Theorem gpg3nbgrvtx1 48661

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 48658	. . 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 6866	. 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 11136	. . . . . . 7 ⊢ 1 ≠ 0
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 1 ≠ 0)
9	8	orcd 884	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
10		1ex 11170	. . . . . 6 ⊢ 1 ∈ V
11		ovex 7424	. . . . . 6 ⊢ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ∈ V
12	10, 11	opthne 5447	. . . . 5 ⊢ (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩ ↔ (1 ≠ 0 ∨ (((2nd ‘𝑋) + 𝐾) mod 𝑁) ≠ (2nd ‘𝑋)))
13	9, 12	sylibr 236	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩)
14		0ne1 12283	. . . . . . 7 ⊢ 0 ≠ 1
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 0 ≠ 1)
16	15	orcd 884	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
17		c0ex 11167	. . . . . 6 ⊢ 0 ∈ V
18		fvex 6875	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5447	. . . . 5 ⊢ (⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ↔ (0 ≠ 1 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 𝐾) mod 𝑁)))
20	16, 19	sylibr 236	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
21		simpll 776	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑁 ∈ (ℤ≥‘3))
22		eqid 2761	. . . . . . . . 9 ⊢ (0..^𝑁) = (0..^𝑁)
23	22, 1, 2, 3	gpgvtxel2 48631	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
24	23	adantrr 727	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		simplr 778	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝐾 ∈ 𝐽)
26	1, 22	modmknepk 47923	. . . . . . 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 1389	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (((2nd ‘𝑋) − 𝐾) mod 𝑁) ≠ (((2nd ‘𝑋) + 𝐾) mod 𝑁))
28	27	olcd 885	. . . . 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 7424	. . . . . 6 ⊢ (((2nd ‘𝑋) − 𝐾) mod 𝑁) ∈ V
30	10, 29	opthne 5447	. . . . 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 236	. . . 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 1140	. . 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 5428	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V
34		opex 5428	. . . 4 ⊢ ⟨0, (2nd ‘𝑋)⟩ ∈ V
35		opex 5428	. . . 4 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V
36		hashtpg 14492	. . . 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 1481	. . 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 220	. 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 2796	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  {ctp 4583  ⟨cop 4585  ‘cfv 6516  (class class class)co 7391  1^st c1st 7963  2^nd c2nd 7964  0cc0 11067  1c1 11068   + caddc 11070   − cmin 11408   / cdiv 11838  2c2 12266  3c3 12267  ℤ_≥cuz 12833  ..^cfzo 13653  ⌈cceil 13795   mod cmo 13873  ♯chash 14337  Vtxcvtx 29154   NeighbVtx cnbgr 29490   gPetersenGr cgpg 48623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-pre-sup 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-oadd 8435  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-inf 9383  df-dju 9853  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-xnn0 12549  df-z 12563  df-dec 12683  df-uz 12834  df-rp 12988  df-ico 13349  df-fz 13507  df-fzo 13654  df-fl 13796  df-ceil 13797  df-mod 13874  df-hash 14338  df-dvds 16278  df-struct 17174  df-slot 17209  df-ndx 17221  df-base 17237  df-edgf 29147  df-vtx 29156  df-iedg 29157  df-edg 29206  df-upgr 29240  df-umgr 29241  df-usgr 29309  df-nbgr 29491  df-gpg 48624

This theorem is referenced by:  gpgcubic  48662  gpg5nbgr3star  48664