Theorem gpg3nbgrvtx0 48080

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 48078	. . 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 6830	. 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 12218	. . . . . . 7 ⊢ 0 ≠ 1
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 0 ≠ 1)
9	8	orcd 873	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (0 ≠ 1 ∨ (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (2nd ‘𝑋)))
10		c0ex 11128	. . . . . 6 ⊢ 0 ∈ V
11		ovex 7386	. . . . . 6 ⊢ (((2nd ‘𝑋) + 1) mod 𝑁) ∈ V
12	10, 11	opthne 5429	. . . . 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 11097	. . . . . . 7 ⊢ 1 ≠ 0
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ≠ 0)
16	15	orcd 873	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (1 ≠ 0 ∨ (2nd ‘𝑋) ≠ (((2nd ‘𝑋) − 1) mod 𝑁)))
17		1ex 11130	. . . . . 6 ⊢ 1 ∈ V
18		fvex 6839	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5429	. . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^𝑁)
24	23, 1, 2, 3	gpgvtxel2 48052	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		elfzoelz 13581	. . . . . . . . . . . . . . 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 12600	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℂ)
28		1cnd 11129	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℂ)
29		2cnd 12225	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℂ)
30	27, 28, 29	subadd23d 11516	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + (2 − 1)))
31		2m1e1 12268	. . . . . . . . . . . . . 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 7369	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + (2 − 1)) = ((2nd ‘𝑋) + 1))
34	30, 33	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + 1))
35	34	eqcomd 2735	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + 1) = (((2nd ‘𝑋) − 1) + 2))
36	35	oveq1d 7368	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
37		1zzd 12525	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℤ)
38	26, 37	zsubcld 12604	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℤ)
39	38	zred 12599	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℝ)
40		2re 12221	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℝ)
42		eluz3nn 12809	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
43	42	nnrpd 12954	. . . . . . . . . . . 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 13834	. . . . . . . . . . 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 2735	. . . . . . . . 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 2764	. . . . . . . 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 13815	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0)
51		modlt 13803	. . . . . . . . . . 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 12420	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℕ0)
56		eluz2 12760	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁))
57	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 ∈ ℝ)
58		3re 12227	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ∈ ℝ)
60		zre 12494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
61	60	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 𝑁 ∈ ℝ)
62		2lt3 12314	. . . . . . . . . . . . . . . . . 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 11295	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
66	65	3adant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
67	56, 66	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁)
68		elfzo0 13622	. . . . . . . . . . . . . 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 13824	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0..^𝑁) → (2 mod 𝑁) = 2)
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) = 2)
72		2nn 12220	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
73	71, 72	eqeltrdi 2836	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) ∈ ℕ)
74	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℝ)
75		modlt 13803	. . . . . . . . . . . 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 47348	. . . . . . . . 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 2992	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
82	81	necomd 2980	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ≠ (((2nd ‘𝑋) + 1) mod 𝑁))
83	82	olcd 874	. . . . 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 7386	. . . . . 6 ⊢ (((2nd ‘𝑋) − 1) mod 𝑁) ∈ V
85	10, 84	opthne 5429	. . . . 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 1128	. . 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 5411	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
89		opex 5411	. . . 4 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
90		opex 5411	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
91		hashtpg 14411	. . . 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 2764	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3438  {ctp 4583  ⟨cop 4585   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168   ≤ cle 11169   − cmin 11366   / cdiv 11796  ℕcn 12147  2c2 12202  3c3 12203  ℕ₀cn0 12403  ℤcz 12490  ℤ_≥cuz 12754  ℝ⁺crp 12912  ..^cfzo 13576  ⌈cceil 13714   mod cmo 13792  ♯chash 14256  Vtxcvtx 28960   NeighbVtx cnbgr 29296   gPetersenGr cgpg 48044

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-xnn0 12477  df-z 12491  df-dec 12611  df-uz 12755  df-rp 12913  df-fz 13430  df-fzo 13577  df-fl 13715  df-ceil 13716  df-mod 13793  df-hash 14257  df-dvds 16183  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17140  df-edgf 28953  df-vtx 28962  df-iedg 28963  df-edg 29012  df-upgr 29046  df-umgr 29047  df-usgr 29115  df-nbgr 29297  df-gpg 48045

This theorem is referenced by:  gpgcubic  48083  gpg5nbgrvtx03star  48084