Theorem n0seo 28808

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

Assertion

Ref	Expression
n0seo	⊢ (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))

Distinct variable group:   𝑥,𝑁

Proof of Theorem n0seo

Dummy variables 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2764	. . . 4 ⊢ (𝑚 = 0s → (𝑚 = (2s ·s 𝑥) ↔ 0s = (2s ·s 𝑥)))
2	1	rexbidv 3186	. . 3 ⊢ (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)))
3		eqeq1 2764	. . . 4 ⊢ (𝑚 = 0s → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 0s = ((2s ·s 𝑥) +s 1s )))
4	3	rexbidv 3186	. . 3 ⊢ (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s )))
5	2, 4	orbi12d 932	. 2 ⊢ (𝑚 = 0s → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s ))))
6		eqeq1 2764	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 = (2s ·s 𝑥) ↔ 𝑛 = (2s ·s 𝑥)))
7	6	rexbidv 3186	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥)))
8		eqeq1 2764	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑛 = ((2s ·s 𝑥) +s 1s )))
9	8	rexbidv 3186	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )))
10	7, 9	orbi12d 932	. 2 ⊢ (𝑚 = 𝑛 → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s ))))
11		eqeq1 2764	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑥)))
12	11	rexbidv 3186	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥)))
13		oveq2 7428	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2s ·s 𝑥) = (2s ·s 𝑦))
14	13	eqeq2d 2771	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑛 +s 1s ) = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑦)))
15	14	cbvrexvw 3241	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
16	12, 15	bitrdi 290	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
17		eqeq1 2764	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
18	17	rexbidv 3186	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
19	13	oveq1d 7435	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
20	19	eqeq2d 2771	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
21	20	cbvrexvw 3241	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
22	18, 21	bitrdi 290	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
23	16, 22	orbi12d 932	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))))
24		eqeq1 2764	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑚 = (2s ·s 𝑥) ↔ 𝑁 = (2s ·s 𝑥)))
25	24	rexbidv 3186	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥)))
26		eqeq1 2764	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑁 = ((2s ·s 𝑥) +s 1s )))
27	26	rexbidv 3186	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
28	25, 27	orbi12d 932	. 2 ⊢ (𝑚 = 𝑁 → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s ))))
29		0n0s 28716	. . . 4 ⊢ 0s ∈ ℕ0s
30		2no 28806	. . . . . 6 ⊢ 2s ∈ No
31		muls01 28499	. . . . . 6 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
32	30, 31	ax-mp 5	. . . . 5 ⊢ (2s ·s 0s ) = 0s
33	32	eqcomi 2769	. . . 4 ⊢ 0s = (2s ·s 0s )
34		oveq2 7428	. . . . 5 ⊢ (𝑥 = 0s → (2s ·s 𝑥) = (2s ·s 0s ))
35	34	rspceeqv 3598	. . . 4 ⊢ (( 0s ∈ ℕ0s ∧ 0s = (2s ·s 0s )) → ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥))
36	29, 33, 35	mp2an 705	. . 3 ⊢ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)
37	36	orci 879	. 2 ⊢ (∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s ))
38		eqid 2760	. . . . . . . 8 ⊢ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )
39		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (2s ·s 𝑦) = (2s ·s 𝑥))
40	39	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((2s ·s 𝑦) +s 1s ) = ((2s ·s 𝑥) +s 1s ))
41	40	rspceeqv 3598	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )) → ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
42	38, 41	mpan2 704	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
43		oveq1 7427	. . . . . . . . 9 ⊢ (𝑛 = (2s ·s 𝑥) → (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ))
44	43	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑛 = (2s ·s 𝑥) → ((𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ↔ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s )))
45	44	rexbidv 3186	. . . . . . 7 ⊢ (𝑛 = (2s ·s 𝑥) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s )))
46	42, 45	syl5ibrcom 250	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → (𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
47	46	rexlimiv 3156	. . . . 5 ⊢ (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
48		peano2n0s 28717	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
49		1p1e2s 28803	. . . . . . . . . . 11 ⊢ ( 1s +s 1s ) = 2s
50		mulsrid 28500	. . . . . . . . . . . 12 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
51	30, 50	ax-mp 5	. . . . . . . . . . 11 ⊢ (2s ·s 1s ) = 2s
52	49, 51	eqtr4i 2786	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
53	52	oveq2i 7431	. . . . . . . . 9 ⊢ ((2s ·s 𝑥) +s ( 1s +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s ))
54	30	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0s → 2s ∈ No)
55		n0no 28710	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No)
56	54, 55	mulscld 28522	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0s → (2s ·s 𝑥) ∈ No)
57		1no 28197	. . . . . . . . . . 11 ⊢ 1s ∈ No
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0s → 1s ∈ No)
59	56, 58, 58	addsassd 28393	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = ((2s ·s 𝑥) +s ( 1s +s 1s )))
60	54, 55, 58	addsdid 28543	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (2s ·s (𝑥 +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s )))
61	53, 59, 60	3eqtr4a 2821	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s )))
62		oveq2 7428	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 +s 1s ) → (2s ·s 𝑦) = (2s ·s (𝑥 +s 1s )))
63	62	rspceeqv 3598	. . . . . . . 8 ⊢ (((𝑥 +s 1s ) ∈ ℕ0s ∧ (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s ))) → ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦))
64	48, 61, 63	syl2anc 596	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦))
65		oveq1 7427	. . . . . . . . 9 ⊢ (𝑛 = ((2s ·s 𝑥) +s 1s ) → (𝑛 +s 1s ) = (((2s ·s 𝑥) +s 1s ) +s 1s ))
66	65	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑛 = ((2s ·s 𝑥) +s 1s ) → ((𝑛 +s 1s ) = (2s ·s 𝑦) ↔ (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦)))
67	66	rexbidv 3186	. . . . . . 7 ⊢ (𝑛 = ((2s ·s 𝑥) +s 1s ) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ↔ ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦)))
68	64, 67	syl5ibrcom 250	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → (𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
69	68	rexlimiv 3156	. . . . 5 ⊢ (∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
70	47, 69	orim12i 922	. . . 4 ⊢ ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
71	70	orcomd 885	. . 3 ⊢ ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
72	71	a1i 11	. 2 ⊢ (𝑛 ∈ ℕ0s → ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))))
73	5, 10, 23, 28, 37, 72	n0sind 28720	1 ⊢ (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  (class class class)co 7420  Nocsur 27998   0_s c0s 28192   1_s c1s 28193   +_s cadds 28346   ·_s cmuls 28493  ℕ_0scn0s 28699  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-2s 28798

This theorem is used by:  zseo  28809  bdaypw2n0bndlem  28850