Theorem gpgvtx0 48057

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 2729	. . . 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 48051	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	3	fveq2i 6829	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
7	4, 6	eqtri 2752	. . . . . . 7 ⊢ 𝑉 = (Vtx‘(𝑁 gPetersenGr 𝐾))
8		eluz3nn 12809	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
9	2, 1	gpgvtx 48047	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
10	8, 9	sylan 580	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
11	10	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
12	7, 11	eqtrid 2776	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → 𝑉 = ({0, 1} × (0..^𝑁)))
13		c0ex 11128	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
14	13	prid1 4716	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 0 ∈ {0, 1})
16		elfzoelz 13581	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
17	16	peano2zd 12602	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℤ)
18		zmodfzo 13817	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
19	17, 8, 18	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
20	15, 19	opelxpd 5662	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
21		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ (0..^𝑁))
22	15, 21	opelxpd 5662	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)))
23		1zzd 12525	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℤ)
24	16, 23	zsubcld 12604	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 − 1) ∈ ℤ)
25		zmodfzo 13817	. . . . . . . . . . . 12 ⊢ (((𝑦 − 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
26	24, 8, 25	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
27	15, 26	opelxpd 5662	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
28	20, 22, 27	3jca 1128	. . . . . . . . 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))))
29	28	ad2ant2rl 749	. . . . . . . 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))))
30	29	adantr 480	. . . . . . 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))))
31		eleq2 2817	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
32		eleq2 2817	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, 𝑦⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁))))
33		eleq2 2817	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
34	31, 32, 33	3anbi123d 1438	. . . . . . . 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)))))
35	34	adantl 481	. . . . . . 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)))))
36	30, 35	mpbird 257	. . . . . 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))
37	12, 36	mpdan 687	. . . . 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))
38		vex 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
39		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
40	38, 39	op2ndd 7942	. . . . . 6 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
41		oveq1 7360	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) + 1) = (𝑦 + 1))
42	41	oveq1d 7368	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) + 1) mod 𝑁) = ((𝑦 + 1) mod 𝑁))
43	42	opeq2d 4834	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
44	43	eleq1d 2813	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉))
45		opeq2 4828	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (2nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
46	45	eleq1d 2813	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ 𝑉))
47		oveq1 7360	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) − 1) = (𝑦 − 1))
48	47	oveq1d 7368	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) − 1) mod 𝑁) = ((𝑦 − 1) mod 𝑁))
49	48	opeq2d 4834	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((𝑦 − 1) mod 𝑁)⟩)
50	49	eleq1d 2813	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
51	44, 46, 50	3anbi123d 1438	. . . . . 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)))
52	40, 51	syl 17	. . . . 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)))
53	37, 52	syl5ibrcom 247	. . . 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)))
54	53	rexlimdvva 3186	. . 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)))
55	5, 54	sylbid 240	. 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)))
56	55	imp 406	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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {cpr 4581  ⟨cop 4585   × cxp 5621  ‘cfv 6486  (class class class)co 7353  2^nd c2nd 7930  0cc0 11028  1c1 11029   + caddc 11031   − cmin 11366   / cdiv 11796  ℕcn 12147  2c2 12202  3c3 12203  ℤcz 12490  ℤ_≥cuz 12754  ..^cfzo 13576  ⌈cceil 13714   mod cmo 13792  Vtxcvtx 28960   gPetersenGr cgpg 48044

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  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

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-nel 3030  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-op 4586  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-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-oadd 8399  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-xnn0 12477  df-z 12491  df-dec 12611  df-uz 12755  df-rp 12913  df-fz 13430  df-fzo 13577  df-fl 13715  df-mod 13793  df-hash 14257  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17140  df-edgf 28953  df-vtx 28962  df-gpg 48045

This theorem is referenced by:  gpgnbgrvtx0  48078  gpgnbgrvtx1  48079