Theorem gpgvtx0 48969

Description: The outside vertices in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.)

Hypotheses

Ref	Expression
gpgvtx0.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgvtx0.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgvtx0.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
gpgvtx0	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))

Proof of Theorem gpgvtx0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (0..^𝑁) = (0..^𝑁)
2		gpgvtx0.j	. . . 4 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3		gpgvtx0.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
4		gpgvtx0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
5	1, 2, 3, 4	gpgvtxel 48963	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	3	fveq2i 6881	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
7	4, 6	eqtri 2783	. . . . . . 7 ⊢ 𝑉 = (Vtx‘(𝑁 gPetersenGr 𝐾))
8		eluz3nn 12938	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
9	2, 1	gpgvtx 48959	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
10	8, 9	sylan 592	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
11	10	adantr 486	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
12	7, 11	eqtrid 2807	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → 𝑉 = ({0, 1} × (0..^𝑁)))
13		0elpr01 11225	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
14	13	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 0 ∈ {0, 1})
15		elfzoelz 13714	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
16	15	peano2zd 12728	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℤ)
17		zmodfzo 13955	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
18	16, 8, 17	syl2anr 609	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
19	14, 18	opelxpd 5694	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
20		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ (0..^𝑁))
21	14, 20	opelxpd 5694	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)))
22		1zzd 12649	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℤ)
23	15, 22	zsubcld 12730	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 − 1) ∈ ℤ)
24		zmodfzo 13955	. . . . . . . . . . . 12 ⊢ (((𝑦 − 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
25	23, 8, 24	syl2anr 609	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
26	14, 25	opelxpd 5694	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
27	19, 21, 26	3jca 1146	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
28	27	ad2ant2rl 762	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
29	28	adantr 486	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) ∧ 𝑉 = ({0, 1} × (0..^𝑁))) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
30		eleq2 2849	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
31		eleq2 2849	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, 𝑦⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁))))
32		eleq2 2849	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
33	30, 31, 32	3anbi123d 1464	. . . . . . . 8 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → ((⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉) ↔ (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))))
34	33	adantl 487	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) ∧ 𝑉 = ({0, 1} × (0..^𝑁))) → ((⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉) ↔ (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)) ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))))
35	29, 34	mpbird 260	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) ∧ 𝑉 = ({0, 1} × (0..^𝑁))) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
36	12, 35	mpdan 700	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
37		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
38		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
39	37, 38	op2ndd 7997	. . . . . 6 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
40		oveq1 7420	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) + 1) = (𝑦 + 1))
41	40	oveq1d 7428	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) + 1) mod 𝑁) = ((𝑦 + 1) mod 𝑁))
42	41	opeq2d 4840	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
43	42	eleq1d 2845	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉))
44		opeq2 4834	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (2nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
45	44	eleq1d 2845	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ 𝑉))
46		oveq1 7420	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) − 1) = (𝑦 − 1))
47	46	oveq1d 7428	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) − 1) mod 𝑁) = ((𝑦 − 1) mod 𝑁))
48	47	opeq2d 4840	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((𝑦 − 1) mod 𝑁)⟩)
49	48	eleq1d 2845	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
50	43, 45, 49	3anbi123d 1464	. . . . . 6 ⊢ ((2nd ‘𝑋) = 𝑦 → ((⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉) ↔ (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉)))
51	39, 50	syl 18	. . . . 5 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉) ↔ (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, 𝑦⟩ ∈ 𝑉 ∧ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉)))
52	36, 51	syl5ibrcom 250	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (𝑋 = ⟨𝑥, 𝑦⟩ → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)))
53	52	rexlimdvva 3219	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩ → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)))
54	5, 53	sylbid 243	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)))
55	54	imp 412	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {cpr 4586  ⟨cop 4590   × cxp 5653  ‘cfv 6533  (class class class)co 7413  2^nd c2nd 7985  0cc0 11124  1c1 11125   + caddc 11127   − cmin 11465   / cdiv 11895  ℕcn 12257  2c2 12319  3c3 12320  ℤcz 12615  ℤ_≥cuz 12887  ..^cfzo 13709  ⌈cceil 13852   mod cmo 13930  Vtxcvtx 29453   gPetersenGr cgpg 48956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12737  df-uz 12888  df-rp 13043  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-hash 14395  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-edgf 29446  df-vtx 29455  df-gpg 48957

This theorem is used by:  gpgnbgrvtx0  48990  gpgnbgrvtx1  48991