Theorem bdaypw2n0bndlem 28879

Description: Lemma for bdaypw2n0bnd 28880. Prove the case with a successor. (Contributed by Scott Fenton, 21-Feb-2026.)

Assertion

Ref	Expression
bdaypw2n0bndlem	⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s(𝑁 +s 1s ))) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))

Proof of Theorem bdaypw2n0bndlem

Dummy variables 𝑎 𝑛 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7450	. . . . . . . . . . 11 ⊢ (𝑚 = 0s → (𝑚 +s 1s ) = ( 0s +s 1s ))
2		1no 28226	. . . . . . . . . . . 12 ⊢ 1s ∈ No
3		addslid 28384	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 0s +s 1s ) = 1s
5	1, 4	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑚 = 0s → (𝑚 +s 1s ) = 1s )
6	5	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = 0s → (2s↑s(𝑚 +s 1s )) = (2s↑s 1s ))
7		2no 28835	. . . . . . . . . 10 ⊢ 2s ∈ No
8		exps1 28844	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ (2s↑s 1s ) = 2s
10	6, 9	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑚 = 0s → (2s↑s(𝑚 +s 1s )) = 2s)
11	10	breq2d 5118	. . . . . . 7 ⊢ (𝑚 = 0s → (𝑎 <s (2s↑s(𝑚 +s 1s )) ↔ 𝑎 <s 2s))
12	10	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝑎 /su (2s↑s(𝑚 +s 1s ))) = (𝑎 /su 2s))
13	12	fveq2d 6912	. . . . . . . 8 ⊢ (𝑚 = 0s → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) = (bday‘(𝑎 /su 2s)))
14	5	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑚 = 0s → (bday‘(𝑚 +s 1s )) = (bday‘ 1s ))
15		bday1 28230	. . . . . . . . . . 11 ⊢ (bday‘ 1s ) = 1o
16	14, 15	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑚 = 0s → (bday‘(𝑚 +s 1s )) = 1o)
17	16	suceqd 6448	. . . . . . . . 9 ⊢ (𝑚 = 0s → suc (bday‘(𝑚 +s 1s )) = suc 1o)
18		df-2o 8505	. . . . . . . . 9 ⊢ 2o = suc 1o
19	17, 18	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑚 = 0s → suc (bday‘(𝑚 +s 1s )) = 2o)
20	13, 19	sseq12d 3946	. . . . . . 7 ⊢ (𝑚 = 0s → ((bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s )) ↔ (bday‘(𝑎 /su 2s)) ⊆ 2o))
21	11, 20	imbi12d 347	. . . . . 6 ⊢ (𝑚 = 0s → ((𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ (𝑎 <s 2s → (bday‘(𝑎 /su 2s)) ⊆ 2o)))
22	21	ralbidv 3189	. . . . 5 ⊢ (𝑚 = 0s → (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ ∀𝑎 ∈ ℕ0s (𝑎 <s 2s → (bday‘(𝑎 /su 2s)) ⊆ 2o)))
23		oveq1 7450	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 +s 1s ) = (𝑛 +s 1s ))
24	23	oveq2d 7459	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (2s↑s(𝑚 +s 1s )) = (2s↑s(𝑛 +s 1s )))
25	24	breq2d 5118	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑎 <s (2s↑s(𝑚 +s 1s )) ↔ 𝑎 <s (2s↑s(𝑛 +s 1s ))))
26	24	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑎 /su (2s↑s(𝑚 +s 1s ))) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	fveq2d 6912	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) = (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))))
28	23	fveq2d 6912	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (bday‘(𝑚 +s 1s )) = (bday‘(𝑛 +s 1s )))
29	28	suceqd 6448	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → suc (bday‘(𝑚 +s 1s )) = suc (bday‘(𝑛 +s 1s )))
30	27, 29	sseq12d 3946	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s )) ↔ (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
31	25, 30	imbi12d 347	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
32	31	ralbidv 3189	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
33		oveq1 7450	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 +s 1s ) = ((𝑛 +s 1s ) +s 1s ))
34	33	oveq2d 7459	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s(𝑚 +s 1s )) = (2s↑s((𝑛 +s 1s ) +s 1s )))
35	34	breq2d 5118	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 <s (2s↑s(𝑚 +s 1s )) ↔ 𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
36	34	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s(𝑚 +s 1s ))) = (𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
37	36	fveq2d 6912	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) = (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))))
38	33	fveq2d 6912	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (bday‘(𝑚 +s 1s )) = (bday‘((𝑛 +s 1s ) +s 1s )))
39	38	suceqd 6448	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → suc (bday‘(𝑚 +s 1s )) = suc (bday‘((𝑛 +s 1s ) +s 1s )))
40	37, 39	sseq12d 3946	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s )) ↔ (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
41	35, 40	imbi12d 347	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
42	41	ralbidv 3189	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
43		oveq1 7450	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚 +s 1s ) = (𝑁 +s 1s ))
44	43	oveq2d 7459	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (2s↑s(𝑚 +s 1s )) = (2s↑s(𝑁 +s 1s )))
45	44	breq2d 5118	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (𝑎 <s (2s↑s(𝑚 +s 1s )) ↔ 𝑎 <s (2s↑s(𝑁 +s 1s ))))
46	44	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑎 /su (2s↑s(𝑚 +s 1s ))) = (𝑎 /su (2s↑s(𝑁 +s 1s ))))
47	46	fveq2d 6912	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) = (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))))
48	43	fveq2d 6912	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (bday‘(𝑚 +s 1s )) = (bday‘(𝑁 +s 1s )))
49	48	suceqd 6448	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → suc (bday‘(𝑚 +s 1s )) = suc (bday‘(𝑁 +s 1s )))
50	47, 49	sseq12d 3946	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s )) ↔ (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s ))))
51	45, 50	imbi12d 347	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ (𝑎 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
52	51	ralbidv 3189	. . . . 5 ⊢ (𝑚 = 𝑁 → (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑚 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑚 +s 1s )))) ⊆ suc (bday‘(𝑚 +s 1s ))) ↔ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
53		1n0s 28764	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
54		n0lesltp1 28782	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℕ0s ∧ 1s ∈ ℕ0s) → (𝑎 ≤s 1s ↔ 𝑎 <s ( 1s +s 1s )))
55	53, 54	mpan2 704	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 1s ↔ 𝑎 <s ( 1s +s 1s )))
56		1p1e2s 28832	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
57	56	breq2i 5114	. . . . . . . . 9 ⊢ (𝑎 <s ( 1s +s 1s ) ↔ 𝑎 <s 2s)
58	55, 57	bitrdi 290	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 1s ↔ 𝑎 <s 2s))
59		n0no 28739	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ0s → 𝑎 ∈ No)
60		lesloe 28141	. . . . . . . . . 10 ⊢ ((𝑎 ∈ No ∧ 1s ∈ No) → (𝑎 ≤s 1s ↔ (𝑎 <s 1s ∨ 𝑎 = 1s )))
61	59, 2, 60	sylancl 598	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 1s ↔ (𝑎 <s 1s ∨ 𝑎 = 1s )))
62		0no 28225	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
63		lestri3 28142	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ No ∧ 0s ∈ No) → (𝑎 = 0s ↔ (𝑎 ≤s 0s ∧ 0s ≤s 𝑎)))
64	59, 62, 63	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ0s → (𝑎 = 0s ↔ (𝑎 ≤s 0s ∧ 0s ≤s 𝑎)))
65		n0sge0 28754	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℕ0s → 0s ≤s 𝑎)
66	65	biantrud 541	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 0s ↔ (𝑎 ≤s 0s ∧ 0s ≤s 𝑎)))
67	64, 66	bitr4d 285	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ0s → (𝑎 = 0s ↔ 𝑎 ≤s 0s ))
68		0n0s 28745	. . . . . . . . . . . . 13 ⊢ 0s ∈ ℕ0s
69		n0lesltp1 28782	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℕ0s ∧ 0s ∈ ℕ0s) → (𝑎 ≤s 0s ↔ 𝑎 <s ( 0s +s 1s )))
70	68, 69	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 0s ↔ 𝑎 <s ( 0s +s 1s )))
71	67, 70	bitrd 282	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ0s → (𝑎 = 0s ↔ 𝑎 <s ( 0s +s 1s )))
72	4	breq2i 5114	. . . . . . . . . . 11 ⊢ (𝑎 <s ( 0s +s 1s ) ↔ 𝑎 <s 1s )
73	71, 72	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ0s → (𝑎 = 0s ↔ 𝑎 <s 1s ))
74	73	orbi1d 930	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0s → ((𝑎 = 0s ∨ 𝑎 = 1s ) ↔ (𝑎 <s 1s ∨ 𝑎 = 1s )))
75	61, 74	bitr4d 285	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ0s → (𝑎 ≤s 1s ↔ (𝑎 = 0s ∨ 𝑎 = 1s )))
76	58, 75	bitr3d 284	. . . . . . 7 ⊢ (𝑎 ∈ ℕ0s → (𝑎 <s 2s ↔ (𝑎 = 0s ∨ 𝑎 = 1s )))
77		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑎 = 0s → (𝑎 /su 2s) = ( 0s /su 2s))
78	9	oveq2i 7454	. . . . . . . . . . . . 13 ⊢ ( 0s /su (2s↑s 1s )) = ( 0s /su 2s)
79	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (⊤ → 1s ∈ ℕ0s)
80	79	pw2divs0d 28871	. . . . . . . . . . . . . 14 ⊢ (⊤ → ( 0s /su (2s↑s 1s )) = 0s )
81	80	mptru 1577	. . . . . . . . . . . . 13 ⊢ ( 0s /su (2s↑s 1s )) = 0s
82	78, 81	eqtr3i 2788	. . . . . . . . . . . 12 ⊢ ( 0s /su 2s) = 0s
83	77, 82	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑎 = 0s → (𝑎 /su 2s) = 0s )
84	83	fveq2d 6912	. . . . . . . . . 10 ⊢ (𝑎 = 0s → (bday‘(𝑎 /su 2s)) = (bday‘ 0s ))
85		bday0 28227	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
86	84, 85	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑎 = 0s → (bday‘(𝑎 /su 2s)) = ∅)
87		0ss 4353	. . . . . . . . 9 ⊢ ∅ ⊆ 2o
88	86, 87	eqsstrdi 3957	. . . . . . . 8 ⊢ (𝑎 = 0s → (bday‘(𝑎 /su 2s)) ⊆ 2o)
89		oveq1 7450	. . . . . . . . . . 11 ⊢ (𝑎 = 1s → (𝑎 /su 2s) = ( 1s /su 2s))
90		nohalf 28840	. . . . . . . . . . 11 ⊢ ( 1s /su 2s) = ({ 0s } |s { 1s })
91	89, 90	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑎 = 1s → (𝑎 /su 2s) = ({ 0s } |s { 1s }))
92	91	fveq2d 6912	. . . . . . . . 9 ⊢ (𝑎 = 1s → (bday‘(𝑎 /su 2s)) = (bday‘({ 0s } |s { 1s })))
93	62	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 0s ∈ No)
94	2	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 1s ∈ No)
95		0lt1s 28228	. . . . . . . . . . . . 13 ⊢ 0s <s 1s
96	95	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 0s <s 1s )
97	93, 94, 96	sltssn 28186	. . . . . . . . . . 11 ⊢ (⊤ → { 0s } <<s { 1s })
98	97	mptru 1577	. . . . . . . . . 10 ⊢ { 0s } <<s { 1s }
99		2on 8518	. . . . . . . . . 10 ⊢ 2o ∈ On
100		df-pr 4591	. . . . . . . . . . . 12 ⊢ {∅, 1o} = ({∅} ∪ {1o})
101		df2o3 8512	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
102		imaundi 6153	. . . . . . . . . . . . 13 ⊢ (bday “ ({ 0s } ∪ { 1s })) = ((bday “ { 0s }) ∪ (bday “ { 1s }))
103		bdayfn 28164	. . . . . . . . . . . . . . . 16 ⊢ bday Fn No
104		fnsnfv 6987	. . . . . . . . . . . . . . . 16 ⊢ ((bday Fn No ∧ 0s ∈ No) → {(bday‘ 0s )} = (bday “ { 0s }))
105	103, 62, 104	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ {(bday‘ 0s )} = (bday “ { 0s })
106	85	sneqi 4595	. . . . . . . . . . . . . . 15 ⊢ {(bday‘ 0s )} = {∅}
107	105, 106	eqtr3i 2788	. . . . . . . . . . . . . 14 ⊢ (bday “ { 0s }) = {∅}
108		fnsnfv 6987	. . . . . . . . . . . . . . . 16 ⊢ ((bday Fn No ∧ 1s ∈ No) → {(bday‘ 1s )} = (bday “ { 1s }))
109	103, 2, 108	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ {(bday‘ 1s )} = (bday “ { 1s })
110	15	sneqi 4595	. . . . . . . . . . . . . . 15 ⊢ {(bday‘ 1s )} = {1o}
111	109, 110	eqtr3i 2788	. . . . . . . . . . . . . 14 ⊢ (bday “ { 1s }) = {1o}
112	107, 111	uneq12i 4168	. . . . . . . . . . . . 13 ⊢ ((bday “ { 0s }) ∪ (bday “ { 1s })) = ({∅} ∪ {1o})
113	102, 112	eqtri 2786	. . . . . . . . . . . 12 ⊢ (bday “ ({ 0s } ∪ { 1s })) = ({∅} ∪ {1o})
114	100, 101, 113	3eqtr4ri 2797	. . . . . . . . . . 11 ⊢ (bday “ ({ 0s } ∪ { 1s })) = 2o
115		ssid 3935	. . . . . . . . . . 11 ⊢ 2o ⊆ 2o
116	114, 115	eqsstri 3959	. . . . . . . . . 10 ⊢ (bday “ ({ 0s } ∪ { 1s })) ⊆ 2o
117		cutbdaybnd 28211	. . . . . . . . . 10 ⊢ (({ 0s } <<s { 1s } ∧ 2o ∈ On ∧ (bday “ ({ 0s } ∪ { 1s })) ⊆ 2o) → (bday‘({ 0s } |s { 1s })) ⊆ 2o)
118	98, 99, 116, 117	mp3an 1491	. . . . . . . . 9 ⊢ (bday‘({ 0s } |s { 1s })) ⊆ 2o
119	92, 118	eqsstrdi 3957	. . . . . . . 8 ⊢ (𝑎 = 1s → (bday‘(𝑎 /su 2s)) ⊆ 2o)
120	88, 119	jaoi 871	. . . . . . 7 ⊢ ((𝑎 = 0s ∨ 𝑎 = 1s ) → (bday‘(𝑎 /su 2s)) ⊆ 2o)
121	76, 120	biimtrdi 256	. . . . . 6 ⊢ (𝑎 ∈ ℕ0s → (𝑎 <s 2s → (bday‘(𝑎 /su 2s)) ⊆ 2o))
122	121	rgen 3082	. . . . 5 ⊢ ∀𝑎 ∈ ℕ0s (𝑎 <s 2s → (bday‘(𝑎 /su 2s)) ⊆ 2o)
123		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑎 𝑛 ∈ ℕ0s
124		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑎∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))
125	123, 124	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑎(𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
126		n0seo 28837	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑎 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑎 = ((2s ·s 𝑥) +s 1s )))
127		breq1 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → (𝑎 <s (2s↑s(𝑛 +s 1s )) ↔ 𝑥 <s (2s↑s(𝑛 +s 1s ))))
128		fvoveq1 7466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑥 → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) = (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))))
129	128	sseq1d 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → ((bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
130	127, 129	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑥 → ((𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) ↔ (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
131	130	rspccv 3577	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → (𝑥 ∈ ℕ0s → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
132	131	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (𝑥 ∈ ℕ0s → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
133	132	imp32 424	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))
134		sssucid 6463	. . . . . . . . . . . . . . 15 ⊢ suc (bday‘(𝑛 +s 1s )) ⊆ suc suc (bday‘(𝑛 +s 1s ))
135	133, 134	sstrdi 3925	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc suc (bday‘(𝑛 +s 1s )))
136		simpll 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 𝑛 ∈ ℕ0s)
137		peano2n0s 28746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
138	136, 137	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑛 +s 1s ) ∈ ℕ0s)
139	138	adantrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → (𝑛 +s 1s ) ∈ ℕ0s)
140		bdayn0p1 28785	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → (bday‘((𝑛 +s 1s ) +s 1s )) = suc (bday‘(𝑛 +s 1s )))
141	139, 140	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → (bday‘((𝑛 +s 1s ) +s 1s )) = suc (bday‘(𝑛 +s 1s )))
142	141	suceqd 6448	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → suc (bday‘((𝑛 +s 1s ) +s 1s )) = suc suc (bday‘(𝑛 +s 1s )))
143	135, 142	sseqtrrd 3950	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ 𝑥 <s (2s↑s(𝑛 +s 1s )))) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))
144	143	expr 462	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
145		expsp1 28845	. . . . . . . . . . . . . . . 16 ⊢ ((2s ∈ No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
146	7, 138, 145	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
147		expscl 28847	. . . . . . . . . . . . . . . . 17 ⊢ ((2s ∈ No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ∈ No)
148	7, 138, 147	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ∈ No)
149	7	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 2s ∈ No)
150	148, 149	mulscomd 28556	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s↑s(𝑛 +s 1s )) ·s 2s) = (2s ·s (2s↑s(𝑛 +s 1s ))))
151	146, 150	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) = (2s ·s (2s↑s(𝑛 +s 1s ))))
152	151	breq2d 5118	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ (2s ·s 𝑥) <s (2s ·s (2s↑s(𝑛 +s 1s )))))
153		n0no 28739	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No)
154	153	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 𝑥 ∈ No)
155		2nns 28834	. . . . . . . . . . . . . . . 16 ⊢ 2s ∈ ℕs
156		nnsgt0 28755	. . . . . . . . . . . . . . . 16 ⊢ (2s ∈ ℕs → 0s <s 2s)
157	155, 156	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0s <s 2s
158	157	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 0s <s 2s)
159	154, 148, 149, 158	ltmuls2d 28588	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 <s (2s↑s(𝑛 +s 1s )) ↔ (2s ·s 𝑥) <s (2s ·s (2s↑s(𝑛 +s 1s )))))
160	152, 159	bitr4d 285	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ 𝑥 <s (2s↑s(𝑛 +s 1s ))))
161	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 1s ∈ ℕ0s)
162	154, 138, 161	pw2divscan4d 28860	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 /su (2s↑s(𝑛 +s 1s ))) = (((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
163	162	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) = (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))))
164	163	sseq1d 3944	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )) ↔ (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
165	164	bicomd 226	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )) ↔ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
166	144, 160, 165	3imtr4d 297	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
167		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑎 = (2s ·s 𝑥) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ (2s ·s 𝑥) <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
168		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (2s ·s 𝑥) → 𝑎 = (2s ·s 𝑥))
169	9	oveq1i 7453	. . . . . . . . . . . . . . . 16 ⊢ ((2s↑s 1s ) ·s 𝑥) = (2s ·s 𝑥)
170	168, 169	eqtr4di 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (2s ·s 𝑥) → 𝑎 = ((2s↑s 1s ) ·s 𝑥))
171	170	oveq1d 7458	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (2s ·s 𝑥) → (𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = (((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
172	171	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ (𝑎 = (2s ·s 𝑥) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) = (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))))
173	172	sseq1d 3944	. . . . . . . . . . . 12 ⊢ (𝑎 = (2s ·s 𝑥) → ((bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )) ↔ (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
174	167, 173	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑎 = (2s ·s 𝑥) → ((𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))) ↔ ((2s ·s 𝑥) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(((2s↑s 1s ) ·s 𝑥) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
175	166, 174	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑎 = (2s ·s 𝑥) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
176	175	rexlimdva 3167	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (∃𝑥 ∈ ℕ0s 𝑎 = (2s ·s 𝑥) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
177		n0zs 28805	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
178	177	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 𝑥 ∈ ℤs)
179	178	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → 𝑥 ∈ ℤs)
180	179	znod 28799	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → 𝑥 ∈ No)
181	138	adantrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑛 +s 1s ) ∈ ℕ0s)
182	180, 181	pw2divscld 28855	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 /su (2s↑s(𝑛 +s 1s ))) ∈ No)
183		1zs 28807	. . . . . . . . . . . . . . . . . . 19 ⊢ 1s ∈ ℤs
184	183	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → 1s ∈ ℤs)
185	179, 184	zaddscld 28811	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 +s 1s ) ∈ ℤs)
186	185	znod 28799	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 +s 1s ) ∈ No)
187	186, 181	pw2divscld 28855	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No)
188	180	ltsp1d 28431	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → 𝑥 <s (𝑥 +s 1s ))
189	180, 186, 181	pw2ltsdiv1d 28868	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 <s (𝑥 +s 1s ) ↔ (𝑥 /su (2s↑s(𝑛 +s 1s ))) <s ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
190	188, 189	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 /su (2s↑s(𝑛 +s 1s ))) <s ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
191	182, 187, 190	sltssn 28186	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → {(𝑥 /su (2s↑s(𝑛 +s 1s )))} <<s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
192		imaundi 6153	. . . . . . . . . . . . . . 15 ⊢ (bday “ ({(𝑥 /su (2s↑s(𝑛 +s 1s )))} ∪ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = ((bday “ {(𝑥 /su (2s↑s(𝑛 +s 1s )))}) ∪ (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
193		fnsnfv 6987	. . . . . . . . . . . . . . . . . 18 ⊢ ((bday Fn No ∧ (𝑥 /su (2s↑s(𝑛 +s 1s ))) ∈ No) → {(bday‘(𝑥 /su (2s↑s(𝑛 +s 1s ))))} = (bday “ {(𝑥 /su (2s↑s(𝑛 +s 1s )))}))
194	103, 182, 193	sylancr 599	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → {(bday‘(𝑥 /su (2s↑s(𝑛 +s 1s ))))} = (bday “ {(𝑥 /su (2s↑s(𝑛 +s 1s )))}))
195		oveq1 7450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑥 → (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑥 /su (2s↑s(𝑛 +s 1s ))))
196	195	fveq2d 6912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑥 → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) = (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))))
197	196	sseq1d 3944	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑥 → ((bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
198	127, 197	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑥 → ((𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) ↔ (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
199	198	rspccv 3577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → (𝑥 ∈ ℕ0s → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
200	199	imp 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
201	200	adantll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
202	201	adantrr 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
203	148	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s(𝑛 +s 1s )) ∈ No)
204	203, 203	addscld 28396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))) ∈ No)
205	154	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → 𝑥 ∈ No)
206	205, 203	addscld 28396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑥 +s (2s↑s(𝑛 +s 1s ))) ∈ No)
207		peano2n0s 28746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
208	207	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 +s 1s ) ∈ ℕ0s)
209	208	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
210	209	n0nod 28741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑥 +s 1s ) ∈ No)
211	205, 210	addscld 28396	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑥 +s (𝑥 +s 1s )) ∈ No)
212		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s(𝑛 +s 1s )) ≤s 𝑥)
213	203, 205, 203	leadds1d 28411	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) ≤s 𝑥 ↔ ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))) ≤s (𝑥 +s (2s↑s(𝑛 +s 1s )))))
214	212, 213	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))) ≤s (𝑥 +s (2s↑s(𝑛 +s 1s ))))
215	205	ltsp1d 28431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → 𝑥 <s (𝑥 +s 1s ))
216	203, 205, 210, 212, 215	leltstrd 28152	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s(𝑛 +s 1s )) <s (𝑥 +s 1s ))
217	203, 210, 216	ltlesd 28160	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s(𝑛 +s 1s )) ≤s (𝑥 +s 1s ))
218	203, 210, 205	leadds2d 28412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) ≤s (𝑥 +s 1s ) ↔ (𝑥 +s (2s↑s(𝑛 +s 1s ))) ≤s (𝑥 +s (𝑥 +s 1s ))))
219	217, 218	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑥 +s (2s↑s(𝑛 +s 1s ))) ≤s (𝑥 +s (𝑥 +s 1s )))
220	204, 206, 211, 214, 219	lestrd 28153	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))) ≤s (𝑥 +s (𝑥 +s 1s )))
221	138	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (𝑛 +s 1s ) ∈ ℕ0s)
222	7, 221, 145	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
223	7	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → 2s ∈ No)
224	203, 223	mulscomd 28556	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s↑s(𝑛 +s 1s )) ·s 2s) = (2s ·s (2s↑s(𝑛 +s 1s ))))
225		no2times 28833	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2s↑s(𝑛 +s 1s )) ∈ No → (2s ·s (2s↑s(𝑛 +s 1s ))) = ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))))
226	203, 225	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s ·s (2s↑s(𝑛 +s 1s ))) = ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))))
227	222, 224, 226	3eqtrd 2802	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) +s (2s↑s(𝑛 +s 1s ))))
228		no2times 28833	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ No → (2s ·s 𝑥) = (𝑥 +s 𝑥))
229	205, 228	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s ·s 𝑥) = (𝑥 +s 𝑥))
230	229	oveq1d 7458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s ·s 𝑥) +s 1s ) = ((𝑥 +s 𝑥) +s 1s ))
231	2	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → 1s ∈ No)
232	205, 205, 231	addsassd 28422	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((𝑥 +s 𝑥) +s 1s ) = (𝑥 +s (𝑥 +s 1s )))
233	230, 232	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → ((2s ·s 𝑥) +s 1s ) = (𝑥 +s (𝑥 +s 1s )))
234	220, 227, 233	3brtr4d 5140	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ≤s 𝑥)) → (2s↑s((𝑛 +s 1s ) +s 1s )) ≤s ((2s ·s 𝑥) +s 1s ))
235	234	expr 462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s↑s(𝑛 +s 1s )) ≤s 𝑥 → (2s↑s((𝑛 +s 1s ) +s 1s )) ≤s ((2s ·s 𝑥) +s 1s )))
236		lenlts 28139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2s↑s(𝑛 +s 1s )) ∈ No ∧ 𝑥 ∈ No) → ((2s↑s(𝑛 +s 1s )) ≤s 𝑥 ↔ ¬ 𝑥 <s (2s↑s(𝑛 +s 1s ))))
237	148, 154, 236	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s↑s(𝑛 +s 1s )) ≤s 𝑥 ↔ ¬ 𝑥 <s (2s↑s(𝑛 +s 1s ))))
238		peano2n0s 28746	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → ((𝑛 +s 1s ) +s 1s ) ∈ ℕ0s)
239	138, 238	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑛 +s 1s ) +s 1s ) ∈ ℕ0s)
240		expscl 28847	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2s ∈ No ∧ ((𝑛 +s 1s ) +s 1s ) ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) ∈ No)
241	7, 239, 240	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) ∈ No)
242		nnn0s 28743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
243	155, 242	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 2s ∈ ℕ0s
244		n0mulscl 28761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2s ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → (2s ·s 𝑥) ∈ ℕ0s)
245	243, 244	mpan 703	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ℕ0s → (2s ·s 𝑥) ∈ ℕ0s)
246	245	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s ·s 𝑥) ∈ ℕ0s)
247		n0addscl 28760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((2s ·s 𝑥) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝑥) +s 1s ) ∈ ℕ0s)
248	246, 53, 247	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) +s 1s ) ∈ ℕ0s)
249	248	n0nod 28741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) +s 1s ) ∈ No)
250		lenlts 28139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2s↑s((𝑛 +s 1s ) +s 1s )) ∈ No ∧ ((2s ·s 𝑥) +s 1s ) ∈ No) → ((2s↑s((𝑛 +s 1s ) +s 1s )) ≤s ((2s ·s 𝑥) +s 1s ) ↔ ¬ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
251	241, 249, 250	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s↑s((𝑛 +s 1s ) +s 1s )) ≤s ((2s ·s 𝑥) +s 1s ) ↔ ¬ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
252	235, 237, 251	3imtr3d 296	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (¬ 𝑥 <s (2s↑s(𝑛 +s 1s )) → ¬ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
253	252	con4d 116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → 𝑥 <s (2s↑s(𝑛 +s 1s ))))
254	253	impr 460	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → 𝑥 <s (2s↑s(𝑛 +s 1s )))
255		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → (𝑥 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
256	202, 254, 255	sylc 66	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))
257		bdayon 28168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ∈ On
258		bdayon 28168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday‘(𝑛 +s 1s )) ∈ On
259	258	onsuci 7874	. . . . . . . . . . . . . . . . . . . 20 ⊢ suc (bday‘(𝑛 +s 1s )) ∈ On
260		onsssuc 6473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ∈ On ∧ suc (bday‘(𝑛 +s 1s )) ∈ On) → ((bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s ))))
261	257, 259, 260	mp2an 705	. . . . . . . . . . . . . . . . . . 19 ⊢ ((bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s )))
262	256, 261	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘(𝑥 /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s )))
263	262	snssd 4747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → {(bday‘(𝑥 /su (2s↑s(𝑛 +s 1s ))))} ⊆ suc suc (bday‘(𝑛 +s 1s )))
264	194, 263	eqsstrrd 3948	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday “ {(𝑥 /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s )))
265	151	breq2d 5118	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ ((2s ·s 𝑥) +s 1s ) <s (2s ·s (2s↑s(𝑛 +s 1s )))))
266		n0expscl 28848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2s ∈ ℕ0s ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ∈ ℕ0s)
267	243, 138, 266	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ∈ ℕ0s)
268		n0mulscl 28761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2s ∈ ℕ0s ∧ (2s↑s(𝑛 +s 1s )) ∈ ℕ0s) → (2s ·s (2s↑s(𝑛 +s 1s ))) ∈ ℕ0s)
269	243, 267, 268	sylancr 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s ·s (2s↑s(𝑛 +s 1s ))) ∈ ℕ0s)
270		n0ltsp1le 28781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((2s ·s 𝑥) +s 1s ) ∈ ℕ0s ∧ (2s ·s (2s↑s(𝑛 +s 1s ))) ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (((2s ·s 𝑥) +s 1s ) +s 1s ) ≤s (2s ·s (2s↑s(𝑛 +s 1s )))))
271	248, 269, 270	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (((2s ·s 𝑥) +s 1s ) +s 1s ) ≤s (2s ·s (2s↑s(𝑛 +s 1s )))))
272	149, 154	mulscld 28551	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s ·s 𝑥) ∈ No)
273	2	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → 1s ∈ No)
274	272, 273, 273	addsassd 28422	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) +s 1s ) = ((2s ·s 𝑥) +s ( 1s +s 1s )))
275		mulsrid 28529	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
276	7, 275	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2s ·s 1s ) = 2s
277	276	eqcomi 2772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2s = (2s ·s 1s )
278	56, 277	eqtri 2786	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
279	278	oveq2i 7454	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2s ·s 𝑥) +s ( 1s +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s ))
280	149, 154, 273	addsdid 28572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (2s ·s (𝑥 +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s )))
281	280	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) +s (2s ·s 1s )) = (2s ·s (𝑥 +s 1s )))
282	279, 281	eqtrid 2810	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s 𝑥) +s ( 1s +s 1s )) = (2s ·s (𝑥 +s 1s )))
283	274, 282	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s )))
284	283	breq1d 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((((2s ·s 𝑥) +s 1s ) +s 1s ) ≤s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (2s ·s (𝑥 +s 1s )) ≤s (2s ·s (2s↑s(𝑛 +s 1s )))))
285	208	n0nod 28741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑥 +s 1s ) ∈ No)
286	285, 148, 149, 158	lemuls2d 28590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s )) ↔ (2s ·s (𝑥 +s 1s )) ≤s (2s ·s (2s↑s(𝑛 +s 1s )))))
287	286	bicomd 226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s ·s (𝑥 +s 1s )) ≤s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s ))))
288	284, 287	bitrd 282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((((2s ·s 𝑥) +s 1s ) +s 1s ) ≤s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s ))))
289	271, 288	bitrd 282	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s ·s (2s↑s(𝑛 +s 1s ))) ↔ (𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s ))))
290	265, 289	bitrd 282	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ (𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s ))))
291		lesloe 28141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 +s 1s ) ∈ No ∧ (2s↑s(𝑛 +s 1s )) ∈ No) → ((𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s )) ↔ ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) ∨ (𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )))))
292	285, 148, 291	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s )) ↔ ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) ∨ (𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )))))
293	285, 138	pw2divscld 28855	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No)
294	293	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No)
295		fnsnfv 6987	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((bday Fn No ∧ ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No) → {(bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))} = (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
296	103, 294, 295	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → {(bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))} = (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
297		breq1 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = (𝑥 +s 1s ) → (𝑎 <s (2s↑s(𝑛 +s 1s )) ↔ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s ))))
298		fvoveq1 7466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 = (𝑥 +s 1s ) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) = (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
299	298	sseq1d 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 = (𝑥 +s 1s ) → ((bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))))
300	297, 299	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = (𝑥 +s 1s ) → ((𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) ↔ ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
301	300	rspccv 3577	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → ((𝑥 +s 1s ) ∈ ℕ0s → ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
302	207, 301	syl5 35	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → (𝑥 ∈ ℕ0s → ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
303	302	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (𝑥 ∈ ℕ0s → ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))))
304	303	imp32 424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))
305		bdayon 28168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ∈ On
306		onsssuc 6473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ∈ On ∧ suc (bday‘(𝑛 +s 1s )) ∈ On) → ((bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s ))))
307	305, 259, 306	mp2an 705	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s )))
308	304, 307	sylib 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → (bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ∈ suc suc (bday‘(𝑛 +s 1s )))
309	308	snssd 4747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → {(bday‘((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))} ⊆ suc suc (bday‘(𝑛 +s 1s )))
310	296, 309	eqsstrrd 3948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ (𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )))) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s )))
311	310	expr 462	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
312	138	pw2divsidd 28872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s ))) = 1s )
313	312	sneqd 4596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))} = { 1s })
314	313	imaeq2d 6064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (bday “ {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))}) = (bday “ { 1s }))
315		df-1o 8504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 1o = suc ∅
316	15, 315	eqtri 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (bday‘ 1s ) = suc ∅
317		0ss 4353	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ∅ ⊆ (bday‘(𝑛 +s 1s ))
318		ord0 6435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Ord ∅
319	258	onordi 6494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ Ord (bday‘(𝑛 +s 1s ))
320		ordsucsssuc 7858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((Ord ∅ ∧ Ord (bday‘(𝑛 +s 1s ))) → (∅ ⊆ (bday‘(𝑛 +s 1s )) ↔ suc ∅ ⊆ suc (bday‘(𝑛 +s 1s ))))
321	318, 319, 320	mp2an 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∅ ⊆ (bday‘(𝑛 +s 1s )) ↔ suc ∅ ⊆ suc (bday‘(𝑛 +s 1s )))
322	317, 321	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ suc ∅ ⊆ suc (bday‘(𝑛 +s 1s ))
323	316, 322	eqsstri 3959	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (bday‘ 1s ) ⊆ suc (bday‘(𝑛 +s 1s ))
324		bdayon 28168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (bday‘ 1s ) ∈ On
325		onsssuc 6473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((bday‘ 1s ) ∈ On ∧ suc (bday‘(𝑛 +s 1s )) ∈ On) → ((bday‘ 1s ) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘ 1s ) ∈ suc suc (bday‘(𝑛 +s 1s ))))
326	324, 259, 325	mp2an 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((bday‘ 1s ) ⊆ suc (bday‘(𝑛 +s 1s )) ↔ (bday‘ 1s ) ∈ suc suc (bday‘(𝑛 +s 1s )))
327	323, 326	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (bday‘ 1s ) ∈ suc suc (bday‘(𝑛 +s 1s ))
328	327	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ℕ0s → (bday‘ 1s ) ∈ suc suc (bday‘(𝑛 +s 1s )))
329	328	snssd 4747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ0s → {(bday‘ 1s )} ⊆ suc suc (bday‘(𝑛 +s 1s )))
330	109, 329	eqsstrrid 3952	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ0s → (bday “ { 1s }) ⊆ suc suc (bday‘(𝑛 +s 1s )))
331	330	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (bday “ { 1s }) ⊆ suc suc (bday‘(𝑛 +s 1s )))
332	331	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (bday “ { 1s }) ⊆ suc suc (bday‘(𝑛 +s 1s )))
333	314, 332	eqsstrd 3947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (bday “ {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s )))
334		oveq1 7450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )) → ((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) = ((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s ))))
335	334	sneqd 4596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )) → {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))} = {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))})
336	335	imaeq2d 6064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) = (bday “ {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))}))
337	336	sseq1d 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )) → ((bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s )) ↔ (bday “ {((2s↑s(𝑛 +s 1s )) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
338	333, 337	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s )) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
339	311, 338	jaod 873	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((𝑥 +s 1s ) <s (2s↑s(𝑛 +s 1s )) ∨ (𝑥 +s 1s ) = (2s↑s(𝑛 +s 1s ))) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
340	292, 339	sylbid 243	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → ((𝑥 +s 1s ) ≤s (2s↑s(𝑛 +s 1s )) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
341	290, 340	sylbid 243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s ))))
342	341	impr 460	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ⊆ suc suc (bday‘(𝑛 +s 1s )))
343	264, 342	unssd 4193	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → ((bday “ {(𝑥 /su (2s↑s(𝑛 +s 1s )))}) ∪ (bday “ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s )))
344	192, 343	eqsstrid 3951	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday “ ({(𝑥 /su (2s↑s(𝑛 +s 1s )))} ∪ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s )))
345	259	onsuci 7874	. . . . . . . . . . . . . . 15 ⊢ suc suc (bday‘(𝑛 +s 1s )) ∈ On
346		cutbdaybnd 28211	. . . . . . . . . . . . . . 15 ⊢ (({(𝑥 /su (2s↑s(𝑛 +s 1s )))} <<s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))} ∧ suc suc (bday‘(𝑛 +s 1s )) ∈ On ∧ (bday “ ({(𝑥 /su (2s↑s(𝑛 +s 1s )))} ∪ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s ))) → (bday‘({(𝑥 /su (2s↑s(𝑛 +s 1s )))} |s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s )))
347	345, 346	mp3an2 1479	. . . . . . . . . . . . . 14 ⊢ (({(𝑥 /su (2s↑s(𝑛 +s 1s )))} <<s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))} ∧ (bday “ ({(𝑥 /su (2s↑s(𝑛 +s 1s )))} ∪ {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s ))) → (bday‘({(𝑥 /su (2s↑s(𝑛 +s 1s )))} |s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s )))
348	191, 344, 347	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘({(𝑥 /su (2s↑s(𝑛 +s 1s )))} |s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ⊆ suc suc (bday‘(𝑛 +s 1s )))
349	179, 181	pw2cutp1 28877	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → ({(𝑥 /su (2s↑s(𝑛 +s 1s )))} |s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
350	349	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘({(𝑥 /su (2s↑s(𝑛 +s 1s )))} |s {((𝑥 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))))
351	181, 140	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘((𝑛 +s 1s ) +s 1s )) = suc (bday‘(𝑛 +s 1s )))
352	351	suceqd 6448	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → suc (bday‘((𝑛 +s 1s ) +s 1s )) = suc suc (bday‘(𝑛 +s 1s )))
353	352	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → suc suc (bday‘(𝑛 +s 1s )) = suc (bday‘((𝑛 +s 1s ) +s 1s )))
354	348, 350, 353	3sstr3d 3967	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ (𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )))) → (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))
355	354	expr 462	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
356		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑎 = ((2s ·s 𝑥) +s 1s ) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) ↔ ((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s ))))
357		oveq1 7450	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ((2s ·s 𝑥) +s 1s ) → (𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
358	357	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ (𝑎 = ((2s ·s 𝑥) +s 1s ) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) = (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))))
359	358	sseq1d 3944	. . . . . . . . . . . 12 ⊢ (𝑎 = ((2s ·s 𝑥) +s 1s ) → ((bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )) ↔ (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
360	356, 359	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑎 = ((2s ·s 𝑥) +s 1s ) → ((𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))) ↔ (((2s ·s 𝑥) +s 1s ) <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(((2s ·s 𝑥) +s 1s ) /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
361	355, 360	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) ∧ 𝑥 ∈ ℕ0s) → (𝑎 = ((2s ·s 𝑥) +s 1s ) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
362	361	rexlimdva 3167	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (∃𝑥 ∈ ℕ0s 𝑎 = ((2s ·s 𝑥) +s 1s ) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
363	176, 362	jaod 873	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → ((∃𝑥 ∈ ℕ0s 𝑎 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑎 = ((2s ·s 𝑥) +s 1s )) → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
364	126, 363	syl5 35	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → (𝑎 ∈ ℕ0s → (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
365	125, 364	ralrimi 3264	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s )))) → ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s ))))
366	365	ex 418	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑛 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑛 +s 1s )))) ⊆ suc (bday‘(𝑛 +s 1s ))) → ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s((𝑛 +s 1s ) +s 1s )) → (bday‘(𝑎 /su (2s↑s((𝑛 +s 1s ) +s 1s )))) ⊆ suc (bday‘((𝑛 +s 1s ) +s 1s )))))
367	22, 32, 42, 52, 122, 366	n0sind 28749	. . . 4 ⊢ (𝑁 ∈ ℕ0s → ∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s ))))
368		breq1 5109	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 <s (2s↑s(𝑁 +s 1s )) ↔ 𝐴 <s (2s↑s(𝑁 +s 1s ))))
369		oveq1 7450	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 /su (2s↑s(𝑁 +s 1s ))) = (𝐴 /su (2s↑s(𝑁 +s 1s ))))
370	369	fveq2d 6912	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) = (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))))
371	370	sseq1d 3944	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )) ↔ (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s ))))
372	368, 371	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑎 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s ))) ↔ (𝐴 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
373	372	rspccv 3577	. . . 4 ⊢ (∀𝑎 ∈ ℕ0s (𝑎 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝑎 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s ))) → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
374	367, 373	syl 18	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
375	374	com12 33	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝑁 ∈ ℕ0s → (𝐴 <s (2s↑s(𝑁 +s 1s )) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))))
376	375	3imp 1128	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ∧ 𝐴 <s (2s↑s(𝑁 +s 1s ))) → (bday‘(𝐴 /su (2s↑s(𝑁 +s 1s )))) ⊆ suc (bday‘(𝑁 +s 1s )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588  {cpr 4590   class class class wbr 5106   “ cima 5658  Ord word 6379  Oncon0 6380  suc csuc 6382   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443  1_oc1o 8497  2_oc2o 8498  Nocsur 28027   <s clts 28028  bdaycbday 28029   ≤s cles 28131   <<s cslts 28173   |s ccuts 28175   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   ·_s cmuls 28522   /_su cdivs 28603  ℕ_0scn0s 28728  ℕ_scnns 28729  ℤ_sczs 28794  2_sc2s 28826  ↑_scexps 28828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-dc 10551

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-divs 28604  df-ons 28668  df-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-2s 28827  df-exps 28829

This theorem is used by:  bdaypw2n0bnd  28880