Theorem gpgvtx0 49424

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 2763	. . . 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 49418	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	3	fveq2i 6919	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
7	4, 6	eqtri 2786	. . . . . . 7 ⊢ 𝑉 = (Vtx‘(𝑁 gPetersenGr 𝐾))
8		eluz3nn 13056	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
9	2, 1	gpgvtx 49414	. . . . . . . . 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 2810	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → 𝑉 = ({0, 1} × (0..^𝑁)))
13		0elpr01 11340	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
14	13	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 0 ∈ {0, 1})
15		elfzoelz 13833	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
16	15	peano2zd 12846	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℤ)
17		zmodfzo 14074	. . . . . . . . . . . 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 5697	. . . . . . . . . 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 5697	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)))
22		1zzd 12767	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℤ)
23	15, 22	zsubcld 12848	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 − 1) ∈ ℤ)
24		zmodfzo 14074	. . . . . . . . . . . 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 5697	. . . . . . . . . 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 2853	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
31		eleq2 2853	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, 𝑦⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁))))
32		eleq2 2853	. . . . . . . . 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 1465	. . . . . . . 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 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
38		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
39	37, 38	op2ndd 8046	. . . . . 6 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
40		oveq1 7458	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) + 1) = (𝑦 + 1))
41	40	oveq1d 7466	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) + 1) mod 𝑁) = ((𝑦 + 1) mod 𝑁))
42	41	opeq2d 4852	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
43	42	eleq1d 2849	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉))
44		opeq2 4846	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (2nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
45	44	eleq1d 2849	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ 𝑉))
46		oveq1 7458	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) − 1) = (𝑦 − 1))
47	46	oveq1d 7466	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) − 1) mod 𝑁) = ((𝑦 − 1) mod 𝑁))
48	47	opeq2d 4852	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((𝑦 − 1) mod 𝑁)⟩)
49	48	eleq1d 2849	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
50	43, 45, 49	3anbi123d 1465	. . . . . 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 3223	. . 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 2147  ∃wrex 3090  {cpr 4590  ⟨cop 4840   × cxp 5656  ‘cfv 6566  (class class class)co 7451  2^nd c2nd 8034  0cc0 11239  1c1 11240   + caddc 11242   − cmin 11580   / cdiv 12013  ℕcn 12375  2c2 12437  3c3 12438  ℤcz 12733  ℤ_≥cuz 13005  ..^cfzo 13828  ⌈cceil 13971   mod cmo 14049  Vtxcvtx 29631   gPetersenGr cgpg 49411

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-mod 14050  df-hash 14516  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-edgf 29624  df-vtx 29633  df-gpg 49412

This theorem is used by:  gpgnbgrvtx0  49445  gpgnbgrvtx1  49446