Theorem zseo 28338

Description: A surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)

Assertion

Ref	Expression
zseo	⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))

Distinct variable group:   𝑥,𝑁

Proof of Theorem zseo

Dummy variables 𝑦 𝑧 𝑤 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elzs 28301	. 2 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑦 ∈ ℕs ∃𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧))
2		nnn0s 28249	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
3		n0seo 28337	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
4	2, 3	syl 17	. . . . 5 ⊢ (𝑦 ∈ ℕs → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
5		nnn0s 28249	. . . . . 6 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ ℕ0s)
6		n0seo 28337	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑧 ∈ ℕs → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
8		reeanv 3202	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)))
9		n0zs 28306	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ ℤs)
10	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ ℤs)
11		n0zs 28306	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ ℤs)
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ ℤs)
13	10, 12	zsubscld 28313	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ ℤs)
14		2sno 28335	. . . . . . . . . . . . . 14 ⊢ 2s ∈ No
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 2s ∈ No )
16		n0sno 28245	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ No )
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ No )
18		n0sno 28245	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ No )
19	18	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ No )
20	15, 17, 19	subsdid 28090	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
21	20	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡)))
22		oveq2 7349	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → (2s ·s 𝑥) = (2s ·s (𝑤 -s 𝑡)))
23	22	rspceeqv 3598	. . . . . . . . . . 11 ⊢ (((𝑤 -s 𝑡) ∈ ℤs ∧ ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡))) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥))
24	13, 21, 23	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥))
25		oveq12 7350	. . . . . . . . . . . 12 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
26	25	eqeq1d 2732	. . . . . . . . . . 11 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ((𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥)))
27	26	rexbidv 3154	. . . . . . . . . 10 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥)))
28	24, 27	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
29	28	rexlimivv 3172	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
30	8, 29	sylbir 235	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
31	30	orcd 873	. . . . . 6 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
32		reeanv 3202	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)))
33	15, 17	mulscld 28067	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑤) ∈ No )
34		1sno 27764	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ No )
36	15, 19	mulscld 28067	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑡) ∈ No )
37	33, 35, 36	addsubsd 28015	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 1s ))
38	21	oveq1d 7356	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
39	37, 38	eqtrd 2765	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
40	22	oveq1d 7356	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
41	40	rspceeqv 3598	. . . . . . . . . . 11 ⊢ (((𝑤 -s 𝑡) ∈ ℤs ∧ (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s )) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
42	13, 39, 41	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
43		oveq12 7350	. . . . . . . . . . . 12 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)))
44	43	eqeq1d 2732	. . . . . . . . . . 11 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ((𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s )))
45	44	rexbidv 3154	. . . . . . . . . 10 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s )))
46	42, 45	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
47	46	rexlimivv 3172	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
48	32, 47	sylbir 235	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
49	48	olcd 874	. . . . . 6 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
50		reeanv 3202	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
51		1zs 28308	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ ℤs)
53	13, 52	zsubscld 28313	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑤 -s 𝑡) -s 1s ) ∈ ℤs)
54	13	znod 28300	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ No )
55	15, 54, 35	subsdid 28090	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s ((𝑤 -s 𝑡) -s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )))
56	55	oveq1d 7356	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ) = (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ))
57		mulsrid 28045	. . . . . . . . . . . . . . . . 17 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
58	14, 57	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (2s ·s 1s ) = 2s
59	58	oveq2i 7352	. . . . . . . . . . . . . . 15 ⊢ ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s 2s)
60	59	oveq1i 7351	. . . . . . . . . . . . . 14 ⊢ (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ) = (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s )
61	15, 54	mulscld 28067	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) ∈ No )
62	61, 35, 15	addsubsd 28015	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) +s 1s ) -s 2s) = (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s ))
63	61, 35, 15	addsubsassd 28014	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) +s 1s ) -s 2s) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
64	62, 63	eqtr3d 2767	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) -s 2s) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
65	60, 64	eqtrid 2777	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
66	56, 65	eqtrd 2765	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)))
67		subscl 27995	. . . . . . . . . . . . . . . . . 18 ⊢ (( 1s ∈ No ∧ 2s ∈ No ) → ( 1s -s 2s) ∈ No )
68	34, 14, 67	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ ( 1s -s 2s) ∈ No
69		negnegs 27979	. . . . . . . . . . . . . . . . 17 ⊢ (( 1s -s 2s) ∈ No → ( -us ‘( -us ‘( 1s -s 2s))) = ( 1s -s 2s))
70	68, 69	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ( -us ‘( -us ‘( 1s -s 2s))) = ( 1s -s 2s)
71	34	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 1s ∈ No )
72	14	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 2s ∈ No )
73	71, 72	negsubsdi2d 28013	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ( -us ‘( 1s -s 2s)) = (2s -s 1s ))
74	73	mptru 1548	. . . . . . . . . . . . . . . . . 18 ⊢ ( -us ‘( 1s -s 2s)) = (2s -s 1s )
75		1p1e2s 28332	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 1s +s 1s ) = 2s
76		subadds 28003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s))
77	14, 34, 34, 76	mp3an 1463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s)
78	75, 77	mpbir 231	. . . . . . . . . . . . . . . . . 18 ⊢ (2s -s 1s ) = 1s
79	74, 78	eqtri 2753	. . . . . . . . . . . . . . . . 17 ⊢ ( -us ‘( 1s -s 2s)) = 1s
80	79	fveq2i 6820	. . . . . . . . . . . . . . . 16 ⊢ ( -us ‘( -us ‘( 1s -s 2s))) = ( -us ‘ 1s )
81	70, 80	eqtr3i 2755	. . . . . . . . . . . . . . 15 ⊢ ( 1s -s 2s) = ( -us ‘ 1s )
82	81	oveq2i 7352	. . . . . . . . . . . . . 14 ⊢ ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s ))
83	61, 35	subsvald 27994	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s )))
84	82, 83	eqtr4id 2784	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) -s 1s ))
85	20	oveq1d 7356	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ))
86	84, 85	eqtrd 2765	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ))
87	33, 36, 35	subsubs4d 28027	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) -s 1s ) = ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )))
88	66, 86, 87	3eqtrrd 2770	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
89		oveq2 7349	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → (2s ·s 𝑥) = (2s ·s ((𝑤 -s 𝑡) -s 1s )))
90	89	oveq1d 7356	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
91	90	rspceeqv 3598	. . . . . . . . . . 11 ⊢ ((((𝑤 -s 𝑡) -s 1s ) ∈ ℤs ∧ ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s )) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
92	53, 88, 91	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
93		oveq12 7350	. . . . . . . . . . . 12 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )))
94	93	eqeq1d 2732	. . . . . . . . . . 11 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ((𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s )))
95	94	rexbidv 3154	. . . . . . . . . 10 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s )))
96	92, 95	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
97	96	rexlimivv 3172	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = (2s ·s 𝑤) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
98	50, 97	sylbir 235	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))
99	98	olcd 874	. . . . . 6 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
100		reeanv 3202	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) ↔ (∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
101	33, 35, 36, 35	addsubs4d 28033	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )))
102		subsid 28002	. . . . . . . . . . . . . . 15 ⊢ ( 1s ∈ No → ( 1s -s 1s ) = 0s )
103	34, 102	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( 1s -s 1s ) = 0s
104	103	oveq2i 7352	. . . . . . . . . . . . 13 ⊢ (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )) = (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s )
105	33, 36	subscld 27996	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) ∈ No )
106	105	addsridd 27901	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
107	106, 21	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = (2s ·s (𝑤 -s 𝑡)))
108	104, 107	eqtrid 2777	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s ( 1s -s 1s )) = (2s ·s (𝑤 -s 𝑡)))
109	101, 108	eqtrd 2765	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡)))
110	22	rspceeqv 3598	. . . . . . . . . . 11 ⊢ (((𝑤 -s 𝑡) ∈ ℤs ∧ (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡))) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
111	13, 109, 110	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
112		oveq12 7350	. . . . . . . . . . . 12 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (𝑦 -s 𝑧) = (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )))
113	112	eqeq1d 2732	. . . . . . . . . . 11 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ((𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥)))
114	113	rexbidv 3154	. . . . . . . . . 10 ⊢ ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥)))
115	111, 114	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
116	115	rexlimivv 3172	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℕ0s ∃𝑡 ∈ ℕ0s (𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
117	100, 116	sylbir 235	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
118	117	orcd 873	. . . . . 6 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s ) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
119	31, 49, 99, 118	ccase 1037	. . . . 5 ⊢ (((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )) ∧ (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s ))) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
120	4, 7, 119	syl2an 596	. . . 4 ⊢ ((𝑦 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
121		eqeq1 2734	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = (2s ·s 𝑥) ↔ (𝑦 -s 𝑧) = (2s ·s 𝑥)))
122	121	rexbidv 3154	. . . . 5 ⊢ (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
123		eqeq1 2734	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
124	123	rexbidv 3154	. . . . 5 ⊢ (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
125	122, 124	orbi12d 918	. . . 4 ⊢ (𝑁 = (𝑦 -s 𝑧) → ((∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s ))))
126	120, 125	syl5ibrcom 247	. . 3 ⊢ ((𝑦 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s ))))
127	126	rexlimivv 3172	. 2 ⊢ (∃𝑦 ∈ ℕs ∃𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
128	1, 127	sylbi 217	1 ⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ⊤wtru 1542   ∈ wcel 2110  ∃wrex 3054  ‘cfv 6477  (class class class)co 7341   No csur 27571   0_s c0s 27759   1_s c1s 27760   +_s cadds 27895   -u_s cnegs 27954   -_s csubs 27955   ·_s cmuls 28038  ℕ_0scnn0s 28235  ℕ_scnns 28236  ℤ_sczs 28295  2_sc2s 28326

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-ot 4583  df-uni 4858  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 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-nadd 8576  df-no 27574  df-slt 27575  df-bday 27576  df-sle 27677  df-sslt 27714  df-scut 27716  df-0s 27761  df-1s 27762  df-made 27781  df-old 27782  df-left 27784  df-right 27785  df-norec 27874  df-norec2 27885  df-adds 27896  df-negs 27956  df-subs 27957  df-muls 28039  df-n0s 28237  df-nns 28238  df-zs 28296  df-2s 28327

This theorem is referenced by:  zs12zodd  28385  zs12bday  28387