Theorem zseo 28439

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 28401	. 2 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑦 ∈ ℕs ∃𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧))
2		nnn0s 28344	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
3		n0seo 28438	. . . . . 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 28344	. . . . . 6 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ ℕ0s)
6		n0seo 28438	. . . . . 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 3212	. . . . . . . 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 28406	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ ℤs)
10	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ ℤs)
11		n0zs 28406	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ ℤs)
12	11	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ ℤs)
13	10, 12	zsubscld 28413	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ ℤs)
14		2no 28436	. . . . . . . . . . . . . 14 ⊢ 2s ∈ No
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 2s ∈ No )
16		n0no 28340	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ No )
17	16	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ No )
18		n0no 28340	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ No )
19	18	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ No )
20	15, 17, 19	subsdid 28175	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
21	20	eqcomd 2746	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡)))
22		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → (2s ·s 𝑥) = (2s ·s (𝑤 -s 𝑡)))
23	22	rspceeqv 3590	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s 𝑥))
25		oveq12 7372	. . . . . . . . . . . 12 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
26	25	eqeq1d 2742	. . . . . . . . . . 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 3164	. . . . . . . . . 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 248	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
29	28	rexlimivv 3182	. . . . . . . 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 236	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥))
31	30	orcd 879	. . . . . 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 3212	. . . . . . . 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 28152	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑤) ∈ No )
34		1no 27827	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ No )
36	15, 19	mulscld 28152	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑡) ∈ No )
37	33, 35, 36	addsubsd 28099	. . . . . . . . . . . 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 7378	. . . . . . . . . . . 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 2775	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
40	22	oveq1d 7378	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
41	40	rspceeqv 3590	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
43		oveq12 7372	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 3164	. . . . . . . . . 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 248	. . . . . . . . 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 3182	. . . . . . . 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 236	. . . . . . 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 880	. . . . . 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 3212	. . . . . . . 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 28408	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ ℤs)
53	13, 52	zsubscld 28413	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑤 -s 𝑡) -s 1s ) ∈ ℤs)
54	13	znod 28400	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ No )
55	15, 54, 35	subsdid 28175	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s ((𝑤 -s 𝑡) -s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )))
56	55	oveq1d 7378	. . . . . . . . . . . . 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 28130	. . . . . . . . . . . . . . . . 17 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
58	14, 57	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (2s ·s 1s ) = 2s
59	58	oveq2i 7374	. . . . . . . . . . . . . . 15 ⊢ ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s 2s)
60	59	oveq1i 7373	. . . . . . . . . . . . . 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 28152	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) ∈ No )
62	61, 35, 15	addsubsd 28099	. . . . . . . . . . . . . . 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 28098	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 28079	. . . . . . . . . . . . . . . . . 18 ⊢ (( 1s ∈ No ∧ 2s ∈ No ) → ( 1s -s 2s) ∈ No )
68	34, 14, 67	mp2an 698	. . . . . . . . . . . . . . . . 17 ⊢ ( 1s -s 2s) ∈ No
69		negnegs 28061	. . . . . . . . . . . . . . . . 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 28097	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ( -us ‘( 1s -s 2s)) = (2s -s 1s ))
74	73	mptru 1554	. . . . . . . . . . . . . . . . . 18 ⊢ ( -us ‘( 1s -s 2s)) = (2s -s 1s )
75		1p1e2s 28433	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 1s +s 1s ) = 2s
76		subadds 28087	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s))
77	14, 34, 34, 76	mp3an 1469	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s -s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s)
78	75, 77	mpbir 232	. . . . . . . . . . . . . . . . . 18 ⊢ (2s -s 1s ) = 1s
79	74, 78	eqtri 2763	. . . . . . . . . . . . . . . . 17 ⊢ ( -us ‘( 1s -s 2s)) = 1s
80	79	fveq2i 6837	. . . . . . . . . . . . . . . 16 ⊢ ( -us ‘( -us ‘( 1s -s 2s))) = ( -us ‘ 1s )
81	70, 80	eqtr3i 2765	. . . . . . . . . . . . . . 15 ⊢ ( 1s -s 2s) = ( -us ‘ 1s )
82	81	oveq2i 7374	. . . . . . . . . . . . . 14 ⊢ ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s ))
83	61, 35	subsvald 28078	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s )))
84	82, 83	eqtr4id 2794	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) -s 1s ))
85	20	oveq1d 7378	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 28111	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 7371	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → (2s ·s 𝑥) = (2s ·s ((𝑤 -s 𝑡) -s 1s )))
90	89	oveq1d 7378	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
91	90	rspceeqv 3590	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) -s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
93		oveq12 7372	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 3164	. . . . . . . . . 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 248	. . . . . . . . 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 3182	. . . . . . . 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 236	. . . . . . 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 880	. . . . . 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 3212	. . . . . . . 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 28118	. . . . . . . . . . . 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 28086	. . . . . . . . . . . . . . 15 ⊢ ( 1s ∈ No → ( 1s -s 1s ) = 0s )
103	34, 102	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( 1s -s 1s ) = 0s
104	103	oveq2i 7374	. . . . . . . . . . . . 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 28080	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) ∈ No )
106	105	addsridd 27982	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = (2s ·s (𝑤 -s 𝑡)))
108	104, 107	eqtrid 2787	. . . . . . . . . . . 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 2775	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡)))
110	22	rspceeqv 3590	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
112		oveq12 7372	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 3164	. . . . . . . . . 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 248	. . . . . . . . 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 3182	. . . . . . . 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 236	. . . . . . 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 879	. . . . . 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 1043	. . . . 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 602	. . . 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 2744	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = (2s ·s 𝑥) ↔ (𝑦 -s 𝑧) = (2s ·s 𝑥)))
122	121	rexbidv 3164	. . . . 5 ⊢ (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
123		eqeq1 2744	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
124	123	rexbidv 3164	. . . . 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 924	. . . 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 248	. . 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 3182	. 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 218	1 ⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  ∃wrex 3064  ‘cfv 6492  (class class class)co 7363   No csur 27628   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976   -u_s cnegs 28036   -_s csubs 28037   ·_s cmuls 28123  ℕ_0scn0s 28329  ℕ_scnns 28330  ℤ_sczs 28395  2_sc2s 28427

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-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-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428

This theorem is referenced by:  z12zsodd  28499