Theorem gpgcubic 48555

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 48558), 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 2736	. . . 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 48523	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩))
6	5	biimp3a 1472	. 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^𝑁)𝑋 = ⟨𝑥, 𝑦⟩)
7		elpri 4591	. . . . . . 7 ⊢ (𝑥 ∈ {0, 1} → (𝑥 = 0 ∨ 𝑥 = 1))
8		opeq1 4816	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
9	8	eqeq2d 2747	. . . . . . . . . . 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 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
13	11, 12	op1std 7952	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1st ‘𝑋) = 0)
14		gpgnbgr.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
15	2, 3, 4, 14	gpg3nbgrvtx0 48552	. . . . . . . . . . . . . 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 1111	. . . . . . . . . . . 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 4816	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
23	22	eqeq2d 2747	. . . . . . . . . . 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 11140	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
26	25, 12	op1std 7952	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1st ‘𝑋) = 1)
27	2, 3, 4, 14	gpg3nbgrvtx1 48554	. . . . . . . . . . . . . 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 1111	. . . . . . . . . . . 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 858	. . . . . . 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 3193	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {cpr 4569  ⟨cop 4573  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  0cc0 11038  1c1 11039   / cdiv 11807  2c2 12236  3c3 12237  ℤ_≥cuz 12788  ..^cfzo 13608  ⌈cceil 13750  ♯chash 14292  Vtxcvtx 29065   NeighbVtx cnbgr 29401   gPetersenGr cgpg 48516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-ico 13304  df-fz 13462  df-fzo 13609  df-fl 13751  df-ceil 13752  df-mod 13829  df-hash 14293  df-dvds 16222  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-edgf 29058  df-vtx 29067  df-iedg 29068  df-edg 29117  df-upgr 29151  df-umgr 29152  df-usgr 29220  df-nbgr 29402  df-gpg 48517

This theorem is referenced by:  gpgvtxdg3  48558