Theorem zseo 28332

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 28295	. 2 ⊢ (𝑁 ∈ ℤs ↔ ∃𝑦 ∈ ℕs ∃𝑧 ∈ ℕs 𝑁 = (𝑦 -s 𝑧))
2		nnn0s 28243	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕ0s)
3		n0seo 28331	. . . . . 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 28243	. . . . . 6 ⊢ (𝑧 ∈ ℕs → 𝑧 ∈ ℕ0s)
6		n0seo 28331	. . . . . 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 3201	. . . . . . . 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 28300	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ ℤs)
10	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ ℤs)
11		n0zs 28300	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ ℤs)
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ ℤs)
13	10, 12	zsubscld 28307	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ ℤs)
14		2sno 28329	. . . . . . . . . . . . . 14 ⊢ 2s ∈ No
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 2s ∈ No )
16		n0sno 28239	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℕ0s → 𝑤 ∈ No )
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑤 ∈ No )
18		n0sno 28239	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℕ0s → 𝑡 ∈ No )
19	18	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 𝑡 ∈ No )
20	15, 17, 19	subsdid 28084	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
21	20	eqcomd 2735	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) = (2s ·s (𝑤 -s 𝑡)))
22		oveq2 7361	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → (2s ·s 𝑥) = (2s ·s (𝑤 -s 𝑡)))
23	22	rspceeqv 3602	. . . . . . . . . . 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 7362	. . . . . . . . . . . 12 ⊢ ((𝑦 = (2s ·s 𝑤) ∧ 𝑧 = (2s ·s 𝑡)) → (𝑦 -s 𝑧) = ((2s ·s 𝑤) -s (2s ·s 𝑡)))
26	25	eqeq1d 2731	. . . . . . . . . . 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 3153	. . . . . . . . . 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 3171	. . . . . . . 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 3201	. . . . . . . 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 28061	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑤) ∈ No )
34		1sno 27759	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ No )
36	15, 19	mulscld 28061	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s 𝑡) ∈ No )
37	33, 35, 36	addsubsd 28009	. . . . . . . . . . . 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 7368	. . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s (2s ·s 𝑡)) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
40	22	oveq1d 7368	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 -s 𝑡) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s 1s ))
41	40	rspceeqv 3602	. . . . . . . . . . 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 7362	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 3153	. . . . . . . . . 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 3171	. . . . . . . 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 3201	. . . . . . . 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 28302	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → 1s ∈ ℤs)
53	13, 52	zsubscld 28307	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((𝑤 -s 𝑡) -s 1s ) ∈ ℤs)
54	13	znod 28294	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (𝑤 -s 𝑡) ∈ No )
55	15, 54, 35	subsdid 28084	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s ((𝑤 -s 𝑡) -s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )))
56	55	oveq1d 7368	. . . . . . . . . . . . 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 28039	. . . . . . . . . . . . . . . . 17 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
58	14, 57	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (2s ·s 1s ) = 2s
59	58	oveq2i 7364	. . . . . . . . . . . . . . 15 ⊢ ((2s ·s (𝑤 -s 𝑡)) -s (2s ·s 1s )) = ((2s ·s (𝑤 -s 𝑡)) -s 2s)
60	59	oveq1i 7363	. . . . . . . . . . . . . 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 28061	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (2s ·s (𝑤 -s 𝑡)) ∈ No )
62	61, 35, 15	addsubsd 28009	. . . . . . . . . . . . . . 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 28008	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 27989	. . . . . . . . . . . . . . . . . 18 ⊢ (( 1s ∈ No ∧ 2s ∈ No ) → ( 1s -s 2s) ∈ No )
68	34, 14, 67	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ ( 1s -s 2s) ∈ No
69		negnegs 27973	. . . . . . . . . . . . . . . . 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 28007	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ( -us ‘( 1s -s 2s)) = (2s -s 1s ))
74	73	mptru 1547	. . . . . . . . . . . . . . . . . 18 ⊢ ( -us ‘( 1s -s 2s)) = (2s -s 1s )
75		1p1e2s 28326	. . . . . . . . . . . . . . . . . . 19 ⊢ ( 1s +s 1s ) = 2s
76		subadds 27997	. . . . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . 17 ⊢ ( -us ‘( 1s -s 2s)) = 1s
80	79	fveq2i 6829	. . . . . . . . . . . . . . . 16 ⊢ ( -us ‘( -us ‘( 1s -s 2s))) = ( -us ‘ 1s )
81	70, 80	eqtr3i 2754	. . . . . . . . . . . . . . 15 ⊢ ( 1s -s 2s) = ( -us ‘ 1s )
82	81	oveq2i 7364	. . . . . . . . . . . . . 14 ⊢ ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s ))
83	61, 35	subsvald 27988	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) -s 1s ) = ((2s ·s (𝑤 -s 𝑡)) +s ( -us ‘ 1s )))
84	82, 83	eqtr4id 2783	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s (𝑤 -s 𝑡)) +s ( 1s -s 2s)) = ((2s ·s (𝑤 -s 𝑡)) -s 1s ))
85	20	oveq1d 7368	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 28021	. . . . . . . . . . . 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 2769	. . . . . . . . . . 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 7361	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → (2s ·s 𝑥) = (2s ·s ((𝑤 -s 𝑡) -s 1s )))
90	89	oveq1d 7368	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑤 -s 𝑡) -s 1s ) → ((2s ·s 𝑥) +s 1s ) = ((2s ·s ((𝑤 -s 𝑡) -s 1s )) +s 1s ))
91	90	rspceeqv 3602	. . . . . . . . . . 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 7362	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 3153	. . . . . . . . . 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 3171	. . . . . . . 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 3201	. . . . . . . 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 28027	. . . . . . . . . . . 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 27996	. . . . . . . . . . . . . . 15 ⊢ ( 1s ∈ No → ( 1s -s 1s ) = 0s )
103	34, 102	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( 1s -s 1s ) = 0s
104	103	oveq2i 7364	. . . . . . . . . . . . 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 27990	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → ((2s ·s 𝑤) -s (2s ·s 𝑡)) ∈ No )
106	105	addsridd 27895	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) -s (2s ·s 𝑡)) +s 0s ) = (2s ·s (𝑤 -s 𝑡)))
108	104, 107	eqtrid 2776	. . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ0s ∧ 𝑡 ∈ ℕ0s) → (((2s ·s 𝑤) +s 1s ) -s ((2s ·s 𝑡) +s 1s )) = (2s ·s (𝑤 -s 𝑡)))
110	22	rspceeqv 3602	. . . . . . . . . . 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 7362	. . . . . . . . . . . 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 2731	. . . . . . . . . . 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 3153	. . . . . . . . . 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 3171	. . . . . . . 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 2733	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = (2s ·s 𝑥) ↔ (𝑦 -s 𝑧) = (2s ·s 𝑥)))
122	121	rexbidv 3153	. . . . 5 ⊢ (𝑁 = (𝑦 -s 𝑧) → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℤs (𝑦 -s 𝑧) = (2s ·s 𝑥)))
123		eqeq1 2733	. . . . . 6 ⊢ (𝑁 = (𝑦 -s 𝑧) → (𝑁 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑦 -s 𝑧) = ((2s ·s 𝑥) +s 1s )))
124	123	rexbidv 3153	. . . . 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 3171	. 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 1540  ⊤wtru 1541   ∈ wcel 2109  ∃wrex 3053  ‘cfv 6486  (class class class)co 7353   No csur 27567   0_s c0s 27754   1_s c1s 27755   +_s cadds 27889   -u_s cnegs 27948   -_s csubs 27949   ·_s cmuls 28032  ℕ_0scnn0s 28229  ℕ_scnns 28230  ℤ_sczs 28289  2_sc2s 28320

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-nadd 8591  df-no 27570  df-slt 27571  df-bday 27572  df-sle 27673  df-sslt 27710  df-scut 27712  df-0s 27756  df-1s 27757  df-made 27775  df-old 27776  df-left 27778  df-right 27779  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27950  df-subs 27951  df-muls 28033  df-n0s 28231  df-nns 28232  df-zs 28290  df-2s 28321

This theorem is referenced by:  zs12zodd  28377  zs12bday  28379