Theorem zs12zodd 28392

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 28383	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑏 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑏)))
2		oveq2 7354	. . . . . . . . . . . 12 ⊢ (𝑐 = 0s → (2s↑s𝑐) = (2s↑s 0s ))
3		2sno 28342	. . . . . . . . . . . . 13 ⊢ 2s ∈ No
4		exps0 28350	. . . . . . . . . . . . 13 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
5	3, 4	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2s↑s 0s ) = 1s
6	2, 5	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑐 = 0s → (2s↑s𝑐) = 1s )
7	6	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑐 = 0s → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su 1s ))
8	7	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑐 = 0s → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su 1s ) ∈ ℤs))
9	7	eqeq1d 2733	. . . . . . . . . 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 3197	. . . . . . . . 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 3155	. . . . . . 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 7354	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (2s↑s𝑐) = (2s↑s𝑤))
14	13	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑤)))
15	14	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑤)) ∈ ℤs))
16	14	eqeq1d 2733	. . . . . . . . . 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 3197	. . . . . . . . 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 3155	. . . . . . 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 7354	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑤 +s 1s ) → (2s↑s𝑐) = (2s↑s(𝑤 +s 1s )))
21	20	oveq2d 7362	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑤 +s 1s ) → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s(𝑤 +s 1s ))))
22	21	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑐 = (𝑤 +s 1s ) → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
23	21	eqeq1d 2733	. . . . . . . . . . 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 3197	. . . . . . . . . 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 3155	. . . . . . . 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 7353	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 /su (2s↑s(𝑤 +s 1s ))) = (𝑏 /su (2s↑s(𝑤 +s 1s ))))
28	27	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑏 /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs))
29	27	eqeq1d 2733	. . . . . . . . . . . 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 3197	. . . . . . . . . . 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 7354	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑝 → (2s ·s 𝑥) = (2s ·s 𝑝))
32	31	oveq1d 7361	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑝 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑝) +s 1s ))
33	32	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑝 → (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)))
34	33	eqeq2d 2742	. . . . . . . . . . . 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 7354	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑞 → (2s↑s𝑦) = (2s↑s𝑞))
36	35	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑞 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑦)) = (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)))
37	36	eqeq2d 2742	. . . . . . . . . . . 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 3215	. . . . . . . . . . 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 3210	. . . . . . . 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 7354	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (2s↑s𝑐) = (2s↑s𝑏))
44	43	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝑎 /su (2s↑s𝑐)) = (𝑎 /su (2s↑s𝑏)))
45	44	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝑎 /su (2s↑s𝑐)) ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
46	44	eqeq1d 2733	. . . . . . . . . 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 3197	. . . . . . . . 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 3155	. . . . . . 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 28306	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
51		divs1 28143	. . . . . . . . . . 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 2831	. . . . . . . . 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 3049	. . . . . . 7 ⊢ ∀𝑎 ∈ ℤs ((𝑎 /su 1s ) ∈ ℤs ∨ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕs (𝑎 /su 1s ) = (((2s ·s 𝑥) +s 1s ) /su (2s↑s𝑦)))
57		zseo 28345	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤs → (∃𝑐 ∈ ℤs 𝑏 = (2s ·s 𝑐) ∨ ∃𝑐 ∈ ℤs 𝑏 = ((2s ·s 𝑐) +s 1s )))
58		oveq1 7353	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑐 → (𝑎 /su (2s↑s𝑤)) = (𝑐 /su (2s↑s𝑤)))
59	58	eleq1d 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑐 → ((𝑎 /su (2s↑s𝑤)) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
60	58	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . 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 3197	. . . . . . . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . . . . . 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 28352	. . . . . . . . . . . . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = (((2s↑s𝑤) ·s 2s) ·s 𝑐))
76		expscl 28354	. . . . . . . . . . . . . . . . . . . . . . . 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 28306	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ ℤs → 𝑐 ∈ No )
80	79	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → 𝑐 ∈ No )
81	77, 78, 80	mulsassd 28106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s𝑤) ·s 2s) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
82	75, 81	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s(𝑤 +s 1s )) ·s 𝑐) = ((2s↑s𝑤) ·s (2s ·s 𝑐)))
83	82	oveq1d 7361	. . . . . . . . . . . . . . . . . . . 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 28259	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕ0s)
85	84	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕ0s)
86	80, 85	pw2divscan3d 28364	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s↑s(𝑤 +s 1s )) ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = 𝑐)
87	78, 80	mulscld 28074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (2s ·s 𝑐) ∈ No )
88		expscl 28354	. . . . . . . . . . . . . . . . . . . . . 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 28343	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2s ≠ 0s
91		expsne0 28359	. . . . . . . . . . . . . . . . . . . . . 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 28361	. . . . . . . . . . . . . . . . . . . . . 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 28142	. . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s↑s𝑤) ·s ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s )))) = 𝑐)
97	87, 85	pw2divscld 28362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ No )
98	80, 97, 72	pw2divsmuld 28363	. . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) = (𝑐 /su (2s↑s𝑤)))
101	100	eleq1d 2816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))) ∈ ℤs ↔ (𝑐 /su (2s↑s𝑤)) ∈ ℤs))
102	100	eqeq1d 2733	. . . . . . . . . . . . . . . . 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 3197	. . . . . . . . . . . . . . . 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 7353	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (2s ·s 𝑐) → (𝑏 /su (2s↑s(𝑤 +s 1s ))) = ((2s ·s 𝑐) /su (2s↑s(𝑤 +s 1s ))))
108	107	eleq1d 2816	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 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 3197	. . . . . . . . . . . . . 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 3133	. . . . . . . . . . 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 7354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑐 → (2s ·s 𝑝) = (2s ·s 𝑐))
115	114	oveq1d 7361	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑐 → ((2s ·s 𝑝) +s 1s ) = ((2s ·s 𝑐) +s 1s ))
116	115	oveq1d 7361	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑐 → (((2s ·s 𝑝) +s 1s ) /su (2s↑s𝑞)) = (((2s ·s 𝑐) +s 1s ) /su (2s↑s𝑞)))
117	116	eqeq2d 2742	. . . . . . . . . . . . . . . 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 7354	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = (𝑤 +s 1s ) → (2s↑s𝑞) = (2s↑s(𝑤 +s 1s )))
119	118	oveq2d 7362	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . 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 28296	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℕ0s → (𝑤 +s 1s ) ∈ ℕs)
123	122	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑐 ∈ ℤs) → (𝑤 +s 1s ) ∈ ℕs)
124		eqidd 2732	. . . . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . . . 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 7353	. . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 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 3197	. . . . . . . . . . . . . 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 3133	. . . . . . . . . . 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 3123	. . . . . . . 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 28261	. . . . . 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 3220	. . . . . 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 2819	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑏)) → (𝐴 ∈ ℤs ↔ (𝑎 /su (2s↑s𝑏)) ∈ ℤs))
144		eqeq1 2735	. . . . . 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 3197	. . . . 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 3174	. 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 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  (class class class)co 7346   No csur 27578   0_s c0s 27766   1_s c1s 27767   +_s cadds 27902   ·_s cmuls 28045   /_su cdivs 28126  ℕ_0scnn0s 28242  ℕ_scnns 28243  ℤ_sczs 28302  2_sc2s 28333  ↑_scexps 28335  ℤ_s[1/2]czs12 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-muls 28046  df-divs 28127  df-seqs 28214  df-n0s 28244  df-nns 28245  df-zs 28303  df-2s 28334  df-exps 28336  df-zs12 28338

This theorem is referenced by: (None)