Theorem n0seo 28364

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 2740	. . . 4 ⊢ (𝑚 = 0s → (𝑚 = (2s ·s 𝑥) ↔ 0s = (2s ·s 𝑥)))
2	1	rexbidv 3165	. . 3 ⊢ (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)))
3		eqeq1 2740	. . . 4 ⊢ (𝑚 = 0s → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 0s = ((2s ·s 𝑥) +s 1s )))
4	3	rexbidv 3165	. . 3 ⊢ (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s )))
5	2, 4	orbi12d 918	. 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 2740	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 = (2s ·s 𝑥) ↔ 𝑛 = (2s ·s 𝑥)))
7	6	rexbidv 3165	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥)))
8		eqeq1 2740	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑛 = ((2s ·s 𝑥) +s 1s )))
9	8	rexbidv 3165	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )))
10	7, 9	orbi12d 918	. 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 2740	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑥)))
12	11	rexbidv 3165	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥)))
13		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2s ·s 𝑥) = (2s ·s 𝑦))
14	13	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑛 +s 1s ) = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑦)))
15	14	cbvrexvw 3225	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
16	12, 15	bitrdi 287	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
17		eqeq1 2740	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
18	17	rexbidv 3165	. . . 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 7425	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
20	19	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
21	20	cbvrexvw 3225	. . . 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 287	. . 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 918	. 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 2740	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑚 = (2s ·s 𝑥) ↔ 𝑁 = (2s ·s 𝑥)))
25	24	rexbidv 3165	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥)))
26		eqeq1 2740	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑁 = ((2s ·s 𝑥) +s 1s )))
27	26	rexbidv 3165	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
28	25, 27	orbi12d 918	. 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 28279	. . . 4 ⊢ 0s ∈ ℕ0s
30		2sno 28362	. . . . . 6 ⊢ 2s ∈ No
31		muls01 28072	. . . . . 6 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
32	30, 31	ax-mp 5	. . . . 5 ⊢ (2s ·s 0s ) = 0s
33	32	eqcomi 2745	. . . 4 ⊢ 0s = (2s ·s 0s )
34		oveq2 7418	. . . . 5 ⊢ (𝑥 = 0s → (2s ·s 𝑥) = (2s ·s 0s ))
35	34	rspceeqv 3629	. . . 4 ⊢ (( 0s ∈ ℕ0s ∧ 0s = (2s ·s 0s )) → ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥))
36	29, 33, 35	mp2an 692	. . 3 ⊢ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)
37	36	orci 865	. 2 ⊢ (∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s ))
38		eqid 2736	. . . . . . . 8 ⊢ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )
39		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (2s ·s 𝑦) = (2s ·s 𝑥))
40	39	oveq1d 7425	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((2s ·s 𝑦) +s 1s ) = ((2s ·s 𝑥) +s 1s ))
41	40	rspceeqv 3629	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
43		oveq1 7417	. . . . . . . . 9 ⊢ (𝑛 = (2s ·s 𝑥) → (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ))
44	43	eqeq1d 2738	. . . . . . . 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 3165	. . . . . . 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 247	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → (𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
47	46	rexlimiv 3135	. . . . 5 ⊢ (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
48		peano2n0s 28280	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
49		1p1e2s 28359	. . . . . . . . . . 11 ⊢ ( 1s +s 1s ) = 2s
50		mulsrid 28073	. . . . . . . . . . . 12 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
51	30, 50	ax-mp 5	. . . . . . . . . . 11 ⊢ (2s ·s 1s ) = 2s
52	49, 51	eqtr4i 2762	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
53	52	oveq2i 7421	. . . . . . . . 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		n0sno 28273	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
56	54, 55	mulscld 28095	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0s → (2s ·s 𝑥) ∈ No )
57		1sno 27796	. . . . . . . . . . 11 ⊢ 1s ∈ No
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0s → 1s ∈ No )
59	56, 58, 58	addsassd 27970	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = ((2s ·s 𝑥) +s ( 1s +s 1s )))
60	54, 55, 58	addsdid 28116	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (2s ·s (𝑥 +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s )))
61	53, 59, 60	3eqtr4a 2797	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s )))
62		oveq2 7418	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 +s 1s ) → (2s ·s 𝑦) = (2s ·s (𝑥 +s 1s )))
63	62	rspceeqv 3629	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦))
65		oveq1 7417	. . . . . . . . 9 ⊢ (𝑛 = ((2s ·s 𝑥) +s 1s ) → (𝑛 +s 1s ) = (((2s ·s 𝑥) +s 1s ) +s 1s ))
66	65	eqeq1d 2738	. . . . . . . 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 3165	. . . . . . 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 247	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → (𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
69	68	rexlimiv 3135	. . . . 5 ⊢ (∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
70	47, 69	orim12i 908	. . . 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 871	. . 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 28282	1 ⊢ (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  (class class class)co 7410   No csur 27608   0_s c0s 27791   1_s c1s 27792   +_s cadds 27923   ·_s cmuls 28066  ℕ_0scnn0s 28263  2_sc2s 28353

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-1s 27794  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec 27902  df-norec2 27913  df-adds 27924  df-negs 27984  df-subs 27985  df-muls 28067  df-n0s 28265  df-nns 28266  df-2s 28354

This theorem is referenced by:  zseo  28365