Theorem z12zsodd 28499

Description: A dyadic fraction is either an integer or an odd number divided by a positive power of two. (Contributed by Scott Fenton, 5-Dec-2025.)

Assertion

Ref	Expression
z12zsodd	⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem z12zsodd

Dummy variables 𝑎 𝑏 𝑐 𝑝 𝑞 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elz12s 28489	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑏 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑏)))
2		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑐 = 0s → (2s↑s𝑐) = (2s↑s 0s ))
3		2no 28436	. . . . . . . . . . . . 13 ⊢ 2s ∈ No
4		exps0 28444	. . . . . . . . . . . . 13 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
5	3, 4	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2s↑s 0s ) = 1s
6	2, 5	eqtrdi 2791	. . . . . . . . . . 11 ⊢ (𝑐 = 0s → (2s↑s𝑐) = 1s )
7	6	oveq2d 7379	. . . . . . . . . 10 ⊢ (𝑐 = 0s → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su 1s ))
8	7	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑐 = 0s → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su 1s ) ∈ ℤs))
9	7	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑐 = 0s → ((𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
10	9	2rexbidv 3205	. . . . . . . . 9 ⊢ (𝑐 = 0s → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
11	8, 10	orbi12d 924	. . . . . . . 8 ⊢ (𝑐 = 0s → (((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
12	11	ralbidv 3163	. . . . . . 7 ⊢ (𝑐 = 0s → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑎 ∈ ℤs ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
13		oveq2 7371	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (2s↑s𝑐) = (2s↑s𝑤))
14	13	oveq2d 7379	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑤)))
15	14	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑤)) ∈ ℤs))
16	14	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → ((𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
17	16	2rexbidv 3205	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
18	15, 17	orbi12d 924	. . . . . . . 8 ⊢ (𝑐 = 𝑤 → (((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
19	18	ralbidv 3163	. . . . . . 7 ⊢ (𝑐 = 𝑤 → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
20		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑤 +s 1s ) → (2s↑s𝑐) = (2s↑s(𝑤 +s 1s )))
21	20	oveq2d 7379	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑤 +s 1s ) → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s(𝑤 +s 1s ))))
22	21	eleq1d 2825	. . . . . . . . . 10 ⊢ (𝑐 = (𝑤 +s 1s ) → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
23	21	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑤 +s 1s ) → ((𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
24	23	2rexbidv 3205	. . . . . . . . . 10 ⊢ (𝑐 = (𝑤 +s 1s ) → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
25	22, 24	orbi12d 924	. . . . . . . . 9 ⊢ (𝑐 = (𝑤 +s 1s ) → (((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
26	25	ralbidv 3163	. . . . . . . 8 ⊢ (𝑐 = (𝑤 +s 1s ) → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
27		oveq1 7370	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (𝑏 /su (2s↑s(𝑤 +s 1s ))))
28	27	eleq1d 2825	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
29	27	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
30	29	2rexbidv 3205	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
31		oveq2 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑝 → (2s ·s 𝑥) = (2s ·s 𝑝))
32	31	oveq1d 7378	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑝 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑝) +s 1s ))
33	32	oveq1d 7378	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑝 → (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)))
34	33	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦))))
35		oveq2 7371	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑞 → (2s↑s𝑦) = (2s↑s𝑞))
36	35	oveq2d 7379	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑞 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
37	36	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑞 → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
38	34, 37	cbvrex2vw 3223	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
39	30, 38	bitrdi 288	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
40	28, 39	orbi12d 924	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
41	40	cbvralvw 3218	. . . . . . . 8 ⊢ (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑏 ∈ ℤs ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
42	26, 41	bitrdi 288	. . . . . . 7 ⊢ (𝑐 = (𝑤 +s 1s ) → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑏 ∈ ℤs ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
43		oveq2 7371	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (2s↑s𝑐) = (2s↑s𝑏))
44	43	oveq2d 7379	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑏)))
45	44	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
46	44	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ((𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
47	46	2rexbidv 3205	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
48	45, 47	orbi12d 924	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
49	48	ralbidv 3163	. . . . . . 7 ⊢ (𝑐 = 𝑏 → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑐)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
50		zno 28399	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
51	50	divs1d 28222	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → (𝑎 /su 1s ) = 𝑎)
52		id 22	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ ℤs)
53	51, 52	eqeltrd 2840	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤs → (𝑎 /su 1s ) ∈ ℤs)
54	53	orcd 879	. . . . . . . 8 ⊢ (𝑎 ∈ ℤs → ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
55	54	rgen 3056	. . . . . . 7 ⊢ ∀𝑎 ∈ ℤs ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))
56		zseo 28439	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤs → (∃𝑐 ∈ ℤs 𝑏 = (2s ·s 𝑐) ∨ ∃𝑐 ∈ ℤs 𝑏 = ((2s ·s 𝑐) +s 1s )))
57		oveq1 7370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑎 /su (2s↑s𝑤)) = (𝑐 /su (2s↑s𝑤)))
58	57	eleq1d 2825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
59	57	eqeq1d 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → ((𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
60	59	2rexbidv 3205	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
61	58, 60	orbi12d 924	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑐 → (((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
62	61	rspcv 3563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ ℤs → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) → ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
63	62	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) → ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
64	33	eqeq2d 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑝 → ((𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦))))
65	36	eqeq2d 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑞 → ((𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
66	64, 65	cbvrex2vw 3223	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
67	66	orbi2i 918	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
68	63, 67	imbitrdi 252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) → ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
69	68	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
70	69	an32s 658	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
71		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑤 ∈ ℕ0s)
72		expsp1 28446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2s ∈ No ∧ 𝑤 ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) = ((2s↑s𝑤) ·s 2s))
73	3, 71, 72	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) = ((2s↑s𝑤) ·s 2s))
74	73	oveq1d 7378	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = (((2s↑s𝑤) ·s 2s) ·s 𝑐))
75		expscl 28448	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2s ∈ No ∧ 𝑤 ∈ ℕ0s) → (2s↑s𝑤) ∈ No )
76	3, 71, 75	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s𝑤) ∈ No )
77	3	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 2s ∈ No )
78		zno 28399	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ ℤs → 𝑐 ∈ No )
79	78	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑐 ∈ No )
80	76, 77, 79	mulsassd 28184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s𝑤) ·s 2s) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
81	74, 80	eqtrd 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
82	81	oveq1d 7378	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s(𝑤 +s 1s )) ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s↑s𝑤) ·s (2s ·s 𝑐)) /su (2s↑s(𝑤 +s 1s ))))
83		peano2n0s 28347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕ0s)
84	83	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕ0s)
85	79, 84	pw2divscan3d 28458	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s(𝑤 +s 1s )) ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = 𝑐)
86	77, 79	mulscld 28152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s ·s 𝑐) ∈ No )
87		expscl 28448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2s ∈ No ∧ (𝑤 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) ∈ No )
88	3, 84, 87	sylancr 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) ∈ No )
89		2ne0s 28437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2s ≠ 0s
90		expsne0 28453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ (𝑤 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) ≠ 0s )
91	3, 89, 84, 90	mp3an12i 1473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) ≠ 0s )
92		pw2recs 28455	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 +s 1s ) ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s(𝑤 +s 1s )) ·s 𝑥) = 1s )
93	84, 92	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ∃𝑥 ∈ No ((2s↑s(𝑤 +s 1s )) ·s 𝑥) = 1s )
94	76, 86, 88, 91, 93	divsasswd 28220	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s𝑤) ·s (2s ·s 𝑐)) /su (2s↑s(𝑤 +s 1s ))) = ((2s↑s𝑤) ·s ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s )))))
95	82, 85, 94	3eqtr3rd 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s𝑤) ·s ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s )))) = 𝑐)
96	86, 84	pw2divscld 28456	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ No )
97	79, 96, 71	pw2divmulsd 28457	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((𝑐 /su (2s↑s𝑤)) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ↔ ((2s↑s𝑤) ·s ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s )))) = 𝑐))
98	95, 97	mpbird 258	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑐 /su (2s↑s𝑤)) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))))
99	98	eqcomd 2746	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (𝑐 /su (2s↑s𝑤)))
100	99	eleq1d 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
101	99	eqeq1d 2742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
102	101	2rexbidv 3205	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
103	100, 102	orbi12d 924	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))) ↔ ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
104	103	adantlr 721	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → ((((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))) ↔ ((𝑐 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑐 /su (2s↑s𝑤)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
105	70, 104	mpbird 258	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
106		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (2s ·s 𝑐) → (𝑏 /su (2s↑s(𝑤 +s 1s ))) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))))
107	106	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (2s ·s 𝑐) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
108	106	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (2s ·s 𝑐) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
109	108	2rexbidv 3205	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (2s ·s 𝑐) → (∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
110	107, 109	orbi12d 924	. . . . . . . . . . . . 13 ⊢ (𝑏 = (2s ·s 𝑐) → (((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))) ↔ (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
111	105, 110	syl5ibrcom 248	. . . . . . . . . . . 12 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → (𝑏 = (2s ·s 𝑐) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
112	111	rexlimdva 3141	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → (∃𝑐 ∈ ℤs 𝑏 = (2s ·s 𝑐) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
113		oveq2 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑐 → (2s ·s 𝑝) = (2s ·s 𝑐))
114	113	oveq1d 7378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑐 → ((2s ·s 𝑝) +s 1s ) = ((2s ·s 𝑐) +s 1s ))
115	114	oveq1d 7378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑐 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)))
116	115	eqeq2d 2751	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑐 → ((((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞))))
117		oveq2 7371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = (𝑤 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑤 +s 1s )))
118	117	oveq2d 7379	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = (𝑤 +s 1s ) → (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))))
119	118	eqeq2d 2751	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = (𝑤 +s 1s ) → ((((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)) ↔ (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s )))))
120		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑐 ∈ ℤs)
121		n0p1nns 28388	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕs)
122	121	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕs)
123		eqidd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))))
124	116, 119, 120, 122, 123	2rspcedvdw 3581	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
125	124	olcd 880	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
126	125	adantlr 721	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → ((((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
127		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((2s ·s 𝑐) +s 1s ) → (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))))
128	127	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((2s ·s 𝑐) +s 1s ) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
129	127	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((2s ·s 𝑐) +s 1s ) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
130	129	2rexbidv 3205	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((2s ·s 𝑐) +s 1s ) → (∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) ↔ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
131	128, 130	orbi12d 924	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((2s ·s 𝑐) +s 1s ) → (((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))) ↔ ((((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
132	126, 131	syl5ibrcom 248	. . . . . . . . . . . 12 ⊢ (((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) ∧ 𝑐 ∈ ℤs) → (𝑏 = ((2s ·s 𝑐) +s 1s ) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
133	132	rexlimdva 3141	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → (∃𝑐 ∈ ℤs 𝑏 = ((2s ·s 𝑐) +s 1s ) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
134	112, 133	jaod 865	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → ((∃𝑐 ∈ ℤs 𝑏 = (2s ·s 𝑐) ∨ ∃𝑐 ∈ ℤs 𝑏 = ((2s ·s 𝑐) +s 1s )) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
135	56, 134	syl5 34	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → (𝑏 ∈ ℤs → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
136	135	ralrimiv 3131	. . . . . . . 8 ⊢ ((𝑤 ∈ ℕ0s ∧ ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))) → ∀𝑏 ∈ ℤs ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞))))
137	136	ex 413	. . . . . . 7 ⊢ (𝑤 ∈ ℕ0s → (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑤)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) → ∀𝑏 ∈ ℤs ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ∨ ∃𝑝 ∈ ℤs ∃𝑞 ∈ ℕs (𝑏 /su (2s↑s(𝑤 +s 1s ))) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))))
138	12, 19, 42, 49, 55, 137	n0sind 28350	. . . . . 6 ⊢ (𝑏 ∈ ℕ0s → ∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
139		rsp 3228	. . . . . 6 ⊢ (∀𝑎 ∈ ℤs ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) → (𝑎 ∈ ℤs → ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
140	138, 139	syl 17	. . . . 5 ⊢ (𝑏 ∈ ℕ0s → (𝑎 ∈ ℤs → ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
141	140	impcom 408	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑏 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
142		eleq1 2828	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
143		eqeq1 2744	. . . . . 6 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
144	143	2rexbidv 3205	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
145	142, 144	orbi12d 924	. . . 4 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → ((𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))) ↔ ((𝑎 /su (2s↑s𝑏)) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su (2s↑s𝑏)) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
146	141, 145	syl5ibrcom 248	. . 3 ⊢ ((𝑎 ∈ ℤs ∧ 𝑏 ∈ ℕ0s) → (𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))))
147	146	rexlimivv 3182	. 2 ⊢ (∃𝑎 ∈ ℤs ∃𝑏 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
148	1, 147	sylbi 218	1 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs 𝐴 = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  (class class class)co 7363   No csur 27628   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976   ·_s cmuls 28123   /_su cdivs 28204  ℕ_0scn0s 28329  ℕ_scnns 28330  ℤ_sczs 28395  2_sc2s 28427  ↑_scexps 28429  ℤ_s[1/2]cz12s 28431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  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 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124  df-divs 28205  df-seqs 28301  df-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428  df-exps 28430  df-z12s 28432

This theorem is referenced by: (None)