Theorem gpgcubic 48998

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 49001), 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 2760	. . . 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 48966	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	5	biimp3a 1498	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩)
7		elpri 4608	. . . . . . 7 ⊢ (𝑥 ∈ {0, 1} → (𝑥 = 0 ∨ 𝑥 = 1))
8		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
9	8	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
10	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
11		c0ex 11227	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
13	11, 12	op1std 7997	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1st ‘𝑋) = 0)
14		gpgnbgr.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
15	2, 3, 4, 14	gpg3nbgrvtx0 48995	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
16	15	exp43 442	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 0 → (♯‘𝑈) = 3))))
17	16	3imp 1128	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 0 → (♯‘𝑈) = 3))
18	13, 17	syl5 35	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨0, 𝑦⟩ → (♯‘𝑈) = 3))
19	18	adantl 487	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨0, 𝑦⟩ → (♯‘𝑈) = 3))
20	10, 19	sylbid 243	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
21	20	ex 418	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
22		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
23	22	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
24	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
25		1ex 11230	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
26	25, 12	op1std 7997	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1st ‘𝑋) = 1)
27	2, 3, 4, 14	gpg3nbgrvtx1 48997	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
28	27	exp43 442	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 1 → (♯‘𝑈) = 3))))
29	28	3imp 1128	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 1 → (♯‘𝑈) = 3))
30	26, 29	syl5 35	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨1, 𝑦⟩ → (♯‘𝑈) = 3))
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨1, 𝑦⟩ → (♯‘𝑈) = 3))
32	24, 31	sylbid 243	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
33	32	ex 418	. . . . . . . 8 ⊢ (𝑥 = 1 → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
34	21, 33	jaoi 871	. . . . . . 7 ⊢ ((𝑥 = 0 ∨ 𝑥 = 1) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
35	7, 34	syl 18	. . . . . 6 ⊢ (𝑥 ∈ {0, 1} → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
36	35	impcom 413	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ {0, 1}) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
37	36	a1d 26	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ {0, 1}) → (𝑦 ∈ (0..^𝑁) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
38	37	expimpd 459	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^𝑁)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3)))
39	38	rexlimdvv 3218	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩ → (♯‘𝑈) = 3))
40	6, 39	mpd 16	1 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (♯‘𝑈) = 3)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {cpr 4586  ⟨cop 4590  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  0cc0 11127  1c1 11128   / cdiv 11898  2c2 12322  3c3 12323  ℤ_≥cuz 12890  ..^cfzo 13712  ⌈cceil 13855  ♯chash 14397  Vtxcvtx 29456   NeighbVtx cnbgr 29795   gPetersenGr cgpg 48959

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-xnn0 12605  df-z 12619  df-dec 12740  df-uz 12891  df-rp 13046  df-ico 13407  df-fz 13565  df-fzo 13713  df-fl 13856  df-ceil 13857  df-mod 13934  df-hash 14398  df-dvds 16346  df-struct 17242  df-slot 17277  df-ndx 17289  df-base 17305  df-edgf 29449  df-vtx 29458  df-iedg 29459  df-edg 29508  df-upgr 29542  df-umgr 29543  df-usgr 29614  df-nbgr 29796  df-gpg 48960

This theorem is used by:  gpgvtxdg3  49001