Theorem gpgcubic 48110

Description: Every generalized Petersen graph is a cubic graph, i.e., it is a 3-regular graph, i.e., every vertex has degree 3 (see gpgvtxdg3 48113), i.e., every vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-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
gpgcubic	⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (♯‘𝑈) = 3)

Proof of Theorem gpgcubic

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (0..^𝑁) = (0..^𝑁)
2		gpgnbgr.j	. . . 4 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3		gpgnbgr.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
4		gpgnbgr.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	5	biimp3a 1471	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩)
7		elpri 4595	. . . . . . 7 ⊢ (𝑥 ∈ {0, 1} → (𝑥 = 0 ∨ 𝑥 = 1))
8		opeq1 4820	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
9	8	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
11		c0ex 11101	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
13	11, 12	op1std 7926	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1st ‘𝑋) = 0)
14		gpgnbgr.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
15	2, 3, 4, 14	gpg3nbgrvtx0 48107	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
16	15	exp43 436	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 0 → (♯‘𝑈) = 3))))
17	16	3imp 1110	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 0 → (♯‘𝑈) = 3))
18	13, 17	syl5 34	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨0, 𝑦⟩ → (♯‘𝑈) = 3))
19	18	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨0, 𝑦⟩ → (♯‘𝑈) = 3))
20	10, 19	sylbid 240	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
21	20	ex 412	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
22		opeq1 4820	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
23	22	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
25		1ex 11103	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
26	25, 12	op1std 7926	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1st ‘𝑋) = 1)
27	2, 3, 4, 14	gpg3nbgrvtx1 48109	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
28	27	exp43 436	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 1 → (♯‘𝑈) = 3))))
29	28	3imp 1110	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 1 → (♯‘𝑈) = 3))
30	26, 29	syl5 34	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨1, 𝑦⟩ → (♯‘𝑈) = 3))
31	30	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨1, 𝑦⟩ → (♯‘𝑈) = 3))
32	24, 31	sylbid 240	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
33	32	ex 412	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
34	21, 33	jaoi 857	. . . . . . 7 ⊢ ((𝑥 = 0 ∨ 𝑥 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
35	7, 34	syl 17	. . . . . 6 ⊢ (𝑥 ∈ {0, 1} → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
36	35	impcom 407	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ {0, 1}) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
37	36	a1d 25	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ {0, 1}) → (𝑦 ∈ (0..^𝑁) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
38	37	expimpd 453	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
39	38	rexlimdvv 3188	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
40	6, 39	mpd 15	1 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {cpr 4573  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  0cc0 11001  1c1 11002   / cdiv 11769  2c2 12175  3c3 12176  ℤ_≥cuz 12727  ..^cfzo 13549  ⌈cceil 13690  ♯chash 14232  Vtxcvtx 28969   NeighbVtx cnbgr 29305   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-tp 4576  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-2o 8381  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-ico 13246  df-fz 13403  df-fzo 13550  df-fl 13691  df-ceil 13692  df-mod 13769  df-hash 14233  df-dvds 16159  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-edgf 28962  df-vtx 28971  df-iedg 28972  df-edg 29021  df-upgr 29055  df-umgr 29056  df-usgr 29124  df-nbgr 29306  df-gpg 48072

This theorem is referenced by:  gpgvtxdg3  48113