Theorem zs12zodd 28359

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
zs12zodd	⊢ (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 zs12zodd

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

Step	Hyp	Ref	Expression
1		elzs12 28350	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑏 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑏)))
2		oveq2 7357	. . . . . . . . . . . 12 ⊢ (𝑐 = 0s → (2s↑s𝑐) = (2s↑s 0s ))
3		2sno 28311	. . . . . . . . . . . . 13 ⊢ 2s ∈ No
4		exps0 28319	. . . . . . . . . . . . 13 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
5	3, 4	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2s↑s 0s ) = 1s
6	2, 5	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑐 = 0s → (2s↑s𝑐) = 1s )
7	6	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑐 = 0s → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su 1s ))
8	7	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑐 = 0s → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su 1s ) ∈ ℤs))
9	7	eqeq1d 2731	. . . . . . . . . 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 3194	. . . . . . . . 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 918	. . . . . . . 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 3152	. . . . . . 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 7357	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (2s↑s𝑐) = (2s↑s𝑤))
14	13	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑤)))
15	14	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑤)) ∈ ℤs))
16	14	eqeq1d 2731	. . . . . . . . . 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 3194	. . . . . . . . 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 918	. . . . . . . 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 3152	. . . . . . 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 7357	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑤 +s 1s ) → (2s↑s𝑐) = (2s↑s(𝑤 +s 1s )))
21	20	oveq2d 7365	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑤 +s 1s ) → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s(𝑤 +s 1s ))))
22	21	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑐 = (𝑤 +s 1s ) → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
23	21	eqeq1d 2731	. . . . . . . . . . 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 3194	. . . . . . . . . 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 918	. . . . . . . . 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 3152	. . . . . . . 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 7356	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (𝑏 /su (2s↑s(𝑤 +s 1s ))))
28	27	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
29	27	eqeq1d 2731	. . . . . . . . . . . 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 3194	. . . . . . . . . . 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 7357	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑝 → (2s ·s 𝑥) = (2s ·s 𝑝))
32	31	oveq1d 7364	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑝 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑝) +s 1s ))
33	32	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑝 → (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)))
34	33	eqeq2d 2740	. . . . . . . . . . . 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 7357	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑞 → (2s↑s𝑦) = (2s↑s𝑞))
36	35	oveq2d 7365	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑞 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
37	36	eqeq2d 2740	. . . . . . . . . . . 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 3212	. . . . . . . . . . 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 287	. . . . . . . . . 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 918	. . . . . . . . 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 3207	. . . . . . . 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 287	. . . . . . 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 7357	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (2s↑s𝑐) = (2s↑s𝑏))
44	43	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑏)))
45	44	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
46	44	eqeq1d 2731	. . . . . . . . . 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 3194	. . . . . . . . 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 918	. . . . . . . 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 3152	. . . . . . 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 28275	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
51		divs1 28112	. . . . . . . . . . 11 ⊢ (𝑎 ∈ No → (𝑎 /su 1s ) = 𝑎)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → (𝑎 /su 1s ) = 𝑎)
53		id 22	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ ℤs)
54	52, 53	eqeltrd 2828	. . . . . . . . 9 ⊢ (𝑎 ∈ ℤs → (𝑎 /su 1s ) ∈ ℤs)
55	54	orcd 873	. . . . . . . 8 ⊢ (𝑎 ∈ ℤs → ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦))))
56	55	rgen 3046	. . . . . . 7 ⊢ ∀𝑎 ∈ ℤs ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))
57		zseo 28314	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤs → (∃𝑐 ∈ ℤs 𝑏 = (2s ·s 𝑐) ∨ ∃𝑐 ∈ ℤs 𝑏 = ((2s ·s 𝑐) +s 1s )))
58		oveq1 7356	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑎 /su (2s↑s𝑤)) = (𝑐 /su (2s↑s𝑤)))
59	58	eleq1d 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
60	58	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . 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))))
61	60	2rexbidv 3194	. . . . . . . . . . . . . . . . . . . 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))))
62	59, 61	orbi12d 918	. . . . . . . . . . . . . . . . . . 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)))))
63	62	rspcv 3573	. . . . . . . . . . . . . . . . . 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)))))
64	63	adantl 481	. . . . . . . . . . . . . . . . 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)))))
65	33	eqeq2d 2740	. . . . . . . . . . . . . . . . . . 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))))
66	36	eqeq2d 2740	. . . . . . . . . . . . . . . . . . 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))))
67	65, 66	cbvrex2vw 3212	. . . . . . . . . . . . . . . . . 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)))
68	67	orbi2i 912	. . . . . . . . . . . . . . . . 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))))
69	64, 68	imbitrdi 251	. . . . . . . . . . . . . . . 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)))))
70	69	imp 406	. . . . . . . . . . . . . . 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))))
71	70	an32s 652	. . . . . . . . . . . . . 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))))
72		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑤 ∈ ℕ0s)
73		expsp1 28321	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2s ∈ No ∧ 𝑤 ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) = ((2s↑s𝑤) ·s 2s))
74	3, 72, 73	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) = ((2s↑s𝑤) ·s 2s))
75	74	oveq1d 7364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = (((2s↑s𝑤) ·s 2s) ·s 𝑐))
76		expscl 28323	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2s ∈ No ∧ 𝑤 ∈ ℕ0s) → (2s↑s𝑤) ∈ No )
77	3, 72, 76	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s𝑤) ∈ No )
78	3	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 2s ∈ No )
79		zno 28275	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ ℤs → 𝑐 ∈ No )
80	79	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑐 ∈ No )
81	77, 78, 80	mulsassd 28075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s𝑤) ·s 2s) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
82	75, 81	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
83	82	oveq1d 7364	. . . . . . . . . . . . . . . . . . . 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 ))))
84		peano2n0s 28228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕ0s)
85	84	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕ0s)
86	80, 85	pw2divscan3d 28333	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s(𝑤 +s 1s )) ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = 𝑐)
87	78, 80	mulscld 28043	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s ·s 𝑐) ∈ No )
88		expscl 28323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2s ∈ No ∧ (𝑤 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) ∈ No )
89	3, 85, 88	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) ∈ No )
90		2ne0s 28312	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2s ≠ 0s
91		expsne0 28328	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ (𝑤 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑤 +s 1s )) ≠ 0s )
92	3, 90, 85, 91	mp3an12i 1467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s↑s(𝑤 +s 1s )) ≠ 0s )
93		pw2recs 28330	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 +s 1s ) ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s(𝑤 +s 1s )) ·s 𝑥) = 1s )
94	85, 93	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ∃𝑥 ∈ No ((2s↑s(𝑤 +s 1s )) ·s 𝑥) = 1s )
95	77, 87, 89, 92, 94	divsasswd 28111	. . . . . . . . . . . . . . . . . . . 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 )))))
96	83, 86, 95	3eqtr3rd 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s𝑤) ·s ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s )))) = 𝑐)
97	87, 85	pw2divscld 28331	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ No )
98	80, 97, 72	pw2divsmuld 28332	. . . . . . . . . . . . . . . . . . 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))
99	96, 98	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑐 /su (2s↑s𝑤)) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))))
100	99	eqcomd 2735	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (𝑐 /su (2s↑s𝑤)))
101	100	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
102	100	eqeq1d 2731	. . . . . . . . . . . . . . . . 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))))
103	102	2rexbidv 3194	. . . . . . . . . . . . . . . 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))))
104	101, 103	orbi12d 918	. . . . . . . . . . . . . . 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)))))
105	104	adantlr 715	. . . . . . . . . . . . . 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)))))
106	71, 105	mpbird 257	. . . . . . . . . . . . 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))))
107		oveq1 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (2s ·s 𝑐) → (𝑏 /su (2s↑s(𝑤 +s 1s ))) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))))
108	107	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (2s ·s 𝑐) → ((𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
109	107	eqeq1d 2731	. . . . . . . . . . . . . . 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))))
110	109	2rexbidv 3194	. . . . . . . . . . . . . 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))))
111	108, 110	orbi12d 918	. . . . . . . . . . . . 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)))))
112	106, 111	syl5ibrcom 247	. . . . . . . . . . . 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)))))
113	112	rexlimdva 3130	. . . . . . . . . . 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)))))
114		oveq2 7357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑐 → (2s ·s 𝑝) = (2s ·s 𝑐))
115	114	oveq1d 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑐 → ((2s ·s 𝑝) +s 1s ) = ((2s ·s 𝑐) +s 1s ))
116	115	oveq1d 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑐 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)))
117	116	eqeq2d 2740	. . . . . . . . . . . . . . . 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))))
118		oveq2 7357	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = (𝑤 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑤 +s 1s )))
119	118	oveq2d 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = (𝑤 +s 1s ) → (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s(𝑤 +s 1s ))))
120	119	eqeq2d 2740	. . . . . . . . . . . . . . . 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 )))))
121		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑐 ∈ ℤs)
122		n0p1nns 28265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕs)
123	122	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕs)
124		eqidd 2730	. . . . . . . . . . . . . . . 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 ))))
125	117, 120, 121, 123, 124	2rspcedvdw 3591	. . . . . . . . . . . . . . 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)))
126	125	olcd 874	. . . . . . . . . . . . . 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))))
127	126	adantlr 715	. . . . . . . . . . . . 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))))
128		oveq1 7356	. . . . . . . . . . . . . . 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 ))))
129	128	eleq1d 2813	. . . . . . . . . . . . . 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))
130	128	eqeq1d 2731	. . . . . . . . . . . . . . 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))))
131	130	2rexbidv 3194	. . . . . . . . . . . . . 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))))
132	129, 131	orbi12d 918	. . . . . . . . . . . . 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)))))
133	127, 132	syl5ibrcom 247	. . . . . . . . . . . 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)))))
134	133	rexlimdva 3130	. . . . . . . . . . 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)))))
135	113, 134	jaod 859	. . . . . . . . . 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)))))
136	57, 135	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)))))
137	136	ralrimiv 3120	. . . . . . . 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))))
138	137	ex 412	. . . . . . 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)))))
139	12, 19, 42, 49, 56, 138	n0sind 28230	. . . . . 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))))
140		rsp 3217	. . . . . 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)))))
141	139, 140	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)))))
142	141	impcom 407	. . . 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))))
143		eleq1 2816	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
144		eqeq1 2733	. . . . . 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))))
145	144	2rexbidv 3194	. . . . 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))))
146	143, 145	orbi12d 918	. . . 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)))))
147	142, 146	syl5ibrcom 247	. . 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)))))
148	147	rexlimivv 3171	. 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))))
149	1, 148	sylbi 217	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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  (class class class)co 7349   No csur 27549   0_s c0s 27736   1_s c1s 27737   +_s cadds 27871   ·_s cmuls 28014   /_su cdivs 28095  ℕ_0scnn0s 28211  ℕ_scnns 28212  ℤ_sczs 28271  2_sc2s 28302  ↑_scexps 28304  ℤ_s[1/2]czs12 28306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27738  df-1s 27739  df-made 27757  df-old 27758  df-left 27760  df-right 27761  df-norec 27850  df-norec2 27861  df-adds 27872  df-negs 27932  df-subs 27933  df-muls 28015  df-divs 28096  df-seqs 28183  df-n0s 28213  df-nns 28214  df-zs 28272  df-2s 28303  df-exps 28305  df-zs12 28307

This theorem is referenced by: (None)