Theorem gpgvtx0 48084

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 2731	. . . 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 48078	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	3	fveq2i 6820	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
7	4, 6	eqtri 2754	. . . . . . 7 ⊢ 𝑉 = (Vtx‘(𝑁 gPetersenGr 𝐾))
8		eluz3nn 12782	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
9	2, 1	gpgvtx 48074	. . . . . . . . 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 2778	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁))) → 𝑉 = ({0, 1} × (0..^𝑁)))
13		c0ex 11101	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
14	13	prid1 4710	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → 0 ∈ {0, 1})
16		elfzoelz 13554	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
17	16	peano2zd 12575	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 + 1) ∈ ℤ)
18		zmodfzo 13793	. . . . . . . . . . . 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 5650	. . . . . . . . . 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 5650	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑦 ∈ (0..^𝑁)) → ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁)))
23		1zzd 12498	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 1 ∈ ℤ)
24	16, 23	zsubcld 12577	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0..^𝑁) → (𝑦 − 1) ∈ ℤ)
25		zmodfzo 13793	. . . . . . . . . . . 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 5650	. . . . . . . . . 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 2820	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ ({0, 1} × (0..^𝑁))))
32		eleq2 2820	. . . . . . . . 9 ⊢ (𝑉 = ({0, 1} × (0..^𝑁)) → (⟨0, 𝑦⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ ({0, 1} × (0..^𝑁))))
33		eleq2 2820	. . . . . . . . 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 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
39		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
40	38, 39	op2ndd 7927	. . . . . 6 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑋) = 𝑦)
41		oveq1 7348	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) + 1) = (𝑦 + 1))
42	41	oveq1d 7356	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) + 1) mod 𝑁) = ((𝑦 + 1) mod 𝑁))
43	42	opeq2d 4827	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, ((𝑦 + 1) mod 𝑁)⟩)
44	43	eleq1d 2816	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ↔ ⟨0, ((𝑦 + 1) mod 𝑁)⟩ ∈ 𝑉))
45		opeq2 4821	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (2nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
46	45	eleq1d 2816	. . . . . . 7 ⊢ ((2nd ‘𝑋) = 𝑦 → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ↔ ⟨0, 𝑦⟩ ∈ 𝑉))
47		oveq1 7348	. . . . . . . . . 10 ⊢ ((2nd ‘𝑋) = 𝑦 → ((2nd ‘𝑋) − 1) = (𝑦 − 1))
48	47	oveq1d 7356	. . . . . . . . 9 ⊢ ((2nd ‘𝑋) = 𝑦 → (((2nd ‘𝑋) − 1) mod 𝑁) = ((𝑦 − 1) mod 𝑁))
49	48	opeq2d 4827	. . . . . . . 8 ⊢ ((2nd ‘𝑋) = 𝑦 → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, ((𝑦 − 1) mod 𝑁)⟩)
50	49	eleq1d 2816	. . . . . . 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 3189	. . 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 1541   ∈ wcel 2111  ∃wrex 3056  {cpr 4573  ⟨cop 4577   × cxp 5609  ‘cfv 6476  (class class class)co 7341  2^nd c2nd 7915  0cc0 11001  1c1 11002   + caddc 11004   − cmin 11339   / cdiv 11769  ℕcn 12120  2c2 12175  3c3 12176  ℤcz 12463  ℤ_≥cuz 12727  ..^cfzo 13549  ⌈cceil 13690   mod cmo 13768  Vtxcvtx 28969   gPetersenGr cgpg 48071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-dju 9789  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-xnn0 12450  df-z 12464  df-dec 12584  df-uz 12728  df-rp 12886  df-fz 13403  df-fzo 13550  df-fl 13691  df-mod 13769  df-hash 14233  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-edgf 28962  df-vtx 28971  df-gpg 48072

This theorem is referenced by:  gpgnbgrvtx0  48105  gpgnbgrvtx1  48106