Theorem gpg3nbgrvtx0 49143

Description: In a generalized Petersen graph 𝐺, every outside vertex has exactly three (different) neighbors. (Contributed by AV, 30-Aug-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
gpg3nbgrvtx0	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)

Proof of Theorem gpg3nbgrvtx0

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	gpgnbgrvtx0 49141	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
6	5	fveq2d 6887	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = (♯‘{⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}))
7		0ne1 12407	. . . . . . 7 ⊢ 0 ≠ 1
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 0 ≠ 1)
9	8	orcd 887	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (0 ≠ 1 ∨ (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (2nd ‘𝑋)))
10		c0ex 11293	. . . . . 6 ⊢ 0 ∈ V
11		ovex 7451	. . . . . 6 ⊢ (((2nd ‘𝑋) + 1) mod 𝑁) ∈ V
12	10, 11	opthne 5451	. . . . 5 ⊢ (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩ ↔ (0 ≠ 1 ∨ (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (2nd ‘𝑋)))
13	9, 12	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
14		ax-1ne0 11262	. . . . . . 7 ⊢ 1 ≠ 0
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ≠ 0)
16	15	orcd 887	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (1 ≠ 0 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 1) mod 𝑁)))
17		1ex 11296	. . . . . 6 ⊢ 1 ∈ V
18		fvex 6896	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5451	. . . . 5 ⊢ (⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ↔ (1 ≠ 0 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 1) mod 𝑁)))
20	16, 19	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)
21		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
22	21	anim2i 629	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
23		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^𝑁)
24	23, 1, 2, 3	gpgvtxel2 49115	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		elfzoelz 13786	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑋) ∈ (0..^𝑁) → (2nd ‘𝑋) ∈ ℤ)
26	22, 24, 25	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℤ)
27	26	zcnd 12797	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℂ)
28		1cnd 11295	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℂ)
29		2cnd 12414	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℂ)
30	27, 28, 29	subadd23d 11684	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + (2 − 1)))
31		2m1e1 12460	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
32	31	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2 − 1) = 1)
33	32	oveq2d 7434	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + (2 − 1)) = ((2nd ‘𝑋) + 1))
34	30, 33	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + 1))
35	34	eqcomd 2767	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + 1) = (((2nd ‘𝑋) − 1) + 2))
36	35	oveq1d 7433	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
37		1zzd 12720	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℤ)
38	26, 37	zsubcld 12801	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℤ)
39	38	zred 12796	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℝ)
40		2re 12410	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℝ)
42		eluz3nn 13009	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
43	42	nnrpd 13155	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℝ+)
44	43	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ ℝ+)
45		modaddabs 14044	. . . . . . . . . . 11 ⊢ ((((2nd ‘𝑋) − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
46	39, 41, 44, 45	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
47	46	eqcomd 2767	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((((2nd ‘𝑋) − 1) + 2) mod 𝑁) = (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁))
48	36, 47	eqtrd 2796	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) = (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁))
49	42	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ ℕ)
50	38, 49	zmodcld 14025	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0)
51		modlt 14013	. . . . . . . . . . 11 ⊢ ((((2nd ‘𝑋) − 1) ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
52	39, 44, 51	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
53	50, 52	jca 521	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0 ∧ (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁))
54		2nn0 12616	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℕ0)
56		eluz2 12964	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁))
57	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 ∈ ℝ)
58		3re 12416	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ∈ ℝ)
60		zre 12690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
61	60	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 𝑁 ∈ ℝ)
62		2lt3 12509	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 3
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 3)
64		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ≤ 𝑁)
65	57, 59, 61, 63, 64	ltletrd 11463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
66	65	3adant1 1148	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
67	56, 66	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁)
68		elfzo0 13828	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (0..^𝑁) ↔ (2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁))
69	55, 42, 67, 68	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ (0..^𝑁))
70		zmodidfzoimp 14034	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0..^𝑁) → (2 mod 𝑁) = 2)
71	69, 70	syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) = 2)
72		2nn 12409	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
73	71, 72	eqeltrdi 2869	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) ∈ ℕ)
74	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℝ)
75		modlt 14013	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (2 mod 𝑁) < 𝑁)
76	74, 43, 75	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) < 𝑁)
77	73, 76	jca 521	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
78	77	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
79		addmodne 48389	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ((((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0 ∧ (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁) ∧ ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁)) → (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
80	49, 53, 78, 79	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((((2nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
81	48, 80	eqnetrd 3023	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
82	81	necomd 3011	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ≠ (((2nd ‘𝑋) + 1) mod 𝑁))
83	82	olcd 888	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (0 ≠ 0 ∨ (((2nd ‘𝑋) − 1) mod 𝑁) ≠ (((2nd ‘𝑋) + 1) mod 𝑁)))
84		ovex 7451	. . . . . 6 ⊢ (((2nd ‘𝑋) − 1) mod 𝑁) ∈ V
85	10, 84	opthne 5451	. . . . 5 ⊢ (⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ (((2nd ‘𝑋) − 1) mod 𝑁) ≠ (((2nd ‘𝑋) + 1) mod 𝑁)))
86	83, 85	sylibr 237	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
87	13, 20, 86	3jca 1146	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩ ∧ ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩))
88		opex 5432	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
89		opex 5432	. . . 4 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
90		opex 5432	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
91		hashtpg 14623	. . . 4 ⊢ ((⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V ∧ ⟨1, (2nd ‘𝑋)⟩ ∈ V ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V) → ((⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩ ∧ ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩) ↔ (♯‘{⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3))
92	88, 89, 90, 91	mp3an 1490	. . 3 ⊢ ((⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩ ∧ ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩) ↔ (♯‘{⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3)
93	87, 92	sylib 221	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘{⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3)
94	6, 93	eqtrd 2796	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451  {ctp 4588  ⟨cop 4590   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   < clt 11336   ≤ cle 11337   − cmin 11534   / cdiv 11966  ℕcn 12328  2c2 12390  3c3 12391  ℕ₀cn0 12599  ℤcz 12686  ℤ_≥cuz 12958  ℝ⁺crp 13113  ..^cfzo 13781  ⌈cceil 13924   mod cmo 14002  ♯chash 14467  Vtxcvtx 29567   NeighbVtx cnbgr 29906   gPetersenGr cgpg 49107

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-ceil 13926  df-mod 14003  df-hash 14468  df-dvds 16416  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-edgf 29560  df-vtx 29569  df-iedg 29570  df-edg 29619  df-upgr 29653  df-umgr 29654  df-usgr 29725  df-nbgr 29907  df-gpg 49108

This theorem is used by:  gpgcubic  49146  gpg5nbgrvtx03star  49147