Theorem gpgvtx0 48702

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 2769	. . . 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 48696	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	3	fveq2i 6882	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
7	4, 6	eqtri 2792	. . . . . . 7 ⊢ 𝑉 = (Vtx‘(𝑁 gPetersenGr 𝐾))
8		eluz3nn 12909	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
9	2, 1	gpgvtx 48692	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
10	8, 9	sylan 591	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
11	10	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
12	7, 11	eqtrid 2816	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → 𝑉 = ({0, 1} × (0..^𝑁)))
13		c0ex 11196	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
14	13	prid1 4730	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 0 ∈ {0, 1})
16		elfzoelz 13683	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
17	16	peano2zd 12699	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℤ)
18		zmodfzo 13923	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
19	17, 8, 18	syl2anr 608	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 + 1) mod 𝑁) ∈ (0..^𝑁))
20	15, 19	opelxpd 5698	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
21		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ (0..^𝑁))
22	15, 21	opelxpd 5698	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)))
23		1zzd 12621	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℤ)
24	16, 23	zsubcld 12701	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 − 1) ∈ ℤ)
25		zmodfzo 13923	. . . . . . . . . . . 12 ⊢ (((𝑦 − 1) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
26	24, 8, 25	syl2anr 608	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝑦 − 1) mod 𝑁) ∈ (0..^𝑁))
27	15, 26	opelxpd 5698	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁)))
28	20, 22, 27	3jca 1144	. . . . . . . . 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 761	. . . . . . . 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 485	. . . . . . 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 2858	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
32		eleq2 2858	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, 𝑦⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁))))
33		eleq2 2858	. . . . . . . . 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 1462	. . . . . . . 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 486	. . . . . . 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 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))
37	12, 36	mpdan 699	. . . . 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 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
39		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
40	38, 39	op2ndd 7993	. . . . . 6 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
41		oveq1 7415	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) + 1) = (𝑦 + 1))
42	41	oveq1d 7423	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) + 1) mod 𝑁) = ((𝑦 + 1) mod 𝑁))
43	42	opeq2d 4846	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
44	43	eleq1d 2854	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉))
45		opeq2 4840	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (2nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
46	45	eleq1d 2854	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ 𝑉))
47		oveq1 7415	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) − 1) = (𝑦 − 1))
48	47	oveq1d 7423	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) − 1) mod 𝑁) = ((𝑦 − 1) mod 𝑁))
49	48	opeq2d 4846	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((𝑦 − 1) mod 𝑁)⟩)
50	49	eleq1d 2854	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 − 1) mod 𝑁)⟩ ∈ 𝑉))
51	44, 46, 50	3anbi123d 1462	. . . . . 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 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)))
53	37, 52	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)))
54	53	rexlimdvva 3228	. . 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 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)))
56	55	imp 411	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {cpr 4593  ⟨cop 4597   × cxp 5657  ‘cfv 6534  (class class class)co 7408  2^nd c2nd 7981  0cc0 11096  1c1 11097   + caddc 11099   − cmin 11437   / cdiv 11867  ℕcn 12229  2c2 12291  3c3 12292  ℤcz 12587  ℤ_≥cuz 12858  ..^cfzo 13678  ⌈cceil 13820   mod cmo 13898  Vtxcvtx 29283   gPetersenGr cgpg 48689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  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 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-dju 9883  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-xnn0 12574  df-z 12588  df-dec 12708  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-fl 13821  df-mod 13899  df-hash 14363  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-edgf 29276  df-vtx 29285  df-gpg 48690

This theorem is referenced by:  gpgnbgrvtx0  48723  gpgnbgrvtx1  48724