Theorem gpg3nbgrvtx1 48240

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 48237	. . 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 6835	. 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 11086	. . . . . . 7 ⊢ 1 ≠ 0
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 1 ≠ 0)
9	8	orcd 873	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
10		1ex 11119	. . . . . 6 ⊢ 1 ∈ V
11		ovex 7388	. . . . . 6 ⊢ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ∈ V
12	10, 11	opthne 5427	. . . . 5 ⊢ (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ↔ (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
13	9, 12	sylibr 234	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩)
14		0ne1 12207	. . . . . . 7 ⊢ 0 ≠ 1
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 0 ≠ 1)
16	15	orcd 873	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
17		c0ex 11117	. . . . . 6 ⊢ 0 ∈ V
18		fvex 6844	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5427	. . . . 5 ⊢ (⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
20	16, 19	sylibr 234	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
21		simpll 766	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑁 ∈ (ℤ≥‘3))
22		eqid 2733	. . . . . . . . 9 ⊢ (0..^𝑁) = (0..^𝑁)
23	22, 1, 2, 3	gpgvtxel2 48210	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
24	23	adantrr 717	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		simplr 768	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝐾 ∈ 𝐽)
26	1, 22	modmknepk 47524	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁))
28	27	olcd 874	. . . . 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 7388	. . . . . 6 ⊢ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ∈ V
30	10, 29	opthne 5427	. . . . 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 234	. . . 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 1128	. . 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 5409	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V
34		opex 5409	. . . 4 ⊢ ⟨0, (2nd ‘𝑋)⟩ ∈ V
35		opex 5409	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V
36		hashtpg 14399	. . . 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 1463	. . 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 218	. 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 2768	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437  {ctp 4581  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  0cc0 11017  1c1 11018   + caddc 11020   − cmin 11355   / cdiv 11785  2c2 12191  3c3 12192  ℤ_≥cuz 12742  ..^cfzo 13561  ⌈cceil 13702   mod cmo 13780  ♯chash 14244  Vtxcvtx 28995   NeighbVtx cnbgr 29331   gPetersenGr cgpg 48202

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094  ax-pre-sup 11095

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9337  df-inf 9338  df-dju 9805  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-div 11786  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-xnn0 12466  df-z 12480  df-dec 12599  df-uz 12743  df-rp 12897  df-ico 13258  df-fz 13415  df-fzo 13562  df-fl 13703  df-ceil 13704  df-mod 13781  df-hash 14245  df-dvds 16171  df-struct 17065  df-slot 17100  df-ndx 17112  df-base 17128  df-edgf 28988  df-vtx 28997  df-iedg 28998  df-edg 29047  df-upgr 29081  df-umgr 29082  df-usgr 29150  df-nbgr 29332  df-gpg 48203

This theorem is referenced by:  gpgcubic  48241  gpg5nbgr3star  48243