Theorem gpg3nbgrvtx1 48876

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 48873	. . 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 6889	. 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 11180	. . . . . . 7 ⊢ 1 ≠ 0
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 1 ≠ 0)
9	8	orcd 887	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
10		1ex 11214	. . . . . 6 ⊢ 1 ∈ V
11		ovex 7449	. . . . . 6 ⊢ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ∈ V
12	10, 11	opthne 5466	. . . . 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 12323	. . . . . . 7 ⊢ 0 ≠ 1
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 0 ≠ 1)
16	15	orcd 887	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
17		c0ex 11211	. . . . . 6 ⊢ 0 ∈ V
18		fvex 6898	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5466	. . . . 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 779	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑁 ∈ (ℤ≥‘3))
22		eqid 2765	. . . . . . . . 9 ⊢ (0..^𝑁) = (0..^𝑁)
23	22, 1, 2, 3	gpgvtxel2 48846	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
24	23	adantrr 730	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		simplr 781	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝐾 ∈ 𝐽)
26	1, 22	modmknepk 48138	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁))
28	27	olcd 888	. . . . 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 7449	. . . . . 6 ⊢ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ∈ V
30	10, 29	opthne 5466	. . . . 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 1146	. . 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 5447	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V
34		opex 5447	. . . 4 ⊢ ⟨0, (2nd ‘𝑋)⟩ ∈ V
35		opex 5447	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V
36		hashtpg 14536	. . . 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 1490	. . 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 2800	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  {ctp 4595  ⟨cop 4597  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  2^nd c2nd 7987  0cc0 11111  1c1 11112   + caddc 11114   − cmin 11452   / cdiv 11882  2c2 12306  3c3 12307  ℤ_≥cuz 12874  ..^cfzo 13695  ⌈cceil 13838   mod cmo 13916  ♯chash 14380  Vtxcvtx 29377   NeighbVtx cnbgr 29716   gPetersenGr cgpg 48838

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-inf 9406  df-dju 9899  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-xnn0 12589  df-z 12603  df-dec 12724  df-uz 12875  df-rp 13029  df-ico 13390  df-fz 13548  df-fzo 13696  df-fl 13839  df-ceil 13840  df-mod 13917  df-hash 14381  df-dvds 16329  df-struct 17225  df-slot 17260  df-ndx 17272  df-base 17288  df-edgf 29370  df-vtx 29379  df-iedg 29380  df-edg 29429  df-upgr 29463  df-umgr 29464  df-usgr 29535  df-nbgr 29717  df-gpg 48839

This theorem is used by:  gpgcubic  48877  gpg5nbgr3star  48879