Theorem zseo 28809

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 28771	. 2 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑦 ∈ ℕs ∃𝑧 ∈ ℕs 𝑁 = (𝑦 −s 𝑧))
2		nnn0s 28714	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
3		n0seo 28808	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
4	2, 3	syl 18	. . . . 5 ⊢ (𝑦 ∈ ℕs → (∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∨ ∃𝑤 ∈ ℕ0s 𝑦 = ((2s ·s 𝑤) +s 1s )))
5		nnn0s 28714	. . . . . 6 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ ℕ0s)
6		n0seo 28808	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
7	5, 6	syl 18	. . . . 5 ⊢ (𝑧 ∈ ℕs → (∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡) ∨ ∃𝑡 ∈ ℕ0s 𝑧 = ((2s ·s 𝑡) +s 1s )))
8		reeanv 3234	. . . . . . . 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 28776	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ ℤs)
10	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ ℤs)
11		n0zs 28776	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ ℤs)
12	11	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ ℤs)
13	10, 12	zsubscld 28783	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 −s 𝑡) ∈ ℤs)
14		2no 28806	. . . . . . . . . . . . . 14 ⊢ 2s ∈ No
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 2s ∈ No)
16		n0no 28710	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ No)
17	16	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ No)
18		n0no 28710	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ No)
19	18	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ No)
20	15, 17, 19	subsdid 28545	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 −s 𝑡)) = ((2s ·s 𝑤) −s (2s ·s 𝑡)))
21	20	eqcomd 2766	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) −s (2s ·s 𝑡)) = (2s ·s (𝑤 −s 𝑡)))
22		oveq2 7428	. . . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) −s (2s ·s 𝑡)) = (2s ·s 𝑥))
25		oveq12 7429	. . . . . . . . . . . 12 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 −s 𝑧) = ((2s ·s 𝑤) −s (2s ·s 𝑡)))
26	25	eqeq1d 2762	. . . . . . . . . . 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 3186	. . . . . . . . . 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 250	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 −s 𝑧) = (2s ·s 𝑥)))
29	28	rexlimivv 3204	. . . . . . . 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 238	. . . . . . 7 ⊢ ((∃𝑤 ∈ ℕ0s 𝑦 = (2s ·s 𝑤) ∧ ∃𝑡 ∈ ℕ0s 𝑧 = (2s ·s 𝑡)) → ∃𝑥 ∈ ℤs (𝑦 −s 𝑧) = (2s ·s 𝑥))
31	30	orcd 887	. . . . . 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 3234	. . . . . . . 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 28522	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑤) ∈ No)
34		1no 28197	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ No)
36	15, 19	mulscld 28522	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑡) ∈ No)
37	33, 35, 36	addsubsd 28469	. . . . . . . . . . . 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 7435	. . . . . . . . . . . 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 2795	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) −s (2s ·s 𝑡)) = ((2s ·s (𝑤 −s 𝑡)) +s 1s ))
40	22	oveq1d 7435	. . . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) −s (2s ·s 𝑡)) = ((2s ·s 𝑥) +s 1s ))
43		oveq12 7429	. . . . . . . . . . . 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 2762	. . . . . . . . . . 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 3186	. . . . . . . . . 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 250	. . . . . . . . 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 3204	. . . . . . . 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 238	. . . . . . 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 888	. . . . . 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 3234	. . . . . . . 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 28778	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ ℤs)
53	13, 52	zsubscld 28783	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑤 −s 𝑡) −s 1s ) ∈ ℤs)
54	13	znod 28770	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 −s 𝑡) ∈ No)
55	15, 54, 35	subsdid 28545	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s ((𝑤 −s 𝑡) −s 1s )) = ((2s ·s (𝑤 −s 𝑡)) −s (2s ·s 1s )))
56	55	oveq1d 7435	. . . . . . . . . . . . 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 28500	. . . . . . . . . . . . . . . . 17 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
58	14, 57	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (2s ·s 1s ) = 2s
59	58	oveq2i 7431	. . . . . . . . . . . . . . 15 ⊢ ((2s ·s (𝑤 −s 𝑡)) −s (2s ·s 1s )) = ((2s ·s (𝑤 −s 𝑡)) −s 2s)
60	59	oveq1i 7430	. . . . . . . . . . . . . 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 28522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 −s 𝑡)) ∈ No)
62	61, 35, 15	addsubsd 28469	. . . . . . . . . . . . . . 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 28468	. . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 28449	. . . . . . . . . . . . . . . . . 18 ⊢ (( 1s ∈ No ∧ 2s ∈ No) → ( 1s −s 2s) ∈ No)
68	34, 14, 67	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ ( 1s −s 2s) ∈ No
69		negnegs 28431	. . . . . . . . . . . . . . . . 17 ⊢ (( 1s −s 2s) ∈ No → ( -s‘( -s‘( 1s −s 2s))) = ( 1s −s 2s))
70	68, 69	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ( -s‘( -s‘( 1s −s 2s))) = ( 1s −s 2s)
71	34	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 1s ∈ No)
72	14	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 2s ∈ No)
73	71, 72	negsubsdi2d 28467	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ( -s‘( 1s −s 2s)) = (2s −s 1s ))
74	73	mptru 1576	. . . . . . . . . . . . . . . . . 18 ⊢ ( -s‘( 1s −s 2s)) = (2s −s 1s )
75		1p1e2s 28803	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 1s +s 1s ) = 2s
76		subadds 28457	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No) → ((2s −s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s))
77	14, 34, 34, 76	mp3an 1490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s −s 1s ) = 1s ↔ ( 1s +s 1s ) = 2s)
78	75, 77	mpbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ (2s −s 1s ) = 1s
79	74, 78	eqtri 2783	. . . . . . . . . . . . . . . . 17 ⊢ ( -s‘( 1s −s 2s)) = 1s
80	79	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ ( -s‘( -s‘( 1s −s 2s))) = ( -s‘ 1s )
81	70, 80	eqtr3i 2785	. . . . . . . . . . . . . . 15 ⊢ ( 1s −s 2s) = ( -s‘ 1s )
82	81	oveq2i 7431	. . . . . . . . . . . . . 14 ⊢ ((2s ·s (𝑤 −s 𝑡)) +s ( 1s −s 2s)) = ((2s ·s (𝑤 −s 𝑡)) +s ( -s‘ 1s ))
83	61, 35	subsvald 28448	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 −s 𝑡)) −s 1s ) = ((2s ·s (𝑤 −s 𝑡)) +s ( -s‘ 1s )))
84	82, 83	eqtr4id 2814	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 −s 𝑡)) +s ( 1s −s 2s)) = ((2s ·s (𝑤 −s 𝑡)) −s 1s ))
85	20	oveq1d 7435	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 28481	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 7428	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑤 −s 𝑡) −s 1s ) → (2s ·s 𝑥) = (2s ·s ((𝑤 −s 𝑡) −s 1s )))
90	89	oveq1d 7435	. . . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ((2s ·s 𝑤) −s ((2s ·s 𝑡) +s 1s )) = ((2s ·s 𝑥) +s 1s ))
93		oveq12 7429	. . . . . . . . . . . 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 2762	. . . . . . . . . . 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 3186	. . . . . . . . . 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 250	. . . . . . . . 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 3204	. . . . . . . 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 238	. . . . . . 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 888	. . . . . 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 3234	. . . . . . . 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 28488	. . . . . . . . . . . 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 28456	. . . . . . . . . . . . . . 15 ⊢ ( 1s ∈ No → ( 1s −s 1s ) = 0s )
103	34, 102	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( 1s −s 1s ) = 0s
104	103	oveq2i 7431	. . . . . . . . . . . . 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 28450	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) −s (2s ·s 𝑡)) ∈ No)
106	105	addsridd 28352	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) −s (2s ·s 𝑡)) +s 0s ) = (2s ·s (𝑤 −s 𝑡)))
108	104, 107	eqtrid 2807	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ∃𝑥 ∈ ℤs (((2s ·s 𝑤) +s 1s ) −s ((2s ·s 𝑡) +s 1s )) = (2s ·s 𝑥))
112		oveq12 7429	. . . . . . . . . . . 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 2762	. . . . . . . . . . 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 3186	. . . . . . . . . 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 250	. . . . . . . . 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 3204	. . . . . . . 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 238	. . . . . . 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 887	. . . . . 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 1053	. . . . 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 608	. . . 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 2764	. . . . . 6 ⊢ (𝑁 = (𝑦 −s 𝑧) → (𝑁 = (2s ·s 𝑥) ↔ (𝑦 −s 𝑧) = (2s ·s 𝑥)))
122	121	rexbidv 3186	. . . . 5 ⊢ (𝑁 = (𝑦 −s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (𝑦 −s 𝑧) = (2s ·s 𝑥)))
123		eqeq1 2764	. . . . . 6 ⊢ (𝑁 = (𝑦 −s 𝑧) → (𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑦 −s 𝑧) = ((2s ·s 𝑥) +s 1s )))
124	123	rexbidv 3186	. . . . 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 932	. . . 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 250	. . 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 3204	. 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 220	1 ⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∃wrex 3086  ‘cfv 6538  (class class class)co 7420  Nocsur 27998   0_s c0s 28192   1_s c1s 28193   +_s cadds 28346   -_scnegs 28406   −_s csubs 28407   ·_s cmuls 28493  ℕ_0scn0s 28699  ℕ_scnns 28700  ℤ_sczs 28765  2_sc2s 28797

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-nadd 8676  df-no 28001  df-lts 28002  df-bday 28003  df-les 28103  df-slts 28145  df-cuts 28147  df-0s 28194  df-1s 28195  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec 28325  df-norec2 28336  df-adds 28347  df-negs 28408  df-subs 28409  df-muls 28494  df-n0s 28701  df-nns 28702  df-zs 28766  df-2s 28798

This theorem is used by:  z12zsodd  28869