Theorem gpg3nbgrvtx1 48727

Description: In a generalized Petersen graph 𝐺, every inside vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 22-Nov-2025.)

Hypotheses

Ref	Expression
gpgnbgr.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgnbgr.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)
gpgnbgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)

Assertion

Ref	Expression
gpg3nbgrvtx1	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)

Proof of Theorem gpg3nbgrvtx1

Step	Hyp	Ref	Expression
1		gpgnbgr.j	. . . 4 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
2		gpgnbgr.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
3		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4		gpgnbgr.u	. . . 4 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
5	1, 2, 3, 4	gpgnbgrvtx1 48724	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
6	5	fveq2d 6883	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = (♯‘{⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}))
7		ax-1ne0 11165	. . . . . . 7 ⊢ 1 ≠ 0
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 1 ≠ 0)
9	8	orcd 886	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
10		1ex 11199	. . . . . 6 ⊢ 1 ∈ V
11		ovex 7441	. . . . . 6 ⊢ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ∈ V
12	10, 11	opthne 5462	. . . . 5 ⊢ (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ↔ (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
13	9, 12	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩)
14		0ne1 12308	. . . . . . 7 ⊢ 0 ≠ 1
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 0 ≠ 1)
16	15	orcd 886	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
17		c0ex 11196	. . . . . 6 ⊢ 0 ∈ V
18		fvex 6892	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5462	. . . . 5 ⊢ (⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
20	16, 19	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
21		simpll 778	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑁 ∈ (ℤ≥‘3))
22		eqid 2769	. . . . . . . . 9 ⊢ (0..^𝑁) = (0..^𝑁)
23	22, 1, 2, 3	gpgvtxel2 48697	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
24	23	adantrr 729	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		simplr 780	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝐾 ∈ 𝐽)
26	1, 22	modmknepk 47989	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (2nd ‘𝑋) ∈ (0..^𝑁) ∧ 𝐾 ∈ 𝐽) → (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁))
27	21, 24, 25, 26	syl3anc 1396	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁))
28	27	olcd 887	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1 ≠ 1 ∨ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁)))
29		ovex 7441	. . . . . 6 ⊢ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ∈ V
30	10, 29	opthne 5462	. . . . 5 ⊢ (⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ↔ (1 ≠ 1 ∨ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁)))
31	28, 30	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
32	13, 20, 31	3jca 1144	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ∧ ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩))
33		opex 5443	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V
34		opex 5443	. . . 4 ⊢ ⟨0, (2nd ‘𝑋)⟩ ∈ V
35		opex 5443	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V
36		hashtpg 14518	. . . 4 ⊢ ((⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ V ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V) → ((⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ∧ ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩) ↔ (♯‘{⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}) = 3))
37	33, 34, 35, 36	mp3an 1487	. . 3 ⊢ ((⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ∧ ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩) ↔ (♯‘{⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}) = 3)
38	32, 37	sylib 221	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘{⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}) = 3)
39	6, 38	eqtrd 2804	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  {ctp 4595  ⟨cop 4597  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  0cc0 11096  1c1 11097   + caddc 11099   − cmin 11437   / cdiv 11867  2c2 12291  3c3 12292  ℤ_≥cuz 12858  ..^cfzo 13678  ⌈cceil 13820   mod cmo 13898  ♯chash 14362  Vtxcvtx 29283   NeighbVtx cnbgr 29619   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-tp 4596  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-2o 8450  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-ico 13374  df-fz 13532  df-fzo 13679  df-fl 13821  df-ceil 13822  df-mod 13899  df-hash 14363  df-dvds 16307  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-edgf 29276  df-vtx 29285  df-iedg 29286  df-edg 29335  df-upgr 29369  df-umgr 29370  df-usgr 29438  df-nbgr 29620  df-gpg 48690

This theorem is referenced by:  gpgcubic  48728  gpg5nbgr3star  48730