1s ))
85	84	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s)) → ((𝑥 +s 𝑝) <s 𝑁 → (𝑥 +s 𝑝) <s (𝑁 +s 1s )))
86	70, 71, 85	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 ))))
87		oveq1 7367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑥 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑥 +s (𝑏 /su (2s↑s𝑞))))
88	87	eqeq2d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞)))))
89		oveq1 7367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑥 → (𝑎 +s 𝑞) = (𝑥 +s 𝑞))
90	89	breq1d 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑥 +s 𝑞) <s (𝑁 +s 1s )))
91	88, 90	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 ))))
92		oveq1 7367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑦 → (𝑏 /su (2s↑s𝑞)) = (𝑦 /su (2s↑s𝑞)))
93	92	oveq2d 7376	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑦 → (𝑥 +s (𝑏 /su (2s↑s𝑞))) = (𝑥 +s (𝑦 /su (2s↑s𝑞))))
94	93	eqeq2d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑤 = (𝑥 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞)))))
95		breq1 5102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝑏 <s (2s↑s𝑞) ↔ 𝑦 <s (2s↑s𝑞)))
96	94, 95	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 ))))
97		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑞 = 𝑝 → (2s↑s𝑞) = (2s↑s𝑝))
98	97	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = 𝑝 → (𝑦 /su (2s↑s𝑞)) = (𝑦 /su (2s↑s𝑝)))
99	98	oveq2d 7376	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝑝 → (𝑥 +s (𝑦 /su (2s↑s𝑞))) = (𝑥 +s (𝑦 /su (2s↑s𝑝))))
100	99	eqeq2d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
101	97	breq2d 5111	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → (𝑦 <s (2s↑s𝑞) ↔ 𝑦 <s (2s↑s𝑝)))
102		oveq2 7368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝑝 → (𝑥 +s 𝑞) = (𝑥 +s 𝑝))
103	102	breq1d 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑝 → ((𝑥 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑥 +s 𝑝) <s (𝑁 +s 1s )))
104	100, 101, 103	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 ))))
105	91, 96, 104	rspc3ev 3594	. . . . . . . . . . . . . . . . 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 )))
106	105	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 ))))
107	106	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 ))))
108	86, 107	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 ))))
109	108	rexlimdvvva 3195	. . . . . . . . . . . . 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 ))))
110	109	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 ))))
111	69, 110	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 ))))
112	27, 111	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 ))))
113	112	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 )))
114	113	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 ))))
115	114	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 )))))
116	115	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 ))))))
117	4	eqeq2d 2748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ↔ ( bday ‘𝑤) = suc ( bday ‘𝑁)))
118		df-ne 2934	. . . . . . . . . . 11 ⊢ (𝑤 ≠ (𝑁 +s 1s ) ↔ ¬ 𝑤 = (𝑁 +s 1s ))
119		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑤 ∈ No )
120		lesloe 27726	. . . . . . . . . . . . . . . . . 18 ⊢ (( 0s ∈ No ∧ 𝑤 ∈ No ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤)))
121	57, 119, 120	sylancr 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤)))
122		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
123	122	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
124	1	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ ℕ0s)
125		n0bday 28352	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 +s 1s ) ∈ ℕ0s → ( bday ‘(𝑁 +s 1s )) ∈ ω)
126	124, 125	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ( bday ‘(𝑁 +s 1s )) ∈ ω)
127	126	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → ( bday ‘(𝑁 +s 1s )) ∈ ω)
128	127	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘(𝑁 +s 1s )) ∈ ω)
129	123, 128	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( bday ‘𝑤) ∈ ω)
130		oldfi 27914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( bday ‘𝑤) ∈ ω → ( O ‘( bday ‘𝑤)) ∈ Fin)
131	129, 130	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( O ‘( bday ‘𝑤)) ∈ Fin)
132		leftssold 27871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( L ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))
133		ssfi 9101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( O ‘( bday ‘𝑤)) ∈ Fin ∧ ( L ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))) → ( L ‘𝑤) ∈ Fin)
134	131, 132, 133	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( L ‘𝑤) ∈ Fin)
135		simplrl 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑤 ∈ No )
136		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 0s <s 𝑤)
137	135, 136	0elleft 27911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 0s ∈ ( L ‘𝑤))
138	137	ne0d 4295	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( L ‘𝑤) ≠ ∅)
139		leftssno 27873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( L ‘𝑤) ⊆ No
140		ltsso 27648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ <s Or No
141		soss 5553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( L ‘𝑤) ⊆ No → ( <s Or No → <s Or ( L ‘𝑤)))
142	139, 140, 141	mp2 9	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ <s Or ( L ‘𝑤)
143		fimax2g 9190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( <s Or ( L ‘𝑤) ∧ ( L ‘𝑤) ∈ Fin ∧ ( L ‘𝑤) ≠ ∅) → ∃𝑐 ∈ ( L ‘𝑤)∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒)
144	142, 143	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( L ‘𝑤) ∈ Fin ∧ ( L ‘𝑤) ≠ ∅) → ∃𝑐 ∈ ( L ‘𝑤)∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒)
145		leftno 27877	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑒 ∈ ( L ‘𝑤) → 𝑒 ∈ No )
146		leftno 27877	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ ( L ‘𝑤) → 𝑐 ∈ No )
147		lenlts 27724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑒 ∈ No ∧ 𝑐 ∈ No ) → (𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒))
148	145, 146, 147	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑐 ∈ ( L ‘𝑤) ∧ 𝑒 ∈ ( L ‘𝑤)) → (𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒))
149	148	ralbidva 3158	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ ( L ‘𝑤) → (∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐 ↔ ∀𝑒 ∈ ( L ‘𝑤) ¬ 𝑐 <s 𝑒))
150	149	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (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))
151		rightssold 27872	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ( R ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))
152		ssfi 9101	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((( O ‘( bday ‘𝑤)) ∈ Fin ∧ ( R ‘𝑤) ⊆ ( O ‘( bday ‘𝑤))) → ( R ‘𝑤) ∈ Fin)
153	131, 151, 152	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → ( R ‘𝑤) ∈ Fin)
154	153	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( R ‘𝑤) ∈ Fin)
155	135	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 ∈ No )
156		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑤 ≠ (𝑁 +s 1s ))
157	156	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑤 ≠ (𝑁 +s 1s ))
158	157	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
159	158	neneqd 2938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = (𝑁 +s 1s ))
160		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ Ons)
161	124	ad4antr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ ℕ0s)
162		n0on 28336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 +s 1s ) ∈ ℕ0s → (𝑁 +s 1s ) ∈ Ons)
163	161, 162	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (𝑁 +s 1s ) ∈ Ons)
164	123	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)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
165	164	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
166		bday11on 28265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑤 ∈ Ons ∧ (𝑁 +s 1s ) ∈ Ons ∧ ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s ))) → 𝑤 = (𝑁 +s 1s ))
167	160, 163, 165, 166	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑤 = (𝑁 +s 1s ))
168	159, 167	mtand 816	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ Ons)
169		elons 28253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 ∈ Ons ↔ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
170	169	notbii 320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ 𝑤 ∈ Ons ↔ ¬ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
171		imnan 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅) ↔ ¬ (𝑤 ∈ No ∧ ( R ‘𝑤) = ∅))
172	170, 171	bitr4i 278	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ 𝑤 ∈ Ons ↔ (𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅))
173	168, 172	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑤 ∈ No → ¬ ( R ‘𝑤) = ∅))
174	155, 173	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 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) = ∅)
175	174	neqned 2940	. . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( R ‘𝑤) ≠ ∅)
176		rightssno 27874	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( R ‘𝑤) ⊆ No
177		soss 5553	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (( R ‘𝑤) ⊆ No → ( <s Or No → <s Or ( R ‘𝑤)))
178	176, 140, 177	mp2 9	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ <s Or ( R ‘𝑤)
179		fimin2g 9406	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (( <s Or ( R ‘𝑤) ∧ ( R ‘𝑤) ∈ Fin ∧ ( R ‘𝑤) ≠ ∅) → ∃𝑑 ∈ ( R ‘𝑤)∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑)
180	178, 179	mp3an1 1451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( R ‘𝑤) ∈ Fin ∧ ( R ‘𝑤) ≠ ∅) → ∃𝑑 ∈ ( R ‘𝑤)∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑)
181		rightno 27878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑑 ∈ ( R ‘𝑤) → 𝑑 ∈ No )
182		rightno 27878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ ( R ‘𝑤) → 𝑓 ∈ No )
183		lenlts 27724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑑 ∈ No ∧ 𝑓 ∈ No ) → (𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑))
184	181, 182, 183	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑑 ∈ ( R ‘𝑤) ∧ 𝑓 ∈ ( R ‘𝑤)) → (𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑))
185	184	ralbidva 3158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑑 ∈ ( R ‘𝑤) → (∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑))
186	185	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)𝑑 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝑤) ¬ 𝑓 <s 𝑑))
187		simp2l 1201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 ∈ ( L ‘𝑤))
188		simp2r 1202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)
189		simp3l 1203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 ∈ ( R ‘𝑤))
190		simp3r 1204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ∀𝑓 ∈ ( R ‘𝑤)𝑑 ≤s 𝑓)
191	187, 188, 189, 190	cutminmax 27936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 = ({𝑐} |s {𝑑}))
192		simpl2l 1228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ( L ‘𝑤))
193	132, 192	sselid 3932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 𝑐 ∈ ( O ‘( bday ‘𝑤)))
194	192	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
195		oldbday 27901	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((( bday ‘𝑤) ∈ On ∧ 𝑐 ∈ No ) → (𝑐 ∈ ( O ‘( bday ‘𝑤)) ↔ ( bday ‘𝑐) ∈ ( bday ‘𝑤)))
196	6, 194, 195	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → (𝑐 ∈ ( O ‘( bday ‘𝑤)) ↔ ( bday ‘𝑐) ∈ ( bday ‘𝑤)))
197	193, 196	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ‘𝑤))
198	123	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
199	198	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
200	1	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) → 𝑁 ∈ ℕ0s)
201	200	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤) → 𝑁 ∈ ℕ0s)
202	201	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑁 ∈ ℕ0s)
203	202	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 𝑁 ∈ ℕ0s)
204	203, 2	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})) → ( bday ‘(𝑁 +s 1s )) = suc ( bday ‘𝑁))
205	199, 204	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ‘𝑤) = suc ( bday ‘𝑁))
206	197, 205	eleqtrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ( bday ‘𝑐) ∈ suc ( bday ‘𝑁))
207		bdayon 27752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ( bday ‘𝑐) ∈ On
208		onsssuc 6410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((( bday ‘𝑐) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ∈ suc ( bday ‘𝑁)))
209	207, 7, 208	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ∈ suc ( bday ‘𝑁))
210	206, 209	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ( bday ‘𝑐) ⊆ ( bday ‘𝑁))
211		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 0s → (𝑒 ≤s 𝑐 ↔ 0s ≤s 𝑐))
212		simpl2r 1229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ∀𝑒 ∈ ( L ‘𝑤)𝑒 ≤s 𝑐)
213		simpl1 1193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ((𝜑 ∧ (𝑤 ∈ No ∧ (( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )) ∧ 𝑤 ≠ (𝑁 +s 1s )))) ∧ 0s <s 𝑤))
214	213, 137	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 0s ∈ ( L ‘𝑤))
215	211, 212, 214	rspcdva 3578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 0s ≤s 𝑐)
216		simp1ll 1238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝜑)
217	216	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})) → 𝜑)
218		n0on 28336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ Ons)
219	1, 218	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝜑 → 𝑁 ∈ Ons)
220	217, 219	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})) → 𝑁 ∈ Ons)
221		simpl3l 1230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ( R ‘𝑤))
222	151, 221	sselid 3932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ( O ‘( bday ‘𝑤)))
223		oldbdayim 27889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑑 ∈ ( O ‘( bday ‘𝑤)) → ( bday ‘𝑑) ∈ ( bday ‘𝑤))
224	222, 223	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})) → ( bday ‘𝑑) ∈ ( bday ‘𝑤))
225	224, 205	eleqtrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ( bday ‘𝑑) ∈ suc ( bday ‘𝑁))
226		bdayon 27752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ( bday ‘𝑑) ∈ On
227		onsssuc 6410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((( bday ‘𝑑) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ∈ suc ( bday ‘𝑁)))
228	226, 7, 227	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ∈ suc ( bday ‘𝑁))
229	225, 228	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ( bday ‘𝑑) ⊆ ( bday ‘𝑁))
230	221	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
231		madebday 27900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((( bday ‘𝑁) ∈ On ∧ 𝑑 ∈ No ) → (𝑑 ∈ ( M ‘( bday ‘𝑁)) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
232	7, 230, 231	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → (𝑑 ∈ ( M ‘( bday ‘𝑁)) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
233	229, 232	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 𝑑 ∈ ( M ‘( bday ‘𝑁)))
234		onsbnd 28281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑁 ∈ Ons ∧ 𝑑 ∈ ( M ‘( bday ‘𝑁))) → 𝑑 ≤s 𝑁)
235	220, 233, 234	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 𝑑 ≤s 𝑁)
236	203	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → 𝑁 ∈ No )
237	230, 236	lesnltd 27728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → (𝑑 ≤s 𝑁 ↔ ¬ 𝑁 <s 𝑑))
238	235, 237	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ¬ 𝑁 <s 𝑑)
239		lltr 27862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ( L ‘𝑤) <<s ( R ‘𝑤)
240	239	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( L ‘𝑤) <<s ( R ‘𝑤))
241	240, 187, 189	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 <s 𝑑)
242	241	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 <s 𝑑)
243		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑐 = 𝑁 → (𝑐 <s 𝑑 ↔ 𝑁 <s 𝑑))
244	242, 243	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → (𝑐 = 𝑁 → 𝑁 <s 𝑑))
245	238, 244	mtod 198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ¬ 𝑐 = 𝑁)
246		fveq2 6835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑧 = 𝑐 → ( bday ‘𝑧) = ( bday ‘𝑐))
247	246	sseq1d 3966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑐 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑐) ⊆ ( bday ‘𝑁)))
248		breq2 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑐 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑐))
249	247, 248	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = 𝑐 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑐) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑐)))
250		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑐 → (𝑧 = 𝑁 ↔ 𝑐 = 𝑁))
251		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
252	251	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (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)))
253	252	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (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)))
254	253	2rexbidv 3202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (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)))
255		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑥 = 𝑔 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑔 +s (𝑦 /su (2s↑s𝑝))))
256	255	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑥 = 𝑔 → (𝑐 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝)))))
257		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑥 = 𝑔 → (𝑥 +s 𝑝) = (𝑔 +s 𝑝))
258	257	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑥 = 𝑔 → ((𝑥 +s 𝑝) <s 𝑁 ↔ (𝑔 +s 𝑝) <s 𝑁))
259	256, 258	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (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)))
260	259	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (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)))
261		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑦 = ℎ → (𝑦 /su (2s↑s𝑝)) = (ℎ /su (2s↑s𝑝)))
262	261	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 = ℎ → (𝑔 +s (𝑦 /su (2s↑s𝑝))) = (𝑔 +s (ℎ /su (2s↑s𝑝))))
263	262	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑦 = ℎ → (𝑐 = (𝑔 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝)))))
264		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑦 = ℎ → (𝑦 <s (2s↑s𝑝) ↔ ℎ <s (2s↑s𝑝)))
265	263, 264	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (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)))
266	265	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (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)))
267		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑝 = 𝑖 → (2s↑s𝑝) = (2s↑s𝑖))
268	267	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑝 = 𝑖 → (ℎ /su (2s↑s𝑝)) = (ℎ /su (2s↑s𝑖)))
269	268	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑝 = 𝑖 → (𝑔 +s (ℎ /su (2s↑s𝑝))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
270	269	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑝 = 𝑖 → (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑝))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖)))))
271	267	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑝 = 𝑖 → (ℎ <s (2s↑s𝑝) ↔ ℎ <s (2s↑s𝑖)))
272		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑝 = 𝑖 → (𝑔 +s 𝑝) = (𝑔 +s 𝑖))
273	272	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑝 = 𝑖 → ((𝑔 +s 𝑝) <s 𝑁 ↔ (𝑔 +s 𝑖) <s 𝑁))
274	270, 271, 273	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (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)))
275	274	cbvrexvw 3216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (∃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))
276	266, 275	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (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)))
277	260, 276	cbvrex2vw 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∃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))
278	254, 277	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (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)))
279	250, 278	orbi12d 919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (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))))
280	249, 279	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (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)))))
281	14	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((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))))
282	281	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((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))))
283	282	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ∀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))))
284	283	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ∀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))))
285	280, 284, 194	rspcdva 3578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (𝑐 = 𝑁 ∨ ∃𝑔 ∈ ℕ0s ∃ℎ ∈ ℕ0s ∃𝑖 ∈ ℕ0s (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))) ∧ ℎ <s (2s↑s𝑖) ∧ (𝑔 +s 𝑖) <s 𝑁))))
286		orel1 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ 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)))
287	245, 285, 286	sylsyld 61	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ((( 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)))
288		simp3l1 1280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ ℕ0s)
289	288	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ℕ0s)
290	289	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
291		simp3l3 1282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑖 ∈ ℕ0s)
292	291	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑖 ∈ ℕ0s)
293		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (𝑔 +s 𝑖) ∈ ℕ0s)
294	289, 292, 293	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑖) ∈ ℕ0s)
295	294	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑖) ∈ No )
296	216	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝜑)
297	296	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝜑)
298	297, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑁 ∈ No )
299		n0sge0 28338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑖 ∈ ℕ0s → 0s ≤s 𝑖)
300	292, 299	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}) ∧ ((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)
301	292	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑖 ∈ No )
302	290, 301	addsge01d 28016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ( 0s ≤s 𝑖 ↔ 𝑔 ≤s (𝑔 +s 𝑖)))
303	300, 302	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑔 +s 𝑖))
304		simp3r3 1285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s 𝑖) <s 𝑁)
305	304	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑖) <s 𝑁)
306	290, 295, 298, 303, 305	leltstrd 27737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑁)
307	297, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑁 ∈ ℕ0s)
308		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑔 <s 𝑁 ↔ (𝑔 +s 1s ) ≤s 𝑁))
309	289, 307, 308	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑁 ↔ (𝑔 +s 1s ) ≤s 𝑁))
310	306, 309	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (𝑔 +s 1s ) ≤s 𝑁)
311		ltsirr 27718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑁 ∈ No → ¬ 𝑁 <s 𝑁)
312	298, 311	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))) ∧ 𝑑 = 𝑁) → ¬ 𝑁 <s 𝑁)
313	289	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ ℕ0s)
314	313	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ No )
315	314, 298	ltsnled 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 ) <s 𝑁 ↔ ¬ 𝑁 ≤s (𝑔 +s 1s )))
316	296	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 ) <s 𝑁)) → 𝜑)
317	124	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ No )
318	316, 317	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 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ∈ No )
319	316, 32	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 ∈ No )
320	52	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 ) <s 𝑁)) → 2s ∈ No )
321	319, 320	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) ∈ No )
322		1no 27810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ 1s ∈ No
323	322	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))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 1s ∈ No )
324	321, 323, 323	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 1s ) +s 1s ) = ((𝑁 -s 2s) +s ( 1s +s 1s )))
325		1p1e2s 28416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ( 1s +s 1s ) = 2s
326	325	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝑁 -s 2s) +s ( 1s +s 1s )) = ((𝑁 -s 2s) +s 2s)
327		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝑁 ∈ No ∧ 2s ∈ No ) → ((𝑁 -s 2s) +s 2s) = 𝑁)
328	319, 52, 327	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 2s) = 𝑁)
329	326, 328	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s ( 1s +s 1s )) = 𝑁)
330	324, 329	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 1s ) +s 1s ) = 𝑁)
331	330, 319	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 1s ) +s 1s ) ∈ No )
332	321, 323	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 1s ) ∈ No )
333	198	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
334	333	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))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
335		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 = ({𝑐} |s {𝑑}))
336	187	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 ∈ No )
337	336	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 ∈ No )
338	337	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)) → 𝑐 ∈ No )
339	288	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)) → 𝑔 ∈ ℕ0s)
340	339	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) ∈ ℕ0s)
341	340	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) ∈ No )
342		subscl 28062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((𝑁 ∈ No ∧ 1s ∈ No ) → (𝑁 -s 1s ) ∈ No )
343	32, 322, 342	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝜑 → (𝑁 -s 1s ) ∈ No )
344	316, 343	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 ) ∈ No )
345		simp3r1 1283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
346	345	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 ) <s 𝑁)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
347		simp3r2 1284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ℎ <s (2s↑s𝑖))
348	347	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)) → ℎ <s (2s↑s𝑖))
349	291	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)) → 𝑖 ∈ ℕ0s)
350		expscl 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((2s ∈ No ∧ 𝑖 ∈ ℕ0s) → (2s↑s𝑖) ∈ No )
351	52, 349, 350	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (2s↑s𝑖) ∈ No )
352	351	mulslidd 28143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (2s↑s𝑖)) = (2s↑s𝑖))
353	348, 352	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ <s ( 1s ·s (2s↑s𝑖)))
354		simp3l2 1281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ℎ ∈ ℕ0s)
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)) → ℎ ∈ ℕ0s)
356	355	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ ∈ No )
357	356, 323, 349	pw2ltdivmuls2d 28457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ ℎ <s ( 1s ·s (2s↑s𝑖))))
358	353, 357	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) <s 1s )
359	356, 349	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) ∈ No )
360	339	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 ∈ No )
361	359, 323, 360	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
362	358, 361	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))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
363	346, 362	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 <s (𝑔 +s 1s ))
364		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((𝑔 +s 1s ) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑔 +s 1s ) <s 𝑁 ↔ ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁))
365	313, 307, 364	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) ≤s 𝑁))
366	365	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 → ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁))
367	366	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑔 +s 1s ) +s 1s ) ≤s 𝑁)
368		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((𝑁 ∈ No ∧ 1s ∈ No ) → ((𝑁 -s 1s ) +s 1s ) = 𝑁)
369	319, 322, 368	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = 𝑁)
370	367, 369	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑔 +s 1s ) +s 1s ) ≤s ((𝑁 -s 1s ) +s 1s ))
371	341, 344, 323	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑔 +s 1s ) ≤s (𝑁 -s 1s ) ↔ ((𝑔 +s 1s ) +s 1s ) ≤s ((𝑁 -s 1s ) +s 1s )))
372	370, 371	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) ≤s (𝑁 -s 1s ))
373	338, 341, 344, 363, 372	ltlestrd 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))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑐 <s (𝑁 -s 1s ))
374	325	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑁 -s ( 1s +s 1s )) = (𝑁 -s 2s)
375	374	oveq1i 7370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((𝑁 -s ( 1s +s 1s )) +s 1s ) = ((𝑁 -s 2s) +s 1s )
376	319, 323, 323	subsubs4d 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = (𝑁 -s ( 1s +s 1s )))
377	376	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) -s 1s ) +s 1s ) = ((𝑁 -s ( 1s +s 1s )) +s 1s ))
378		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((𝑁 -s 1s ) ∈ No ∧ 1s ∈ No ) → (((𝑁 -s 1s ) -s 1s ) +s 1s ) = (𝑁 -s 1s ))
379	344, 322, 378	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) -s 1s ) +s 1s ) = (𝑁 -s 1s ))
380	377, 379	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 +s 1s )) +s 1s ) = (𝑁 -s 1s ))
381	375, 380	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (𝑁 -s 1s ))
382	373, 381	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 <s ((𝑁 -s 2s) +s 1s ))
383	338, 332, 382	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → {𝑐} <<s {((𝑁 -s 2s) +s 1s )})
384	189	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 ∈ No )
385	384	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
386	385	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)) → 𝑑 ∈ No )
387		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 = 𝑁)
388	387	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑑 -s 1s ) = (𝑁 -s 1s ))
389	386	ltsm1d 28102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑑 -s 1s ) <s 𝑑)
390	388, 389	eqbrtrrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑑)
391	381, 390	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑑)
392	332, 386, 391	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 {𝑑})
393	335, 332, 383, 392	sltsbday 27917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( bday ‘𝑤) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s )))
394	334, 393	eqsstrrd 3970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s )))
395	124, 162	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ Ons)
396	316, 395	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 +s 1s ) ∈ Ons)
397	319, 323, 320	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) = ((𝑁 -s 2s) +s 1s ))
398		n0sge0 28338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝑔 ∈ ℕ0s → 0s ≤s 𝑔)
399	339, 398	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)) → 0s ≤s 𝑔)
400	323, 360	addsge01d 28016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 0s ≤s 𝑔 ↔ 1s ≤s ( 1s +s 𝑔)))
401	399, 400	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 𝑔))
402	360, 323	addscomd 27967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = ( 1s +s 𝑔))
403	401, 402	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s ≤s (𝑔 +s 1s ))
404		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) <s 𝑁)
405	323, 341, 319, 403, 404	leltstrd 27737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s <s 𝑁)
406	323, 319, 405	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s ≤s 𝑁)
407	323, 319, 323	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 1s ≤s 𝑁 ↔ ( 1s +s 1s ) ≤s (𝑁 +s 1s )))
408	406, 407	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 +s 1s ) ≤s (𝑁 +s 1s ))
409	325, 408	eqbrtrrid 5135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 2s ≤s (𝑁 +s 1s ))
410		2nns 28418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ 2s ∈ ℕs
411		nnn0s 28327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
412	410, 411	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ 2s ∈ ℕ0s
413	296, 124	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))) → (𝑁 +s 1s ) ∈ ℕ0s)
414	413	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)) → (𝑁 +s 1s ) ∈ ℕ0s)
415		n0subs 28363	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((2s ∈ ℕ0s ∧ (𝑁 +s 1s ) ∈ ℕ0s) → (2s ≤s (𝑁 +s 1s ) ↔ ((𝑁 +s 1s ) -s 2s) ∈ ℕ0s))
416	412, 414, 415	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (2s ≤s (𝑁 +s 1s ) ↔ ((𝑁 +s 1s ) -s 2s) ∈ ℕ0s))
417	409, 416	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) ∈ ℕ0s)
418	397, 417	eqeltrrd 2838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ ℕ0s)
419		n0on 28336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((𝑁 -s 2s) +s 1s ) ∈ ℕ0s → ((𝑁 -s 2s) +s 1s ) ∈ Ons)
420	418, 419	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 -s 2s) +s 1s ) ∈ Ons)
421	396, 420	onlesd 28270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 2s) +s 1s ) ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘((𝑁 -s 2s) +s 1s ))))
422	394, 421	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 1s ) ≤s ((𝑁 -s 2s) +s 1s ))
423	332	ltsp1d 28015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) +s 1s ) <s (((𝑁 -s 2s) +s 1s ) +s 1s ))
424	318, 332, 331, 422, 423	leltstrd 27737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) <s (((𝑁 -s 2s) +s 1s ) +s 1s ))
425	318, 331, 424	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) <s 𝑁)) → (𝑁 +s 1s ) ≤s (((𝑁 -s 2s) +s 1s ) +s 1s ))
426	425, 330	breqtrd 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))) ∧ (𝑑 = 𝑁 ∧ (𝑔 +s 1s ) <s 𝑁)) → (𝑁 +s 1s ) ≤s 𝑁)
427	316, 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 ∧ (𝑔 +s 1s ) <s 𝑁)) → 𝑁 ∈ ℕ0s)
428		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 <s 𝑁 ↔ (𝑁 +s 1s ) ≤s 𝑁))
429	427, 427, 428	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 1s ) <s 𝑁)) → (𝑁 <s 𝑁 ↔ (𝑁 +s 1s ) ≤s 𝑁))
430	426, 429	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑁)
431	430	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑁))
432	315, 431	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (¬ 𝑁 ≤s (𝑔 +s 1s ) → 𝑁 <s 𝑁))
433	312, 432	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑁 ≤s (𝑔 +s 1s ))
434	314, 298	lestri3d 27731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ((𝑔 +s 1s ) = 𝑁 ↔ ((𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s (𝑔 +s 1s ))))
435	310, 433, 434	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (𝑔 +s 1s ) = 𝑁)
436	304	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 𝑖) <s 𝑁)
437		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) = 𝑁)
438	436, 437	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 𝑖) <s (𝑔 +s 1s ))
439	291	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 1s ) = 𝑁)) → 𝑖 ∈ ℕ0s)
440	439	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
441	322	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s ∈ No )
442	288	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 1s ) = 𝑁)) → 𝑔 ∈ ℕ0s)
443	442	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 ∈ No )
444	440, 441, 443	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 <s 1s ↔ (𝑔 +s 𝑖) <s (𝑔 +s 1s )))
445	438, 444	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑖 <s 1s )
446		n0lts1e0 28368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑖 ∈ ℕ0s → (𝑖 <s 1s ↔ 𝑖 = 0s ))
447	439, 446	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑖 <s 1s ↔ 𝑖 = 0s ))
448	445, 447	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑖 = 0s )
449	345	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
450	347	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))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ <s (2s↑s𝑖))
451		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑖 = 0s → (2s↑s𝑖) = (2s↑s 0s ))
452	451	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) → (2s↑s𝑖) = (2s↑s 0s ))
453	452	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (2s↑s𝑖) = (2s↑s 0s ))
454	453, 54	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (2s↑s𝑖) = 1s )
455	450, 454	breqtrd 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 <s 1s )
456	354	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))) ∧ ((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s )) → ℎ ∈ ℕ0s)
457		n0lts1e0 28368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (ℎ ∈ ℕ0s → (ℎ <s 1s ↔ ℎ = 0s ))
458	456, 457	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 <s 1s ↔ ℎ = 0s ))
459	455, 458	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = 0s )
460	459, 454	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (ℎ /su (2s↑s𝑖)) = ( 0s /su 1s ))
461	460, 59	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (ℎ /su (2s↑s𝑖)) = 0s )
462	461	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (𝑔 +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s 0s ))
463	288	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
464	463	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 1s ) = 𝑁) ∧ 𝑖 = 0s )) → 𝑔 ∈ No )
465	464	addsridd 27965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (𝑔 +s 0s ) = 𝑔)
466	449, 462, 465	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → 𝑐 = 𝑔)
467		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 = ({𝑐} |s {𝑑}))
468	54	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑔 /su (2s↑s 0s )) = (𝑔 /su 1s )
469	463	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 ∈ No )
470	469	divs1d 28205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = 𝑔)
471		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)) → 𝑐 = 𝑔)
472	470, 471	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = 𝑐)
473	468, 472	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = 𝑐)
474	473	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → {(𝑔 /su (2s↑s 0s ))} = {𝑐})
475	54	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝑔 +s 1s ) /su (2s↑s 0s )) = ((𝑔 +s 1s ) /su 1s )
476		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) → (𝑔 +s 1s ) = 𝑁)
477	476	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = 𝑁)
478	288	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s 1s ) ∈ ℕ0s)
479	478	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))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → (𝑔 +s 1s ) ∈ ℕ0s)
480	479	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ No )
481	480	divs1d 28205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = (𝑔 +s 1s ))
482		simplll 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) → 𝑑 = 𝑁)
483	482	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 = 𝑁)
484	477, 481, 483	3eqtr4d 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) = 𝑑)
485	475, 484	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 +s 1s ) /su (2s↑s 0s )) = 𝑑)
486	485	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → {((𝑔 +s 1s ) /su (2s↑s 0s ))} = {𝑑})
487	474, 486	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ({(𝑔 /su (2s↑s 0s ))} |s {((𝑔 +s 1s ) /su (2s↑s 0s ))}) = ({𝑐} |s {𝑑}))
488	288	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))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 𝑔 ∈ ℕ0s)
489	488	n0zsd 28390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 ∈ ℤs)
490	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s ∈ ℕ0s)
491	489, 490	pw2cutp1 28461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ({(𝑔 /su (2s↑s 0s ))} |s {((𝑔 +s 1s ) /su (2s↑s 0s ))}) = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))))
492	467, 487, 491	3eqtr2d 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))))
493		mulscl 28134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((2s ∈ No ∧ 𝑔 ∈ No ) → (2s ·s 𝑔) ∈ No )
494	52, 469, 493	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
495	322	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))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → 1s ∈ No )
496		addslid 27968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
497	322, 496	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ( 0s +s 1s ) = 1s
498		1n0s 28348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ 1s ∈ ℕ0s
499	497, 498	eqeltri 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ( 0s +s 1s ) ∈ ℕ0s
500	499	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))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ( 0s +s 1s ) ∈ ℕ0s)
501	494, 495, 500	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) = (((2s ·s 𝑔) /su (2s↑s( 0s +s 1s ))) +s ( 1s /su (2s↑s( 0s +s 1s )))))
502		exps1 28428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
503	52, 502	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (2s↑s 1s ) = 2s
504	503	oveq1i 7370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((2s↑s 1s ) ·s 𝑔) = (2s ·s 𝑔)
505	504	oveq1i 7370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((2s↑s 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))) = ((2s ·s 𝑔) /su (2s↑s( 0s +s 1s )))
506	498	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s ∈ ℕ0s)
507	469, 490, 506	pw2divscan4d 28444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (2s↑s 0s )) = (((2s↑s 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))))
508	468, 507, 470	3eqtr3a 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) ·s 𝑔) /su (2s↑s( 0s +s 1s ))) = 𝑔)
509	505, 508	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s( 0s +s 1s ))) = 𝑔)
510	497	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (2s↑s( 0s +s 1s )) = (2s↑s 1s )
511	510, 503	eqtri 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (2s↑s( 0s +s 1s )) = 2s
512	511	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ( 1s /su (2s↑s( 0s +s 1s ))) = ( 1s /su 2s)
513	512	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))) ∧ (((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔)) → ( 1s /su (2s↑s( 0s +s 1s ))) = ( 1s /su 2s))
514	509, 513	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s( 0s +s 1s ))) +s ( 1s /su (2s↑s( 0s +s 1s )))) = (𝑔 +s ( 1s /su 2s)))
515	501, 514	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) = (𝑔 +s ( 1s /su 2s)))
516	515	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑤 = (((2s ·s 𝑔) +s 1s ) /su (2s↑s( 0s +s 1s ))) ↔ 𝑤 = (𝑔 +s ( 1s /su 2s))))
517	288	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))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑔 ∈ ℕ0s)
518	498	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 1s ∈ ℕ0s)
519		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))) ∧ ((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑤 = (𝑔 +s ( 1s /su 2s)))
520		ltadds1 27992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (( 0s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s )))
521	57, 322, 322, 520	mp3an 1464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ))
522	36, 521	mpbi 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ( 0s +s 1s ) <s ( 1s +s 1s )
523	522, 497, 325	3brtr3i 5128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ 1s <s 2s
524	523	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 1s <s 2s)
525		simp-4r 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((((𝑑 = 𝑁 ∧ (𝑔 +s 1s ) = 𝑁) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s))) → (𝑔 +s 1s ) = 𝑁)
526	525	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑔 +s 1s ) = 𝑁)
527	296	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) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝜑)
528	527, 32	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 ) ∧ 𝑐 = 𝑔) ∧ 𝑤 = (𝑔 +s ( 1s /su 2s)))) → 𝑁 ∈ No )
529	528	ltsp1d 28015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑁 <s (𝑁 +s 1s ))
530	526, 529	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑔 +s 1s ) <s (𝑁 +s 1s ))
531		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑎 = 𝑔 → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (𝑏 /su (2s↑s𝑞))))
532	531	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑎 = 𝑔 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞)))))
533		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑎 = 𝑔 → (𝑎 +s 𝑞) = (𝑔 +s 𝑞))
534	533	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑎 = 𝑔 → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s 𝑞) <s (𝑁 +s 1s )))
535	532, 534	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (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 ))))
536		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑏 = 1s → (𝑏 /su (2s↑s𝑞)) = ( 1s /su (2s↑s𝑞)))
537	536	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑏 = 1s → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s ( 1s /su (2s↑s𝑞))))
538	537	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑏 = 1s → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞)))))
539		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑏 = 1s → (𝑏 <s (2s↑s𝑞) ↔ 1s <s (2s↑s𝑞)))
540	538, 539	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (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 ))))
541		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑞 = 1s → (2s↑s𝑞) = (2s↑s 1s ))
542	541, 503	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑞 = 1s → (2s↑s𝑞) = 2s)
543	542	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝑞 = 1s → ( 1s /su (2s↑s𝑞)) = ( 1s /su 2s))
544	543	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑞 = 1s → (𝑔 +s ( 1s /su (2s↑s𝑞))) = (𝑔 +s ( 1s /su 2s)))
545	544	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑞 = 1s → (𝑤 = (𝑔 +s ( 1s /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s ( 1s /su 2s))))
546	542	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑞 = 1s → ( 1s <s (2s↑s𝑞) ↔ 1s <s 2s))
547		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝑞 = 1s → (𝑔 +s 𝑞) = (𝑔 +s 1s ))
548	547	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑞 = 1s → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s 1s ) <s (𝑁 +s 1s )))
549	545, 546, 548	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (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 ))))
550	535, 540, 549	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((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 )))
551	517, 518, 518, 519, 524, 530, 550	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = (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 )))
552	551	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 ) = 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 ))))
553	516, 552	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (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 ))))
554	492, 553	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 ) ∧ 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 )))
555	554	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 )) → (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 ))))
556	466, 555	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) ∧ 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 )))
557	556	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)) → (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 ))))
558	448, 557	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 ∧ (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 )))
559	558	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ((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 ))))
560	435, 559	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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 )))
561	560	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (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 ))))
562		simprr1 1223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
563		simprr2 1224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 <s (2s↑s𝑙))
564		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑙 ∈ ℕ0s)
565		expscl 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((2s ∈ No ∧ 𝑙 ∈ ℕ0s) → (2s↑s𝑙) ∈ No )
566	52, 564, 565	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (2s↑s𝑙) ∈ No )
567	566	mulslidd 28143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( 1s ·s (2s↑s𝑙)) = (2s↑s𝑙))
568	563, 567	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑘 <s ( 1s ·s (2s↑s𝑙)))
569		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ℕ0s)
570	569	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
571	322	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 1s ∈ No )
572	570, 571, 564	pw2ltdivmuls2d 28457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑘 /su (2s↑s𝑙)) <s 1s ↔ 𝑘 <s ( 1s ·s (2s↑s𝑙))))
573	568, 572	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑘 /su (2s↑s𝑙)) <s 1s )
574	570, 564	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑘 /su (2s↑s𝑙)) ∈ No )
575		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑗 ∈ ℕ0s)
576	575	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
577	574, 571, 576	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑘 /su (2s↑s𝑙)) <s 1s ↔ (𝑗 +s (𝑘 /su (2s↑s𝑙))) <s (𝑗 +s 1s )))
578	573, 577	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑗 +s (𝑘 /su (2s↑s𝑙))) <s (𝑗 +s 1s ))
579	562, 578	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 <s (𝑗 +s 1s ))
580	288	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))) → 𝑔 ∈ ℕ0s)
581	580	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 )) → 𝑔 ∈ ℕ0s)
582	581	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
583	582	addsridd 27965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (𝑔 +s 0s ) = 𝑔)
584	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))) → ℎ ∈ ℕ0s)
585	584	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 )) → ℎ ∈ ℕ0s)
586		n0sge0 28338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (ℎ ∈ ℕ0s → 0s ≤s ℎ)
587	585, 586	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ 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)
588	585	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → ℎ ∈ No )
589	291	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) → 𝑖 ∈ ℕ0s)
590	589	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))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑖 ∈ ℕ0s)
591	588, 590	pw2ge0divsd 28446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → ( 0s ≤s ℎ ↔ 0s ≤s (ℎ /su (2s↑s𝑖))))
592	57	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))) ∧ 𝑑 <s (𝑗 +s 1s )) → 0s ∈ No )
593	588, 590	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → (ℎ /su (2s↑s𝑖)) ∈ No )
594	592, 593, 582	leadds2d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → ( 0s ≤s (ℎ /su (2s↑s𝑖)) ↔ (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖)))))
595	591, 594	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → ( 0s ≤s ℎ ↔ (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖)))))
596	587, 595	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))) ∧ 𝑑 <s (𝑗 +s 1s )) → (𝑔 +s 0s ) ≤s (𝑔 +s (ℎ /su (2s↑s𝑖))))
597	583, 596	eqbrtrrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → 𝑔 ≤s (𝑔 +s (ℎ /su (2s↑s𝑖))))
598	345	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
599	598	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
600	597, 599	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 𝑑 <s (𝑗 +s 1s )) → 𝑔 ≤s 𝑐)
601	580	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))) ∧ (𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ∈ ℕ0s)
602	601	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 ∈ No )
603	337	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 ∈ No )
604	603	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 ) ∧ 𝑔 ≤s 𝑐)) → 𝑐 ∈ No )
605	385	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))) → 𝑑 ∈ No )
606	605	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 ) ∧ 𝑔 ≤s 𝑐)) → 𝑑 ∈ No )
607		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 ≤s 𝑐)
608	241	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))) → 𝑐 <s 𝑑)
609	608	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 <s 𝑑)
610	609	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 ) ∧ 𝑔 ≤s 𝑐)) → 𝑐 <s 𝑑)
611	602, 604, 606, 607, 610	leltstrd 27737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 <s 𝑑)
612	580	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) ∧ 𝑔 <s 𝑑)) → 𝑔 ∈ ℕ0s)
613	612	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
614	605	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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 )
615	575	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 <s 𝑑)) → 𝑗 ∈ ℕ0s)
616	615	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑗 +s 1s ) ∈ ℕ0s)
617	616	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑗 +s 1s ) ∈ No )
618		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑑)
619		simprll 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 <s (𝑗 +s 1s ))
620	613, 614, 617, 618, 619	ltstrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑑)) → 𝑔 <s (𝑗 +s 1s ))
621		n0lesltp1 28366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s) → (𝑔 ≤s 𝑗 ↔ 𝑔 <s (𝑗 +s 1s )))
622	612, 615, 621	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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 )))
623	620, 622	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ 𝑔 <s 𝑑)) → 𝑔 ≤s 𝑗)
624	623	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)) → (𝑔 <s 𝑑 → 𝑔 ≤s 𝑗))
625	611, 624	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 ) ∧ 𝑔 ≤s 𝑐)) → 𝑔 ≤s 𝑗)
626	576	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 ) ∧ 𝑔 ≤s 𝑐)) → 𝑗 ∈ No )
627	602, 626	lesloed 27730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 ≤s 𝑗 ↔ (𝑔 <s 𝑗 ∨ 𝑔 = 𝑗)))
628	575	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)) → 𝑗 ∈ ℕ0s)
629		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s) → (𝑔 <s 𝑗 ↔ (𝑔 +s 1s ) ≤s 𝑗))
630	601, 628, 629	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 𝑗 ↔ (𝑔 +s 1s ) ≤s 𝑗))
631	630	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 → (𝑔 +s 1s ) ≤s 𝑗))
632	631	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) ≤s 𝑗)
633	478	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))) → (𝑔 +s 1s ) ∈ ℕ0s)
634	633	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 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ∈ ℕ0s)
635	634	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) ∈ No )
636	575	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 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑗 ∈ ℕ0s)
637	636	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑗 ∈ No )
638	605	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 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑑 ∈ No )
639		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 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ≤s 𝑗)
640	569	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)) → 𝑘 ∈ ℕ0s)
641		n0sge0 28338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝑘 ∈ ℕ0s → 0s ≤s 𝑘)
642	640, 641	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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)
643	640	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑘 ∈ No )
644	564	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)) → 𝑙 ∈ ℕ0s)
645	643, 644	pw2ge0divsd 28446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 0s ≤s 𝑘 ↔ 0s ≤s (𝑘 /su (2s↑s𝑙))))
646	643, 644	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑘 /su (2s↑s𝑙)) ∈ No )
647	637, 646	addsge01d 28016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 0s ≤s (𝑘 /su (2s↑s𝑙)) ↔ 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
648	645, 647	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 0s ≤s 𝑘 ↔ 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
649	642, 648	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑗 ≤s (𝑗 +s (𝑘 /su (2s↑s𝑙))))
650	562	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 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
651	649, 650	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑗 ≤s 𝑑)
652	635, 637, 638, 639, 651	lestrd 27738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (𝑔 +s 1s ) ≤s 𝑗)) → (𝑔 +s 1s ) ≤s 𝑑)
653	575	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑗 ∈ ℕ0s)
654	564	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑙 ∈ ℕ0s)
655		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑗 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (𝑗 +s 𝑙) ∈ ℕ0s)
656	653, 654, 655	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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)
657	656	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑗 +s 𝑙) ∈ No )
658	296	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))) → 𝜑)
659	658	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝜑)
660	659, 32	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑁 ∈ No )
661	659, 317	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑁 +s 1s ) ∈ No )
662		simprr3 1225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑗 +s 𝑙) <s 𝑁)
663	662	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑗 +s 𝑙) <s 𝑁)
664	660	ltsp1d 28015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑁 <s (𝑁 +s 1s ))
665	657, 660, 661, 663, 664	ltstrd 27735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑗 +s 𝑙) <s (𝑁 +s 1s ))
666	657, 661	ltsnled 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑗 +s 𝑙) <s (𝑁 +s 1s ) ↔ ¬ (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
667	665, 666	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ¬ (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙))
668	633	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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → (𝑔 +s 1s ) ∈ ℕ0s)
669	668	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ No )
670	605	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑑 ∈ No )
671	669, 670	ltsnled 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑔 +s 1s ) <s 𝑑 ↔ ¬ 𝑑 ≤s (𝑔 +s 1s )))
672	661	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ∈ No )
673	576	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑗 ∈ No )
674	657	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑗 +s 𝑙) ∈ No )
675	633	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) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ∈ ℕ0s)
676	675	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 1s ) ∈ No )
677	333	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))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
678	677	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)) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
679		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑤 = ({𝑐} |s {𝑑}))
680	603	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)) → 𝑐 ∈ No )
681	598	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)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
682	347	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))) → ℎ <s (2s↑s𝑖))
683	682	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)) → ℎ <s (2s↑s𝑖))
684	589	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑖 ∈ ℕ0s)
685	52, 684, 350	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (2s↑s𝑖) ∈ No )
686	685	mulslidd 28143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 1s ·s (2s↑s𝑖)) = (2s↑s𝑖))
687	683, 686	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ <s ( 1s ·s (2s↑s𝑖)))
688	584	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → ℎ ∈ ℕ0s)
689	688	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ ∈ No )
690	322	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 1s ∈ No )
691	689, 690, 684	pw2ltdivmuls2d 28457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ ℎ <s ( 1s ·s (2s↑s𝑖))))
692	687, 691	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) <s 1s )
693	689, 684	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) ∈ No )
694	580	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)) → 𝑔 ∈ ℕ0s)
695	694	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
696	693, 690, 695	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((ℎ /su (2s↑s𝑖)) <s 1s ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
697	692, 696	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
698	681, 697	eqbrtrd 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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑐 <s (𝑔 +s 1s ))
699	680, 676, 698	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → {𝑐} <<s {(𝑔 +s 1s )})
700	605	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)) → 𝑑 ∈ No )
701		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) <s 𝑑)
702	676, 700, 701	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 {𝑑})
703	679, 676, 699, 702	sltsbday 27917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s 1s )))
704	678, 703	eqsstrrd 3970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s 1s )))
705	658	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) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝜑)
706	705, 395	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ∈ Ons)
707		n0on 28336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑔 +s 1s ) ∈ ℕ0s → (𝑔 +s 1s ) ∈ Ons)
708	675, 707	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) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ∈ Ons)
709	706, 708	onlesd 28270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑔 +s 1s ) ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s 1s ))))
710	704, 709	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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 ))
711		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((((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)
712	711	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑑) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑔 +s 1s ) ≤s 𝑗)
713	672, 676, 673, 710, 712	lestrd 27738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (𝑔 +s 1s ) <s 𝑑)) → (𝑁 +s 1s ) ≤s 𝑗)
714	564	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) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑙 ∈ ℕ0s)
715		n0sge0 28338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑙 ∈ ℕ0s → 0s ≤s 𝑙)
716	714, 715	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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)
717	714	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑙 ∈ No )
718	673, 717	addsge01d 28016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ( 0s ≤s 𝑙 ↔ 𝑗 ≤s (𝑗 +s 𝑙)))
719	716, 718	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) ∧ (𝑔 +s 1s ) <s 𝑑)) → 𝑗 ≤s (𝑗 +s 𝑙))
720	672, 673, 674, 713, 719	lestrd 27738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙))
721	720	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝑔 +s 1s ) <s 𝑑 → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
722	671, 721	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (¬ 𝑑 ≤s (𝑔 +s 1s ) → (𝑁 +s 1s ) ≤s (𝑗 +s 𝑙)))
723	667, 722	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 ≤s (𝑔 +s 1s ))
724		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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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)
725	670, 669	lestri3d 27731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑑 = (𝑔 +s 1s ) ↔ (𝑑 ≤s (𝑔 +s 1s ) ∧ (𝑔 +s 1s ) ≤s 𝑑)))
726	723, 724, 725	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (𝑔 +s 1s ) ≤s 𝑑)) → 𝑑 = (𝑔 +s 1s ))
727	682	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → ℎ <s (2s↑s𝑖))
728	584	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) ∧ 𝑑 = (𝑔 +s 1s ))) → ℎ ∈ ℕ0s)
729	589	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) ∧ 𝑑 = (𝑔 +s 1s ))) → 𝑖 ∈ ℕ0s)
730		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (2s↑s𝑖) ∈ ℕ0s)
731	412, 729, 730	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (2s↑s𝑖) ∈ ℕ0s)
732		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((ℎ ∈ ℕ0s ∧ (2s↑s𝑖) ∈ ℕ0s) → (ℎ <s (2s↑s𝑖) ↔ (ℎ +s 1s ) ≤s (2s↑s𝑖)))
733	728, 731, 732	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ <s (2s↑s𝑖) ↔ (ℎ +s 1s ) ≤s (2s↑s𝑖)))
734	727, 733	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) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ +s 1s ) ≤s (2s↑s𝑖))
735	354	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ +s 1s ) ∈ ℕ0s)
736	735	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))) → (ℎ +s 1s ) ∈ ℕ0s)
737	736	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) ∧ 𝑑 = (𝑔 +s 1s ))) → (ℎ +s 1s ) ∈ ℕ0s)
738	737	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → (ℎ +s 1s ) ∈ No )
739	731	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → (2s↑s𝑖) ∈ No )
740	738, 739	lesloed 27730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → ((ℎ +s 1s ) ≤s (2s↑s𝑖) ↔ ((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖))))
741	658	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 ))) → 𝜑)
742	32, 317	ltsnled 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝜑 → (𝑁 <s (𝑁 +s 1s ) ↔ ¬ (𝑁 +s 1s ) ≤s 𝑁))
743	38, 742	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (𝜑 → ¬ (𝑁 +s 1s ) ≤s 𝑁)
744	741, 743	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) ∧ 𝑑 = (𝑔 +s 1s ))) → ¬ (𝑁 +s 1s ) ≤s 𝑁)
745	677	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 )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
746		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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑤 = ({𝑐} |s {𝑑}))
747	580	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 ∈ ℕ0s)
748	747	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ No )
749	736	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))) → (ℎ +s 1s ) ∈ ℕ0s)
750	749	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ +s 1s ) ∈ No )
751	589	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))) → 𝑖 ∈ ℕ0s)
752	750, 751	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) ∈ No )
753	748, 752	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) ∈ No )
754	603	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑐 ∈ No )
755	598	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))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
756	584	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 ∈ ℕ0s)
757	756	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
758	757	ltsp1d 28015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (ℎ +s 1s ))
759	757, 750, 751	pw2ltsdiv1d 28452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (ℎ +s 1s ) ↔ (ℎ /su (2s↑s𝑖)) <s ((ℎ +s 1s ) /su (2s↑s𝑖))))
760	758, 759	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ /su (2s↑s𝑖)) <s ((ℎ +s 1s ) /su (2s↑s𝑖)))
761	757, 751	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /su (2s↑s𝑖)) ∈ No )
762	761, 752, 748	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((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)))))
763	760, 762	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) ∧ 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))))
764	755, 763	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 <s (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))))
765	754, 753, 764	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {𝑐} <<s {(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))})
766	605	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑑 ∈ No )
767		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) <s (2s↑s𝑖))
768	52, 751, 350	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (2s↑s𝑖) ∈ No )
769	768	mulslidd 28143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( 1s ·s (2s↑s𝑖)) = (2s↑s𝑖))
770	767, 769	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖)))
771	322	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 1s ∈ No )
772	750, 771, 751	pw2ltdivmuls2d 28457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) <s 1s ↔ (ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖))))
773	752, 771, 748	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) <s 1s ↔ (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
774	772, 773	bitr3d 281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((ℎ +s 1s ) <s ( 1s ·s (2s↑s𝑖)) ↔ (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s )))
775	770, 774	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s (𝑔 +s 1s ))
776		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (𝑔 +s 1s ))
777	775, 776	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖))) <s 𝑑)
778	753, 766, 777	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))} <<s {𝑑})
779	746, 753, 765, 778	sltsbday 27917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))))
780	745, 779	eqsstrrd 3970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))))
781	658	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 ) <s (2s↑s𝑖))) → 𝜑)
782	781, 1	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → 𝑁 ∈ ℕ0s)
783	304	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))) → (𝑔 +s 𝑖) <s 𝑁)
784	783	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 )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑔 +s 𝑖) <s 𝑁)
785	782, 747, 749, 751, 767, 784	bdaypw2bnd 28465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘(𝑔 +s ((ℎ +s 1s ) /su (2s↑s𝑖)))) ⊆ ( bday ‘𝑁))
786	780, 785	sstrd 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁))
787	219, 395	onltsd 28269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝜑 → (𝑁 <s (𝑁 +s 1s ) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
788	781, 787	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → (𝑁 <s (𝑁 +s 1s ) ↔ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
789	788	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑁 +s 1s ) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
790	317, 32	lesnltd 27728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝜑 → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑁 +s 1s )))
791	781, 790	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 )) ∧ (ℎ +s 1s ) <s (2s↑s𝑖))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s (𝑁 +s 1s )))
792		bdayon 27752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ( bday ‘(𝑁 +s 1s )) ∈ On
793		ontri1 6352	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((( bday ‘(𝑁 +s 1s )) ∈ On ∧ ( bday ‘𝑁) ∈ On) → (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
794	792, 7, 793	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s )))
795	794	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁) ↔ ¬ ( bday ‘𝑁) ∈ ( bday ‘(𝑁 +s 1s ))))
796	789, 791, 795	3bitr4d 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
797	786, 796	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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)
798	797	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → (𝑁 +s 1s ) ≤s 𝑁))
799	744, 798	mtod 198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
800		orel1 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (¬ (ℎ +s 1s ) <s (2s↑s𝑖) → (((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖)) → (ℎ +s 1s ) = (2s↑s𝑖)))
801	799, 800	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → (((ℎ +s 1s ) <s (2s↑s𝑖) ∨ (ℎ +s 1s ) = (2s↑s𝑖)) → (ℎ +s 1s ) = (2s↑s𝑖)))
802	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑔 ∈ ℕ0s)
803	584	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 ) = (2s↑s𝑖))) → ℎ ∈ ℕ0s)
804		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → (2s ·s ℎ) ∈ ℕ0s)
805	412, 803, 804	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 )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s ·s ℎ) ∈ ℕ0s)
806	805	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) ∈ ℕ0s)
807	589	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑖 +s 1s ) ∈ ℕ0s)
808	807	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 1s ) = (2s↑s𝑖))) → (𝑖 +s 1s ) ∈ ℕ0s)
809		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑤 = ({𝑐} |s {𝑑}))
810	802	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ No )
811	589	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))) → 𝑖 ∈ ℕ0s)
812	810, 811	pw2divscan3d 28441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑖)) = 𝑔)
813	812	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑔) /su (2s↑s𝑖)) +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
814	52, 589, 350	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))) → (2s↑s𝑖) ∈ No )
815	814	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))) → (2s↑s𝑖) ∈ No )
816	815, 810	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
817	584	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ℎ ∈ No )
818	817	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ℎ ∈ No )
819	816, 818, 811	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑔) +s ℎ) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s (ℎ /su (2s↑s𝑖))))
820	598	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))) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
821	813, 819, 820	3eqtr4rd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 = ((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖)))
822	821	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {𝑐} = {((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))})
823		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ +s 1s ) = (2s↑s𝑖))
824	823, 815	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (ℎ +s 1s ) ∈ No )
825	816, 824, 811	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 +s 1s )) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) /su (2s↑s𝑖)) +s ((ℎ +s 1s ) /su (2s↑s𝑖))))
826	823	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) = ((2s↑s𝑖) /su (2s↑s𝑖)))
827	811	pw2divsidd 28456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑖)) = 1s )
828	826, 827	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((ℎ +s 1s ) /su (2s↑s𝑖)) = 1s )
829	812, 828	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑖)) +s ((ℎ +s 1s ) /su (2s↑s𝑖))) = (𝑔 +s 1s ))
830	825, 829	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑔) +s (ℎ +s 1s )) /su (2s↑s𝑖)) = (𝑔 +s 1s ))
831	322	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 1s ∈ No )
832	816, 818, 831	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) +s 1s ) = (((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )))
833	832	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖)) = ((((2s↑s𝑖) ·s 𝑔) +s (ℎ +s 1s )) /su (2s↑s𝑖)))
834		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (𝑔 +s 1s ))
835	830, 833, 834	3eqtr4rd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖)))
836	835	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {𝑑} = {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))})
837	822, 836	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ({𝑐} |s {𝑑}) = ({((((2s↑s𝑖) ·s 𝑔) +s ℎ) /su (2s↑s𝑖))} |s {(((((2s↑s𝑖) ·s 𝑔) +s ℎ) +s 1s ) /su (2s↑s𝑖))}))
838	412, 811, 730	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → (2s↑s𝑖) ∈ ℕ0s)
839		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ⊢ (((2s↑s𝑖) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s)
840	838, 802, 839	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 <s 𝑗) ∧ (𝑔 +s 1s ) ≤s 𝑗) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s)
841		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ ((((2s↑s𝑖) ·s 𝑔) ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → (((2s↑s𝑖) ·s 𝑔) +s ℎ) ∈ ℕ0s)
842	840, 803, 841	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑔) +s ℎ) ∈ ℕ0s)
843	842	n0zsd 28390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑s𝑖) ·s 𝑔) +s ℎ) ∈ ℤs)
844	843, 811	pw2cutp1 28461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑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 ))))
845	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
846	845, 816, 818	addsdid 28156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ·s 𝑔) +s ℎ)) = ((2s ·s ((2s↑s𝑖) ·s 𝑔)) +s (2s ·s ℎ)))
847		expsp1 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((2s ∈ No ∧ 𝑖 ∈ ℕ0s) → (2s↑s(𝑖 +s 1s )) = ((2s↑s𝑖) ·s 2s))
848	52, 811, 847	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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(𝑖 +s 1s )) = ((2s↑s𝑖) ·s 2s))
849	815, 845	mulscomd 28140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) = (2s ·s (2s↑s𝑖)))
850	848, 849	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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(𝑖 +s 1s )) = (2s ·s (2s↑s𝑖)))
851	850	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) ·s 𝑔) = ((2s ·s (2s↑s𝑖)) ·s 𝑔))
852	845, 815, 810	mulsassd 28167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = (2s ·s ((2s↑s𝑖) ·s 𝑔)))
853	851, 852	eqtr2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) = ((2s↑s(𝑖 +s 1s )) ·s 𝑔))
854	853	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ·s 𝑔)) +s (2s ·s ℎ)) = (((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)))
855	846, 854	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (((2s↑s𝑖) ·s 𝑔) +s ℎ)) = (((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)))
856	855	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (((2s↑s𝑖) ·s 𝑔) +s ℎ)) +s 1s ) = ((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ))
857	856	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (((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 ))))
858	844, 857	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ({((((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 ))))
859	809, 837, 858	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑤 = (((((2s↑s(𝑖 +s 1s )) ·s 𝑔) +s (2s ·s ℎ)) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
860		expscl 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((2s ∈ No ∧ (𝑖 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑖 +s 1s )) ∈ No )
861	52, 808, 860	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(𝑖 +s 1s )) ∈ No )
862	861, 810	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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(𝑖 +s 1s )) ·s 𝑔) ∈ No )
863	805	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
864	862, 863, 831	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((((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 )))
865	864	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (((((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 ))))
866	806	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ No )
867	862, 866, 808	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((((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 )))))
868	810, 808	pw2divscan3d 28441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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(𝑖 +s 1s )) ·s 𝑔) /su (2s↑s(𝑖 +s 1s ))) = 𝑔)
869	868	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((((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 )))))
870	867, 869	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((((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 )))))
871	859, 865, 870	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (2s↑s𝑖))) → 𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s )))))
872	831, 845, 863	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( 1s <s 2s ↔ ((2s ·s ℎ) +s 1s ) <s ((2s ·s ℎ) +s 2s)))
873	523, 872	mpbii 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((2s ·s ℎ) +s 1s ) <s ((2s ·s ℎ) +s 2s))
874	823	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = (2s ·s (2s↑s𝑖)))
875	845, 818, 831	addsdid 28156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = ((2s ·s ℎ) +s (2s ·s 1s )))
876		mulsrid 28113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
877	52, 876	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (2s ·s 1s ) = 2s
878	877	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((2s ·s ℎ) +s (2s ·s 1s )) = ((2s ·s ℎ) +s 2s)
879	875, 878	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = ((2s ·s ℎ) +s 2s))
880	849, 874, 879	3eqtr2d 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((2s↑s𝑖) ·s 2s) = ((2s ·s ℎ) +s 2s))
881	848, 880	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (2s↑s(𝑖 +s 1s )) = ((2s ·s ℎ) +s 2s))
882	873, 881	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (2s↑s𝑖))) → ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s )))
883	811	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑖 ∈ No )
884	810, 883, 831	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑔 +s 𝑖) +s 1s ) = (𝑔 +s (𝑖 +s 1s )))
885	783	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 ) = (2s↑s𝑖))) → (𝑔 +s 𝑖) <s 𝑁)
886	810, 883	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
887	658	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 )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝜑)
888	887, 32	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 )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → 𝑁 ∈ No )
889	886, 888, 831	ltadds1d 27998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑔 +s 𝑖) <s 𝑁 ↔ ((𝑔 +s 𝑖) +s 1s ) <s (𝑁 +s 1s )))
890	885, 889	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) ∧ 𝑑 = (𝑔 +s 1s )) ∧ (ℎ +s 1s ) = (2s↑s𝑖))) → ((𝑔 +s 𝑖) +s 1s ) <s (𝑁 +s 1s ))
891	884, 890	eqbrtrrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (2s↑s𝑖))) → (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s ))
892		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑏 /su (2s↑s𝑞)) = (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)))
893	892	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞))))
894	893	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)))))
895		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑏 = ((2s ·s ℎ) +s 1s ) → (𝑏 <s (2s↑s𝑞) ↔ ((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞)))
896	894, 895	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (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 ))))
897		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ (𝑞 = (𝑖 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑖 +s 1s )))
898	897	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ (𝑞 = (𝑖 +s 1s ) → (((2s ·s ℎ) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s ℎ) +s 1s ) /su (2s↑s(𝑖 +s 1s ))))
899	898	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (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 )))))
900	899	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (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 ))))))
901	897	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑞 = (𝑖 +s 1s ) → (((2s ·s ℎ) +s 1s ) <s (2s↑s𝑞) ↔ ((2s ·s ℎ) +s 1s ) <s (2s↑s(𝑖 +s 1s ))))
902		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (𝑞 = (𝑖 +s 1s ) → (𝑔 +s 𝑞) = (𝑔 +s (𝑖 +s 1s )))
903	902	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ (𝑞 = (𝑖 +s 1s ) → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s (𝑖 +s 1s )) <s (𝑁 +s 1s )))
904	900, 901, 903	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (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 ))))
905	535, 896, 904	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((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 )))
906	802, 806, 808, 871, 882, 891, 905	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (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 )))
907	906	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → ((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 ))))
908	801, 907	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → (((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 ))))
909	740, 908	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = (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 ))))
910	734, 909	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) ∧ 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 )))
911	910	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)) → (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 ))))
912	911	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 ))))
913	726, 912	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 )))
914	913	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 ))))
915	652, 914	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 )))
916	915	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 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 ))))
917	632, 916	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)) → ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑁 +s 1s )))
918	917	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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 ))))
919	609	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 = 𝑗)) → 𝑐 <s 𝑑)
920	598	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 = 𝑗)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
921	562	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 = 𝑗)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
922		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = 𝑗)
923	922	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s (𝑘 /su (2s↑s𝑙))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
924	921, 923	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
925	919, 920, 924	3brtr3d 5130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s (𝑘 /su (2s↑s𝑙))))
926	817	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 = 𝑗)) → ℎ ∈ No )
927	589	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 = 𝑗)) → 𝑖 ∈ ℕ0s)
928	926, 927	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) ∈ No )
929	570	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 = 𝑗)) → 𝑘 ∈ No )
930	564	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 = 𝑗)) → 𝑙 ∈ ℕ0s)
931	929, 930	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑘 /su (2s↑s𝑙)) ∈ No )
932	580	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ No )
933	932	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 = 𝑗)) → 𝑔 ∈ No )
934	928, 931, 933	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)) ↔ (𝑔 +s (ℎ /su (2s↑s𝑖))) <s (𝑔 +s (𝑘 /su (2s↑s𝑙)))))
935	925, 934	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁))) ∧ ((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗)) → (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
936	584	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) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ ∈ ℕ0s)
937	564	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 ∈ ℕ0s)
938	589	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 ∈ ℕ0s)
939		n0subs 28363	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑙 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (𝑙 ≤s 𝑖 ↔ (𝑖 -s 𝑙) ∈ ℕ0s))
940	937, 938, 939	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (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))
941	940	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑙) ∈ ℕ0s))
942	941	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑖 -s 𝑙) ∈ ℕ0s)
943		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((2s ∈ ℕ0s ∧ (𝑖 -s 𝑙) ∈ ℕ0s) → (2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s)
944	412, 942, 943	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s)
945	569	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑘 ∈ ℕ0s)
946		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((2s↑s(𝑖 -s 𝑙)) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ ℕ0s)
947	944, 945, 946	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ∈ ℕ0s)
948	589	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) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑖 ∈ ℕ0s)
949		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
950	945	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑘 ∈ No )
951	564	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)) → 𝑙 ∈ ℕ0s)
952	950, 951, 942	pw2divscan4d 28444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /su (2s↑s𝑙)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s(𝑙 +s (𝑖 -s 𝑙)))))
953	951	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑙 ∈ No )
954	942	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑖 -s 𝑙) ∈ No )
955	953, 954	addscomd 27967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑙 +s (𝑖 -s 𝑙)) = ((𝑖 -s 𝑙) +s 𝑙))
956	948	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑖 ∈ No )
957		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((𝑖 ∈ No ∧ 𝑙 ∈ No ) → ((𝑖 -s 𝑙) +s 𝑙) = 𝑖)
958	956, 953, 957	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) +s 𝑙) = 𝑖)
959	955, 958	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑙 +s (𝑖 -s 𝑙)) = 𝑖)
960	959	oveq2d 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)) → (2s↑s(𝑙 +s (𝑖 -s 𝑙))) = (2s↑s𝑖))
961	960	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s(𝑙 +s (𝑖 -s 𝑙)))) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
962	952, 961	eqtr2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑖)) = (𝑘 /su (2s↑s𝑙)))
963	949, 962	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) <s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
964	936	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
965	947	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
966	964, 965, 948	pw2ltsdiv1d 28452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) ↔ (ℎ /su (2s↑s𝑖)) <s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))))
967	963, 966	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘))
968	682	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ℎ <s (2s↑s𝑖))
969	563	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)) → 𝑘 <s (2s↑s𝑙))
970		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((2s ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (2s↑s𝑙) ∈ ℕ0s)
971	412, 951, 970	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))) ∧ 𝑙 ≤s 𝑖)) → (2s↑s𝑙) ∈ ℕ0s)
972	971	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑s𝑙) ∈ No )
973	944	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) ∈ No )
974		nnsgt0 28339	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (2s ∈ ℕs → 0s <s 2s)
975	410, 974	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ 0s <s 2s
976		expsgt0 28437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((2s ∈ No ∧ (𝑖 -s 𝑙) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑖 -s 𝑙)))
977	52, 975, 976	mp3an13 1455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑖 -s 𝑙) ∈ ℕ0s → 0s <s (2s↑s(𝑖 -s 𝑙)))
978	942, 977	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s <s (2s↑s(𝑖 -s 𝑙)))
979	950, 972, 973, 978	ltmuls2d 28172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑘 <s (2s↑s𝑙) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙))))
980	969, 979	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 = 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)))
981		expadds 28435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((2s ∈ No ∧ (𝑖 -s 𝑙) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (2s↑s((𝑖 -s 𝑙) +s 𝑙)) = ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
982	52, 942, 951, 981	mp3an2i 1469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑙)) = ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)))
983	958	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑙)) = (2s↑s𝑖))
984	982, 983	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((2s↑s(𝑖 -s 𝑙)) ·s (2s↑s𝑙)) = (2s↑s𝑖))
985	980, 984	breqtrd 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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖))
986	967, 968, 985	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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)))
987	598	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
988	562	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
989		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖) → 𝑔 = 𝑗)
990	989	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 = 𝑗)
991	990, 962	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
992	988, 991	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 = (𝑔 +s (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖))))
993	783	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑙 ≤s 𝑖)) → (𝑔 +s 𝑖) <s 𝑁)
994	987, 992, 993	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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))
995		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑚 = ℎ → (𝑚 <s 𝑛 ↔ ℎ <s 𝑛))
996		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑚 = ℎ → (𝑚 <s (2s↑s𝑜) ↔ ℎ <s (2s↑s𝑜)))
997	995, 996	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑚 = ℎ → ((𝑚 <s 𝑛 ∧ 𝑚 <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜)) ↔ (ℎ <s 𝑛 ∧ ℎ <s (2s↑s𝑜) ∧ 𝑛 <s (2s↑s𝑜))))
998		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ℎ → (𝑚 /su (2s↑s𝑜)) = (ℎ /su (2s↑s𝑜)))
999	998	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑚 = ℎ → (𝑔 +s (𝑚 /su (2s↑s𝑜))) = (𝑔 +s (ℎ /su (2s↑s𝑜))))
1000	999	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑚 = ℎ → (𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜)))))
1001	1000	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1002	997, 1001	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1003		breq2 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (ℎ <s 𝑛 ↔ ℎ <s ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘)))
1004		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑛 <s (2s↑s𝑜) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜)))
1005	1003, 1004	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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))))
1006		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) → (𝑛 /su (2s↑s𝑜)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜)))
1007	1006	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (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))))
1008	1007	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (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)))))
1009	1008	3anbi2d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1010	1005, 1009	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1011		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑖 → (2s↑s𝑜) = (2s↑s𝑖))
1012	1011	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑖 → (ℎ <s (2s↑s𝑜) ↔ ℎ <s (2s↑s𝑖)))
1013	1011	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑖 → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑜) ↔ ((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) <s (2s↑s𝑖)))
1014	1012, 1013	3anbi23d 1442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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))))
1015	1011	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑖 → (ℎ /su (2s↑s𝑜)) = (ℎ /su (2s↑s𝑖)))
1016	1015	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑖 → (𝑔 +s (ℎ /su (2s↑s𝑜))) = (𝑔 +s (ℎ /su (2s↑s𝑖))))
1017	1016	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑖 → (𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑜))) ↔ 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖)))))
1018	1011	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑖 → (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑜)) = (((2s↑s(𝑖 -s 𝑙)) ·s 𝑘) /su (2s↑s𝑖)))
1019	1018	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (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))))
1020	1019	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (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)))))
1021		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑖 → (𝑔 +s 𝑜) = (𝑔 +s 𝑖))
1022	1021	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑖 → ((𝑔 +s 𝑜) <s 𝑁 ↔ (𝑔 +s 𝑖) <s 𝑁))
1023	1017, 1020, 1022	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1024	1014, 1023	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1025	1002, 1010, 1024	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((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)))
1026	936, 947, 948, 986, 994, 1025	syl32anc 1381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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)))
1027	1026	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)))) → (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))))
1028		n0subs 28363	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑖 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) → (𝑖 ≤s 𝑙 ↔ (𝑙 -s 𝑖) ∈ ℕ0s))
1029	938, 937, 1028	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (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))
1030	1029	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑖) ∈ ℕ0s))
1031	1030	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑙 -s 𝑖) ∈ ℕ0s)
1032		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((2s ∈ ℕ0s ∧ (𝑙 -s 𝑖) ∈ ℕ0s) → (2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s)
1033	412, 1031, 1032	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s)
1034	584	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ℎ ∈ ℕ0s)
1035		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (((2s↑s(𝑙 -s 𝑖)) ∈ ℕ0s ∧ ℎ ∈ ℕ0s) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ ℕ0s)
1036	1033, 1034, 1035	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) ∈ ℕ0s)
1037	569	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) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑘 ∈ ℕ0s)
1038	564	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) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑙 ∈ ℕ0s)
1039	1034	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ℎ ∈ No )
1040	589	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)) → 𝑖 ∈ ℕ0s)
1041	1039, 1040, 1031	pw2divscan4d 28444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /su (2s↑s𝑖)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s(𝑖 +s (𝑙 -s 𝑖)))))
1042	1040	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑖 ∈ No )
1043	1031	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1044	1042, 1043	addscomd 27967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑖 +s (𝑙 -s 𝑖)) = ((𝑙 -s 𝑖) +s 𝑖))
1045	1044	oveq2d 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))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s(𝑖 +s (𝑙 -s 𝑖))) = (2s↑s((𝑙 -s 𝑖) +s 𝑖)))
1046	1045	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s(𝑖 +s (𝑙 -s 𝑖)))) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s((𝑙 -s 𝑖) +s 𝑖))))
1047	1038	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
1048		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ ((𝑙 ∈ No ∧ 𝑖 ∈ No ) → ((𝑙 -s 𝑖) +s 𝑖) = 𝑙)
1049	1047, 1042, 1048	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)) → ((𝑙 -s 𝑖) +s 𝑖) = 𝑙)
1050	1049	oveq2d 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))) ∧ 𝑖 ≤s 𝑙)) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = (2s↑s𝑙))
1051	1050	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s((𝑙 -s 𝑖) +s 𝑖))) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1052	1041, 1046, 1051	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1053		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))
1054	1052, 1053	eqbrtrrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)) <s (𝑘 /su (2s↑s𝑙)))
1055		expscl 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s) → (2s↑s(𝑙 -s 𝑖)) ∈ No )
1056	52, 1031, 1055	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)) ∈ No )
1057	1056, 1039	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1058	1037	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
1059	1057, 1058, 1038	pw2ltsdiv1d 28452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘 ↔ (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)) <s (𝑘 /su (2s↑s𝑙))))
1060	1054, 1059	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘)
1061	682	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)) → ℎ <s (2s↑s𝑖))
1062	52, 1040, 350	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↑s𝑖) ∈ No )
1063		expsgt0 28437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑙 -s 𝑖)))
1064	52, 975, 1063	mp3an13 1455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑙 -s 𝑖) ∈ ℕ0s → 0s <s (2s↑s(𝑙 -s 𝑖)))
1065	1031, 1064	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s <s (2s↑s(𝑙 -s 𝑖)))
1066	1039, 1062, 1056, 1065	ltmuls2d 28172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (ℎ <s (2s↑s𝑖) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖))))
1067	1061, 1066	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 = 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)))
1068		expadds 28435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((2s ∈ No ∧ (𝑙 -s 𝑖) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s) → (2s↑s((𝑙 -s 𝑖) +s 𝑖)) = ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1069	52, 1031, 1040, 1068	mp3an2i 1469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑖)) = ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)))
1070	1069, 1050	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((2s↑s(𝑙 -s 𝑖)) ·s (2s↑s𝑖)) = (2s↑s𝑙))
1071	1067, 1070	breqtrd 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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙))
1072	563	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑘 <s (2s↑s𝑙))
1073	1060, 1071, 1072	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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)))
1074	598	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑐 = (𝑔 +s (ℎ /su (2s↑s𝑖))))
1075	1052	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s (ℎ /su (2s↑s𝑖))) = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))))
1076	1074, 1075	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑐 = (𝑔 +s (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙))))
1077	562	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1078		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((((𝑑 <s (𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐) ∧ 𝑔 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙) → 𝑔 = 𝑗)
1079	1078	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑔 = 𝑗)
1080	1079	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s (𝑘 /su (2s↑s𝑙))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1081	1077, 1080	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
1082	1079	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 𝑙) = (𝑗 +s 𝑙))
1083	662	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙))) ∧ 𝑖 ≤s 𝑙)) → (𝑗 +s 𝑙) <s 𝑁)
1084	1082, 1083	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (𝑔 +s 𝑙) <s 𝑁)
1085	1076, 1081, 1084	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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))
1086		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 <s 𝑛 ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛))
1087		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 <s (2s↑s𝑜) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜)))
1088	1086, 1087	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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))))
1089		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑚 = ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) → (𝑚 /su (2s↑s𝑜)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜)))
1090	1089	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (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))))
1091	1090	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (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)))))
1092	1091	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1093	1088, 1092	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1094		breq2 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑛 = 𝑘 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑛 ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s 𝑘))
1095		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑛 = 𝑘 → (𝑛 <s (2s↑s𝑜) ↔ 𝑘 <s (2s↑s𝑜)))
1096	1094, 1095	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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))))
1097		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑛 = 𝑘 → (𝑛 /su (2s↑s𝑜)) = (𝑘 /su (2s↑s𝑜)))
1098	1097	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑛 = 𝑘 → (𝑔 +s (𝑛 /su (2s↑s𝑜))) = (𝑔 +s (𝑘 /su (2s↑s𝑜))))
1099	1098	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑛 = 𝑘 → (𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜)))))
1100	1099	3anbi2d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1101	1096, 1100	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1102		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑙 → (2s↑s𝑜) = (2s↑s𝑙))
1103	1102	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑙 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑜) ↔ ((2s↑s(𝑙 -s 𝑖)) ·s ℎ) <s (2s↑s𝑙)))
1104	1102	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑙 → (𝑘 <s (2s↑s𝑜) ↔ 𝑘 <s (2s↑s𝑙)))
1105	1103, 1104	3anbi23d 1442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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))))
1106	1102	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑙 → (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑜)) = (((2s↑s(𝑙 -s 𝑖)) ·s ℎ) /su (2s↑s𝑙)))
1107	1106	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (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))))
1108	1107	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (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)))))
1109	1102	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑜 = 𝑙 → (𝑘 /su (2s↑s𝑜)) = (𝑘 /su (2s↑s𝑙)))
1110	1109	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑙 → (𝑔 +s (𝑘 /su (2s↑s𝑜))) = (𝑔 +s (𝑘 /su (2s↑s𝑙))))
1111	1110	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑙 → (𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑜))) ↔ 𝑑 = (𝑔 +s (𝑘 /su (2s↑s𝑙)))))
1112		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑜 = 𝑙 → (𝑔 +s 𝑜) = (𝑔 +s 𝑙))
1113	1112	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑜 = 𝑙 → ((𝑔 +s 𝑜) <s 𝑁 ↔ (𝑔 +s 𝑙) <s 𝑁))
1114	1108, 1111, 1113	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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)))
1115	1105, 1114	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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))))
1116	1093, 1101, 1115	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((((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)))
1117	1036, 1037, 1038, 1073, 1085, 1116	syl32anc 1381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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)))
1118	1117	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)))) → (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))))
1119	937	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑙 ∈ No )
1120	938	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑖 ∈ No )
1121		lestric 27740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((𝑙 ∈ No ∧ 𝑖 ∈ No ) → (𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙))
1122	1119, 1120, 1121	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙))
1123	1027, 1118, 1122	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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)))
1124	580	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 /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝑔 ∈ ℕ0s)
1125	1124	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) ∧ (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)
1126		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑚 ∈ ℕ0s)
1127		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((2s ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (2s ·s 𝑚) ∈ ℕ0s)
1128	412, 1126, 1127	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s ·s 𝑚) ∈ ℕ0s)
1129	1128	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((2s ·s 𝑚) +s 1s ) ∈ ℕ0s)
1130		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑜 ∈ ℕ0s)
1131	1130	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑜 +s 1s ) ∈ ℕ0s)
1132		simpll2 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑤 = ({𝑐} |s {𝑑}))
1133	1132	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)))) → 𝑤 = ({𝑐} |s {𝑑}))
1134	1125	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑔 ∈ No )
1135	1134, 1130	pw2divscan3d 28441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑜)) = 𝑔)
1136	1135	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑜)) +s (𝑚 /su (2s↑s𝑜))) = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1137		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ⊢ ((2s ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) → (2s↑s𝑜) ∈ ℕ0s)
1138	412, 1130, 1137	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ ℕ0s)
1139		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ⊢ (((2s↑s𝑜) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s𝑜) ·s 𝑔) ∈ ℕ0s)
1140	1138, 1125, 1139	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) ∈ ℕ0s)
1141	1140	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1142	1126	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
1143	1141, 1142, 1130	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) = ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s (𝑚 /su (2s↑s𝑜))))
1144		simprr1 1223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((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))))
1145	1144	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)))) ∧ ((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))))
1146	1136, 1143, 1145	3eqtr4rd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑐 = ((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜)))
1147	1146	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → {𝑐} = {((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜))})
1148	1126	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑚 +s 1s ) ∈ ℕ0s)
1149	1148	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑚 +s 1s ) ∈ No )
1150	1141, 1149, 1130	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 +s 1s )) /su (2s↑s𝑜)) = ((((2s↑s𝑜) ·s 𝑔) /su (2s↑s𝑜)) +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1151	1135	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) /su (2s↑s𝑜)) +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1152	1150, 1151	eqtr2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ((𝑚 +s 1s ) /su (2s↑s𝑜))) = ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)))
1153		simprr2 1224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1154	1153	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)))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1155	658	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 = 𝑗) ∧ (ℎ /su (2s↑s𝑖)) <s (𝑘 /su (2s↑s𝑙)))) → 𝜑)
1156	1155	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)))) ∧ ((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)
1157	1156, 743	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 = 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)
1158	322	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 1s ∈ No )
1159		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ℕ0s)
1160	1159	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
1161	1160, 1142	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1162	1158, 1161	ltsnled 27729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ( 1s <s (𝑛 -s 𝑚) ↔ ¬ (𝑛 -s 𝑚) ≤s 1s ))
1163	677	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)))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
1164	1163	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))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘𝑤) = ( bday ‘(𝑁 +s 1s )))
1165	1132	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)))) ∧ (((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}))
1166	1124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ ℕ0s)
1167	1166	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑔 ∈ No )
1168	1126	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))) → 𝑚 ∈ ℕ0s)
1169	1168	peano2n0sd 28331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∈ ℕ0s)
1170	1169	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑚 +s 1s ) ∈ No )
1171	1130	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑜 ∈ ℕ0s)
1172	1170, 1171	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑚 +s 1s ) /su (2s↑s𝑜)) ∈ No )
1173	1167, 1172	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) ∈ No )
1174	603	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑐 ∈ No )
1175	1174	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 = 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 )
1176	1144	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 = (𝑔 +s (𝑚 /su (2s↑s𝑜))))
1177	1142	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))) → 𝑚 ∈ No )
1178	1177	ltsp1d 28015	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑚 +s 1s ))
1179	1177, 1170, 1171	pw2ltsdiv1d 28452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑚 +s 1s ) ↔ (𝑚 /su (2s↑s𝑜)) <s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1180	1177, 1171	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /su (2s↑s𝑜)) ∈ No )
1181	1180, 1172, 1167	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 /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)))))
1182	1179, 1181	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑚 +s 1s ) ↔ (𝑔 +s (𝑚 /su (2s↑s𝑜))) <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1183	1178, 1182	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s (𝑚 /su (2s↑s𝑜))) <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1184	1176, 1183	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑐 <s (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1185	1175, 1173, 1184	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {𝑐} <<s {(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))})
1186	605	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑑 ∈ No )
1187	1186	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 = 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 )
1188	1142, 1158, 1160	ltaddsubs2d 28092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑚 +s 1s ) <s 𝑛 ↔ 1s <s (𝑛 -s 𝑚)))
1189	1188	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
1190	1189	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑛)
1191	1159	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))) → 𝑛 ∈ ℕ0s)
1192	1191	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑛 ∈ No )
1193	1170, 1192, 1171	pw2ltsdiv1d 28452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ↔ ((𝑚 +s 1s ) /su (2s↑s𝑜)) <s (𝑛 /su (2s↑s𝑜))))
1194	1192, 1171	pw2divscld 28439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑛 /su (2s↑s𝑜)) ∈ No )
1195	1172, 1194, 1167	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) /su (2s↑s𝑜)) <s (𝑛 /su (2s↑s𝑜)) ↔ (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜)))))
1196	1193, 1195	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑛 ↔ (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜)))))
1197	1190, 1196	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1198	1153	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑑 = (𝑔 +s (𝑛 /su (2s↑s𝑜))))
1199	1197, 1198	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))) <s 𝑑)
1200	1173, 1187, 1199	sltssn 27770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → {(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))} <<s {𝑑})
1201	1165, 1173, 1185, 1200	sltsbday 27917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘𝑤) ⊆ ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1202	1164, 1201	eqsstrrd 3970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))))
1203	1155	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)))) ∧ (((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)
1204	1203, 1	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ℕ0s)
1205		expscl 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((2s ∈ No ∧ 𝑜 ∈ ℕ0s) → (2s↑s𝑜) ∈ No )
1206	52, 1130, 1205	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 = 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 )
1207	1206	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (2s↑s𝑜) ∈ No )
1208		simprl1 1220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ (((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)
1209	1208	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → 𝑚 <s 𝑛)
1210		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ⊢ ((𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s 𝑛))
1211	1168, 1191, 1210	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))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑚 <s 𝑛 ↔ (𝑚 +s 1s ) ≤s 𝑛))
1212	1209, 1211	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑚 +s 1s ) ≤s 𝑛)
1213		simprl3 1222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ⊢ (((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))
1214	1213	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (2s↑s𝑜))
1215	1170, 1192, 1207, 1212, 1214	leltstrd 27737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑚 +s 1s ) <s (2s↑s𝑜))
1216		simprr3 1225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ⊢ (((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)
1217	1216	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑔 +s 𝑜) <s 𝑁)
1218	1204, 1166, 1169, 1171, 1215, 1217	bdaypw2bnd 28465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 1s <s (𝑛 -s 𝑚))) → ( bday ‘(𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜)))) ⊆ ( bday ‘𝑁))
1219	1202, 1218	sstrd 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁))
1220	395, 219	onlesd 28270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ (𝜑 → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
1221	1203, 1220	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘(𝑁 +s 1s )) ⊆ ( bday ‘𝑁)))
1222	1219, 1221	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 1s <s (𝑛 -s 𝑚))) → (𝑁 +s 1s ) ≤s 𝑁)
1223	1222	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ( 1s <s (𝑛 -s 𝑚) → (𝑁 +s 1s ) ≤s 𝑁))
1224	1162, 1223	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (¬ (𝑛 -s 𝑚) ≤s 1s → (𝑁 +s 1s ) ≤s 𝑁))
1225	1157, 1224	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑚) ≤s 1s )
1226	1208	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑛)
1227		npcans 28075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ⊢ ((𝑛 ∈ No ∧ 1s ∈ No ) → ((𝑛 -s 1s ) +s 1s ) = 𝑛)
1228	1160, 322, 1227	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) +s 1s ) = 𝑛)
1229	1228	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑚 +s 1s ) ≤s ((𝑛 -s 1s ) +s 1s ) ↔ (𝑚 +s 1s ) ≤s 𝑛))
1230	1160, 1158	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1s ) ∈ No )
1231	1142, 1230, 1158	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑚 ≤s (𝑛 -s 1s ) ↔ (𝑚 +s 1s ) ≤s ((𝑛 -s 1s ) +s 1s )))
1232	1126, 1159, 1210	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 = 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))
1233	1229, 1231, 1232	3bitr4rd 312	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑚 <s 𝑛 ↔ 𝑚 ≤s (𝑛 -s 1s )))
1234	1142, 1160, 1158	lesubsd 28096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑚 ≤s (𝑛 -s 1s ) ↔ 1s ≤s (𝑛 -s 𝑚)))
1235	1233, 1234	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑛 ↔ 1s ≤s (𝑛 -s 𝑚)))
1236	1226, 1235	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 1s ≤s (𝑛 -s 𝑚))
1237	1161, 1158	lestri3d 27731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑚) = 1s ↔ ((𝑛 -s 𝑚) ≤s 1s ∧ 1s ≤s (𝑛 -s 𝑚))))
1238	1225, 1236, 1237	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑚) = 1s )
1239	1160, 1142, 1158	subaddsd 28071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 -s 𝑚) = 1s ↔ (𝑚 +s 1s ) = 𝑛))
1240	1238, 1239	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 = 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)
1241	1240	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑛 = (𝑚 +s 1s ))
1242	1241	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑛 /su (2s↑s𝑜)) = ((𝑚 +s 1s ) /su (2s↑s𝑜)))
1243	1242	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (𝑛 /su (2s↑s𝑜))) = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1244	1154, 1243	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑑 = (𝑔 +s ((𝑚 +s 1s ) /su (2s↑s𝑜))))
1245	1141, 1142, 1158	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) +s 1s ) = (((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )))
1246	1245	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) +s 1s ) /su (2s↑s𝑜)) = ((((2s↑s𝑜) ·s 𝑔) +s (𝑚 +s 1s )) /su (2s↑s𝑜)))
1247	1152, 1244, 1246	3eqtr4d 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑑 = (((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜)))
1248	1247	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → {𝑑} = {(((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜))})
1249	1147, 1248	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ({𝑐} |s {𝑑}) = ({((((2s↑s𝑜) ·s 𝑔) +s 𝑚) /su (2s↑s𝑜))} |s {(((((2s↑s𝑜) ·s 𝑔) +s 𝑚) +s 1s ) /su (2s↑s𝑜))}))
1250		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ((((2s↑s𝑜) ·s 𝑔) ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s) → (((2s↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℕ0s)
1251	1140, 1126, 1250	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℕ0s)
1252	1251	n0zsd 28390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑s𝑜) ·s 𝑔) +s 𝑚) ∈ ℤs)
1253	1252, 1130	pw2cutp1 28461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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↑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 ))))
1254	1133, 1249, 1253	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑤 = (((2s ·s (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))
1255	52	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)))) → 2s ∈ No )
1256	1255, 1141, 1142	addsdid 28156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) = ((2s ·s ((2s↑s𝑜) ·s 𝑔)) +s (2s ·s 𝑚)))
1257		expsp1 28429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ⊢ ((2s ∈ No ∧ 𝑜 ∈ ℕ0s) → (2s↑s(𝑜 +s 1s )) = ((2s↑s𝑜) ·s 2s))
1258	52, 1130, 1257	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 )) = ((2s↑s𝑜) ·s 2s))
1259	1206, 1255	mulscomd 28140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2s) = (2s ·s (2s↑s𝑜)))
1260	1258, 1259	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = (2s ·s (2s↑s𝑜)))
1261	1260	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = ((2s ·s (2s↑s𝑜)) ·s 𝑔))
1262	1255, 1206, 1134	mulsassd 28167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = (2s ·s ((2s↑s𝑜) ·s 𝑔)))
1263	1261, 1262	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = (2s ·s ((2s↑s𝑜) ·s 𝑔)))
1264	1263	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)) = ((2s ·s ((2s↑s𝑜) ·s 𝑔)) +s (2s ·s 𝑚)))
1265	1256, 1264	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)))
1266	1265	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s (2s ·s 𝑚)) +s 1s ))
1267		n0expscl 28432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ⊢ ((2s ∈ ℕ0s ∧ (𝑜 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑜 +s 1s )) ∈ ℕ0s)
1268	412, 1131, 1267	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)))) ∧ ((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)
1269		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (((2s↑s(𝑜 +s 1s )) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s) → ((2s↑s(𝑜 +s 1s )) ·s 𝑔) ∈ ℕ0s)
1270	1268, 1125, 1269	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)))) ∧ ((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)
1271	1270	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1272	1128	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1273	1271, 1272, 1158	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )))
1274	1266, 1273	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (((2s↑s𝑜) ·s 𝑔) +s 𝑚)) +s 1s ) = (((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )))
1275	1274	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (((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 ))))
1276	1254, 1275	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑤 = ((((2s↑s(𝑜 +s 1s )) ·s 𝑔) +s ((2s ·s 𝑚) +s 1s )) /su (2s↑s(𝑜 +s 1s ))))
1277	1129	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((2s ·s 𝑚) +s 1s ) ∈ No )
1278	1271, 1277, 1131	pw2divsdird 28448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((((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 )))))
1279	1134, 1131	pw2divscan3d 28441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (((2s↑s(𝑜 +s 1s )) ·s 𝑔) /su (2s↑s(𝑜 +s 1s ))) = 𝑔)
1280	1279	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((((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 )))))
1281	1276, 1278, 1280	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s )))))
1282		n0mulscl 28345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((2s ∈ ℕ0s ∧ (𝑚 +s 1s ) ∈ ℕ0s) → (2s ·s (𝑚 +s 1s )) ∈ ℕ0s)
1283	412, 1148, 1282	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s ·s (𝑚 +s 1s )) ∈ ℕ0s)
1284	1283	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s ·s (𝑚 +s 1s )) ∈ No )
1285	1268	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s↑s(𝑜 +s 1s )) ∈ No )
1286	1158, 1255, 1272	ltadds2d 27997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ( 1s <s 2s ↔ ((2s ·s 𝑚) +s 1s ) <s ((2s ·s 𝑚) +s 2s)))
1287	523, 1286	mpbii 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((2s ·s 𝑚) +s 1s ) <s ((2s ·s 𝑚) +s 2s))
1288	1255, 1142, 1158	addsdid 28156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )) = ((2s ·s 𝑚) +s (2s ·s 1s )))
1289	877	oveq2i 7371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((2s ·s 𝑚) +s (2s ·s 1s )) = ((2s ·s 𝑚) +s 2s)
1290	1288, 1289	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s ·s (𝑚 +s 1s )) = ((2s ·s 𝑚) +s 2s))
1291	1287, 1290	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((2s ·s 𝑚) +s 1s ) <s (2s ·s (𝑚 +s 1s )))
1292		simprl2 1221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((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))
1293	1292	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑚 <s (2s↑s𝑜))
1294		n0ltsp1le 28365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑚 ∈ ℕ0s ∧ (2s↑s𝑜) ∈ ℕ0s) → (𝑚 <s (2s↑s𝑜) ↔ (𝑚 +s 1s ) ≤s (2s↑s𝑜)))
1295	1126, 1138, 1294	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (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)))
1296	1293, 1295	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (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))
1297	975	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 0s <s 2s)
1298	1149, 1206, 1255, 1297	lemuls2d 28174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑚 +s 1s ) ≤s (2s↑s𝑜) ↔ (2s ·s (𝑚 +s 1s )) ≤s (2s ·s (2s↑s𝑜))))
1299	1296, 1298	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 = 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)))
1300	1299, 1260	breqtrrd 5127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (2s ·s (𝑚 +s 1s )) ≤s (2s↑s(𝑜 +s 1s )))
1301	1277, 1284, 1285, 1291, 1300	ltlestrd 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 = 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 )))
1302	1130	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑜 ∈ No )
1303	1134, 1302, 1158	addsassd 28006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑔 +s 𝑜) +s 1s ) = (𝑔 +s (𝑜 +s 1s )))
1304	1216	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑔 +s 𝑜) <s 𝑁)
1305		n0addscl 28344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((𝑔 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s) → (𝑔 +s 𝑜) ∈ ℕ0s)
1306	1125, 1130, 1305	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (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)
1307	1306	n0nod 28325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∈ No )
1308	1156, 32	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)))) → 𝑁 ∈ No )
1309	1307, 1308, 1158	ltadds1d 27998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑔 +s 𝑜) <s 𝑁 ↔ ((𝑔 +s 𝑜) +s 1s ) <s (𝑁 +s 1s )))
1310	1304, 1309	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → ((𝑔 +s 𝑜) +s 1s ) <s (𝑁 +s 1s ))
1311	1303, 1310	eqbrtrrd 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s ))
1312		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑏 /su (2s↑s𝑞)) = (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)))
1313	1312	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑔 +s (𝑏 /su (2s↑s𝑞))) = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞))))
1314	1313	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑤 = (𝑔 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑔 +s (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)))))
1315		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑏 = ((2s ·s 𝑚) +s 1s ) → (𝑏 <s (2s↑s𝑞) ↔ ((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞)))
1316	1314, 1315	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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 ))))
1317		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑞 = (𝑜 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑜 +s 1s )))
1318	1317	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑞 = (𝑜 +s 1s ) → (((2s ·s 𝑚) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑚) +s 1s ) /su (2s↑s(𝑜 +s 1s ))))
1319	1318	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (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 )))))
1320	1319	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (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 ))))))
1321	1317	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑞 = (𝑜 +s 1s ) → (((2s ·s 𝑚) +s 1s ) <s (2s↑s𝑞) ↔ ((2s ·s 𝑚) +s 1s ) <s (2s↑s(𝑜 +s 1s ))))
1322		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑞 = (𝑜 +s 1s ) → (𝑔 +s 𝑞) = (𝑔 +s (𝑜 +s 1s )))
1323	1322	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (𝑞 = (𝑜 +s 1s ) → ((𝑔 +s 𝑞) <s (𝑁 +s 1s ) ↔ (𝑔 +s (𝑜 +s 1s )) <s (𝑁 +s 1s )))
1324	1320, 1321, 1323	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (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 ))))
1325	535, 1316, 1324	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (((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 )))
1326	1125, 1129, 1131, 1281, 1301, 1311, 1325	syl33anc 1388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) ∧ ((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 )))
1327	1326	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)))) ∧ (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 ))))
1328	1327	rexlimdvvva 3195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 ))))
1329	1123, 1328	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ (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 )))
1330	1329	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 = 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 ))))
1331	935, 1330	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 = 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 )))
1332	1331	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (𝑑 = (𝑗 +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 ))))
1333	918, 1332	jaod 860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((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 ))))
1334	627, 1333	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ∧ 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 ))))
1335	625, 1334	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (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 )))
1336	1335	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s) ∧ (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 ))))
1337	600, 1336	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))) ∧ ((𝑗 ∈ ℕ0s ∧ 𝑘 ∈ 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 )))
1338	579, 1337	mpdan 688	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
1339	1338	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ (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 ))))
1340	1339	rexlimdvvva 3195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (∃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 ))))
1341	229	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ( bday ‘𝑑) ⊆ ( bday ‘𝑁))
1342	57	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s ∈ No )
1343	135	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ No )
1344		simp1r 1200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑤)
1345	1343, 1344	0elleft 27911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ( L ‘𝑤))
1346	240, 1345, 189	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s <s 𝑑)
1347	1342, 384, 1346	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → 0s ≤s 𝑑)
1348	1347	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)) ∧ 𝑤 = ({𝑐} |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)
1349		fveq2 6835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑧 = 𝑑 → ( bday ‘𝑧) = ( bday ‘𝑑))
1350	1349	sseq1d 3966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑑 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑑) ⊆ ( bday ‘𝑁)))
1351		breq2 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑑 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑑))
1352	1350, 1351	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑧 = 𝑑 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑑) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑑)))
1353		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑧 = 𝑑 → (𝑧 = 𝑁 ↔ 𝑑 = 𝑁))
1354		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑧 = 𝑑 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
1355	1354	3anbi1d 1443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (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)))
1356	1355	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (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)))
1357	1356	2rexbidv 3202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (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)))
1358		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑥 = 𝑗 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑗 +s (𝑦 /su (2s↑s𝑝))))
1359	1358	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑥 = 𝑗 → (𝑑 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝)))))
1360		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑥 = 𝑗 → (𝑥 +s 𝑝) = (𝑗 +s 𝑝))
1361	1360	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑥 = 𝑗 → ((𝑥 +s 𝑝) <s 𝑁 ↔ (𝑗 +s 𝑝) <s 𝑁))
1362	1359, 1361	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (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)))
1363	1362	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (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)))
1364		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = 𝑘 → (𝑦 /su (2s↑s𝑝)) = (𝑘 /su (2s↑s𝑝)))
1365	1364	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑦 = 𝑘 → (𝑗 +s (𝑦 /su (2s↑s𝑝))) = (𝑗 +s (𝑘 /su (2s↑s𝑝))))
1366	1365	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑦 = 𝑘 → (𝑑 = (𝑗 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝)))))
1367		breq1 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑦 = 𝑘 → (𝑦 <s (2s↑s𝑝) ↔ 𝑘 <s (2s↑s𝑝)))
1368	1366, 1367	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (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)))
1369	1368	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (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)))
1370		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑝 = 𝑙 → (2s↑s𝑝) = (2s↑s𝑙))
1371	1370	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑝 = 𝑙 → (𝑘 /su (2s↑s𝑝)) = (𝑘 /su (2s↑s𝑙)))
1372	1371	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑝 = 𝑙 → (𝑗 +s (𝑘 /su (2s↑s𝑝))) = (𝑗 +s (𝑘 /su (2s↑s𝑙))))
1373	1372	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑝 = 𝑙 → (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑝))) ↔ 𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙)))))
1374	1370	breq2d 5111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑝 = 𝑙 → (𝑘 <s (2s↑s𝑝) ↔ 𝑘 <s (2s↑s𝑙)))
1375		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑝 = 𝑙 → (𝑗 +s 𝑝) = (𝑗 +s 𝑙))
1376	1375	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝑝 = 𝑙 → ((𝑗 +s 𝑝) <s 𝑁 ↔ (𝑗 +s 𝑙) <s 𝑁))
1377	1373, 1374, 1376	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (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)))
1378	1377	cbvrexvw 3216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (∃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))
1379	1369, 1378	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (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)))
1380	1363, 1379	cbvrex2vw 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∃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))
1381	1357, 1380	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (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)))
1382	1353, 1381	orbi12d 919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (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))))
1383	1352, 1382	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (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)))))
1384	296, 14	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ∀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))))
1385	1383, 1384, 385	rspcdva 3578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((( 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))))
1386	1341, 1348, 1385	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑑 = 𝑁 ∨ ∃𝑗 ∈ ℕ0s ∃𝑘 ∈ ℕ0s ∃𝑙 ∈ ℕ0s (𝑑 = (𝑗 +s (𝑘 /su (2s↑s𝑙))) ∧ 𝑘 <s (2s↑s𝑙) ∧ (𝑗 +s 𝑙) <s 𝑁)))
1387	561, 1340, 1386	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
1388	1387	3expa 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
1389	1388	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) ∧ (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 ))))
1390	1389	rexlimdvvva 3195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → (∃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 ))))
1391	287, 1390	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) → ((( 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 ))))
1392	210, 215, 1391	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) ∧ 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 )))
1393	191, 1392	mpdan 688	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
1394	1393	3expa 1119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
1395	1394	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)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 ))))
1396	186, 1395	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))))
1397	1396	rexlimdva 3138	. . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (∃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 ))))
1398	180, 1397	syl5 34	. . . . . . . . . . . . . . . . . . . . . . . . 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)) → ((( 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 ))))
1399	154, 175, 1398	mp2and 700	. . . . . . . . . . . . . . . . . . . . . . . 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 )))
1400	1399	expr 456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (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 ))))
1401	150, 1400	sylbird 260	. . . . . . . . . . . . . . . . . . . . . 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 ))))
1402	1401	rexlimdva 3138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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 ))))
1403	144, 1402	syl5 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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 ))))
1404	134, 138, 1403	mp2and 700	. . . . . . . . . . . . . . . . . . 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 )))
1405	1404	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 ))))
1406		addslid 27968	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
1407	57, 1406	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( 0s +s 0s ) = 0s
1408	1407	eqcomi 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0s = ( 0s +s 0s )
1409		n0p1nns 28371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0s → (𝑁 +s 1s ) ∈ ℕs)
1410	1, 1409	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑁 +s 1s ) ∈ ℕs)
1411		nnsgt0 28339	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 +s 1s ) ∈ ℕs → 0s <s (𝑁 +s 1s ))
1412	1410, 1411	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0s <s (𝑁 +s 1s ))
1413	29, 29, 29	3pm3.2i 1341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s)
1414		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 0s → (𝑎 +s (𝑏 /su (2s↑s𝑞))) = ( 0s +s (𝑏 /su (2s↑s𝑞))))
1415	1414	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 0s → ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 0s = ( 0s +s (𝑏 /su (2s↑s𝑞)))))
1416		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 0s → (𝑎 +s 𝑞) = ( 0s +s 𝑞))
1417	1416	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 0s → ((𝑎 +s 𝑞) <s (𝑁 +s 1s ) ↔ ( 0s +s 𝑞) <s (𝑁 +s 1s )))
1418	1415, 1417	3anbi13d 1441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 ))))
1419	46	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 0s → ( 0s +s (𝑏 /su (2s↑s𝑞))) = ( 0s +s ( 0s /su (2s↑s𝑞))))
1420	1419	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 0s → ( 0s = ( 0s +s (𝑏 /su (2s↑s𝑞))) ↔ 0s = ( 0s +s ( 0s /su (2s↑s𝑞)))))
1421	1420, 49	3anbi12d 1440	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 ))))
1422	60	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑞 = 0s → ( 0s +s ( 0s /su (2s↑s𝑞))) = ( 0s +s 0s ))
1423	1422	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑞 = 0s → ( 0s = ( 0s +s ( 0s /su (2s↑s𝑞))) ↔ 0s = ( 0s +s 0s )))
1424		oveq2 7368	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑞 = 0s → ( 0s +s 𝑞) = ( 0s +s 0s ))
1425	1424, 1407	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑞 = 0s → ( 0s +s 𝑞) = 0s )
1426	1425	breq1d 5109	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑞 = 0s → (( 0s +s 𝑞) <s (𝑁 +s 1s ) ↔ 0s <s (𝑁 +s 1s )))
1427	1423, 63, 1426	3anbi123d 1439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 ))))
1428	1418, 1421, 1427	rspc3ev 3594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( 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 )))
1429	1413, 1428	mpan 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( 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 )))
1430	1408, 36, 1412, 1429	mp3an12i 1468	. . . . . . . . . . . . . . . . . . . 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 )))
1431		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( 0s = 𝑤 → ( 0s = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞)))))
1432	1431	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 ))))
1433	1432	rexbidv 3161	. . . . . . . . . . . . . . . . . . . . 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 ))))
1434	1433	2rexbidv 3202	. . . . . . . . . . . . . . . . . . . 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 ))))
1435	1430, 1434	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 ))))
1436	1435	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 ))))
1437	1405, 1436	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 ))))
1438	121, 1437	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 ))))
1439	1438	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 )))))
1440	1439	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 ))))))
1441	1440	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 ))))))
1442	1441	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 )))))
1443	1442	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 ))))
1444	118, 1443	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 ))))
1445	1444	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 ))))
1446	1445	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 )))))
1447	1446	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 ))))))
1448	117, 1447	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 ))))))
1449	116, 1448	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 ))))))
1450	13, 1449	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 ))))))
1451	5, 1450	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 ))))))
1452	1451	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 )))))
1453	1452	ralrimiva 3129	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 2933  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902  ∅c0 4286  {csn 4581   class class class wbr 5099   Or wor 5532  Oncon0 6318  suc csuc 6320  ‘cfv 6493  (class class class)co 7360  ωcom 7810  Fincfn 8887   No csur 27611   <s clts 27612   bday cbday 27613   ≤s cles 27716   <<s cslts 27757   |s ccuts 27759   0_s c0s 27805   1_s c1s 27806   M cmade 27822   O cold 27823   L cleft 27825   R cright 27826   +_s cadds 27959   -_s csubs 28020   ·_s cmuls 28106   /_su cdivs 28187  On_scons 28251  ℕ_0scn0s 28312  ℕ_scnns 28313  2_sc2s 28410  ↑_scexps 28412

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-dc 10360  ax-ac2 10377

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9855  df-acn 9858  df-ac 10030  df-no 27614  df-lts 27615  df-bday 27616  df-les 27717  df-slts 27758  df-cuts 27760  df-0s 27807  df-1s 27808  df-made 27827  df-old 27828  df-left 27830  df-right 27831  df-norec 27938  df-norec2 27949  df-adds 27960  df-negs 28021  df-subs 28022  df-muls 28107  df-divs 28188  df-ons 28252  df-seqs 28284  df-n0s 28314  df-nns 28315  df-zs 28379  df-2s 28411  df-exps 28413

This theorem is referenced by:  bdayfinbndlem2  28468