Theorem gpg3nbgrvtx0 48005

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 48003	. . 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 6908	. 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 12333	. . . . . . 7 ⊢ 0 ≠ 1
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 0 ≠ 1)
9	8	orcd 874	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (0 ≠ 1 ∨ (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (2nd ‘𝑋)))
10		c0ex 11251	. . . . . 6 ⊢ 0 ∈ V
11		ovex 7462	. . . . . 6 ⊢ (((2nd ‘𝑋) + 1) mod 𝑁) ∈ V
12	10, 11	opthne 5485	. . . . 5 ⊢ (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩ ↔ (0 ≠ 1 ∨ (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (2nd ‘𝑋)))
13	9, 12	sylibr 234	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
14		ax-1ne0 11220	. . . . . . 7 ⊢ 1 ≠ 0
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ≠ 0)
16	15	orcd 874	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (1 ≠ 0 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 1) mod 𝑁)))
17		1ex 11253	. . . . . 6 ⊢ 1 ∈ V
18		fvex 6917	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5485	. . . . 5 ⊢ (⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ↔ (1 ≠ 0 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 1) mod 𝑁)))
20	16, 19	sylibr 234	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)
21		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
22	21	anim2i 617	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
23		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^𝑁)
24	23, 1, 2, 3	gpgvtxel2 47979	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		elfzoelz 13695	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑋) ∈ (0..^𝑁) → (2nd ‘𝑋) ∈ ℤ)
26	22, 24, 25	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℤ)
27	26	zcnd 12719	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℂ)
28		1cnd 11252	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℂ)
29		2cnd 12340	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℂ)
30	27, 28, 29	subadd23d 11638	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + (2 − 1)))
31		2m1e1 12388	. . . . . . . . . . . . . 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 7445	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + (2 − 1)) = ((2nd ‘𝑋) + 1))
34	30, 33	eqtrd 2776	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + 1))
35	34	eqcomd 2742	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + 1) = (((2nd ‘𝑋) − 1) + 2))
36	35	oveq1d 7444	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
37		1zzd 12644	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℤ)
38	26, 37	zsubcld 12723	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℤ)
39	38	zred 12718	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℝ)
40		2re 12336	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℝ)
42		eluzge3nn 12928	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
43	42	nnrpd 13071	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℝ+)
44	43	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ ℝ+)
45		modaddabs 13945	. . . . . . . . . . 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 1373	. . . . . . . . . 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 2742	. . . . . . . . 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 2776	. . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ ℕ)
50	38, 49	zmodcld 13928	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0)
51		modlt 13916	. . . . . . . . . . 11 ⊢ ((((2nd ‘𝑋) − 1) ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
52	39, 44, 51	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
53	50, 52	jca 511	. . . . . . . . 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 12539	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℕ0)
56		eluz2 12880	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁))
57	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 ∈ ℝ)
58		3re 12342	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ∈ ℝ)
60		zre 12613	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
61	60	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 𝑁 ∈ ℝ)
62		2lt3 12434	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 3
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 3)
64		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ≤ 𝑁)
65	57, 59, 61, 63, 64	ltletrd 11417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
66	65	3adant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
67	56, 66	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁)
68		elfzo0 13736	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (0..^𝑁) ↔ (2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁))
69	55, 42, 67, 68	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ (0..^𝑁))
70		zmodidfzoimp 13937	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0..^𝑁) → (2 mod 𝑁) = 2)
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) = 2)
72		2nn 12335	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
73	71, 72	eqeltrdi 2848	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) ∈ ℕ)
74	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℝ)
75		modlt 13916	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (2 mod 𝑁) < 𝑁)
76	74, 43, 75	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) < 𝑁)
77	73, 76	jca 511	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
78	77	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
79		addmodne 47319	. . . . . . . . 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 1373	. . . . . . . 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 3007	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
82	81	necomd 2995	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ≠ (((2nd ‘𝑋) + 1) mod 𝑁))
83	82	olcd 875	. . . . 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 7462	. . . . . 6 ⊢ (((2nd ‘𝑋) − 1) mod 𝑁) ∈ V
85	10, 84	opthne 5485	. . . . 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 234	. . . 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 1129	. . 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 5467	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
89		opex 5467	. . . 4 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
90		opex 5467	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
91		hashtpg 14520	. . . 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 1463	. . 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 218	. 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 2776	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2939  Vcvv 3479  {ctp 4628  ⟨cop 4630   class class class wbr 5141  ‘cfv 6559  (class class class)co 7429  1^st c1st 8008  2^nd c2nd 8009  ℝcr 11150  0cc0 11151  1c1 11152   + caddc 11154   < clt 11291   ≤ cle 11292   − cmin 11488   / cdiv 11916  ℕcn 12262  2c2 12317  3c3 12318  ℕ₀cn0 12522  ℤcz 12609  ℤ_≥cuz 12874  ℝ⁺crp 13030  ..^cfzo 13690  ⌈cceil 13827   mod cmo 13905  ♯chash 14365  Vtxcvtx 29003   NeighbVtx cnbgr 29339   gPetersenGr cgpg 47972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228  ax-pre-sup 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-2o 8503  df-oadd 8506  df-er 8741  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-sup 9478  df-inf 9479  df-dju 9937  df-card 9975  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-div 11917  df-nn 12263  df-2 12325  df-3 12326  df-4 12327  df-5 12328  df-6 12329  df-7 12330  df-8 12331  df-9 12332  df-n0 12523  df-xnn0 12596  df-z 12610  df-dec 12730  df-uz 12875  df-rp 13031  df-fz 13544  df-fzo 13691  df-fl 13828  df-ceil 13829  df-mod 13906  df-hash 14366  df-dvds 16287  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17244  df-edgf 28994  df-vtx 29005  df-iedg 29006  df-edg 29055  df-upgr 29089  df-umgr 29090  df-usgr 29158  df-nbgr 29340  df-gpg 47973

This theorem is referenced by:  gpgcubic  48008  gpg5nbgrvtx03star  48009