Theorem gpg3nbgrvtx0 49441

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 49439	. . 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 6918	. 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 12452	. . . . . . 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 11337	. . . . . 6 ⊢ 0 ∈ V
11		ovex 7482	. . . . . 6 ⊢ (((2nd ‘𝑋) + 1) mod 𝑁) ∈ V
12	10, 11	opthne 5456	. . . . 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 11306	. . . . . . 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 11340	. . . . . 6 ⊢ 1 ∈ V
18		fvex 6927	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
19	17, 18	opthne 5456	. . . . 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 2763	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^𝑁)
24	23, 1, 2, 3	gpgvtxel2 49413	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
25		elfzoelz 13831	. . . . . . . . . . . . . . 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 12842	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℂ)
28		1cnd 11339	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℂ)
29		2cnd 12459	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℂ)
30	27, 28, 29	subadd23d 11728	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + (2 − 1)))
31		2m1e1 12505	. . . . . . . . . . . . . 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 7465	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + (2 − 1)) = ((2nd ‘𝑋) + 1))
34	30, 33	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) + 2) = ((2nd ‘𝑋) + 1))
35	34	eqcomd 2769	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) + 1) = (((2nd ‘𝑋) − 1) + 2))
36	35	oveq1d 7464	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) = ((((2nd ‘𝑋) − 1) + 2) mod 𝑁))
37		1zzd 12765	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 1 ∈ ℤ)
38	26, 37	zsubcld 12846	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℤ)
39	38	zred 12841	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((2nd ‘𝑋) − 1) ∈ ℝ)
40		2re 12455	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 2 ∈ ℝ)
42		eluz3nn 13054	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
43	42	nnrpd 13200	. . . . . . . . . . . 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 14089	. . . . . . . . . . 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 1399	. . . . . . . . . 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 2769	. . . . . . . . 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 2798	. . . . . . . 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 14070	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ0)
51		modlt 14058	. . . . . . . . . . 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 12661	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ0
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℕ0)
56		eluz2 13009	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁))
57	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 ∈ ℝ)
58		3re 12461	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ∈ ℝ)
60		zre 12735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
61	60	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 𝑁 ∈ ℝ)
62		2lt3 12554	. . . . . . . . . . . . . . . . . 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 11507	. . . . . . . . . . . . . . . 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 13873	. . . . . . . . . . . . . 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 14079	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0..^𝑁) → (2 mod 𝑁) = 2)
71	69, 70	syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) = 2)
72		2nn 12454	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
73	71, 72	eqeltrdi 2872	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 mod 𝑁) ∈ ℕ)
74	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℝ)
75		modlt 14058	. . . . . . . . . . . 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 48687	. . . . . . . . 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 1399	. . . . . . . 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 3026	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (((2nd ‘𝑋) + 1) mod 𝑁) ≠ (((2nd ‘𝑋) − 1) mod 𝑁))
82	81	necomd 3014	. . . . . 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 7482	. . . . . 6 ⊢ (((2nd ‘𝑋) − 1) mod 𝑁) ∈ V
85	10, 84	opthne 5456	. . . . 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 5437	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
89		opex 5437	. . . 4 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
90		opex 5437	. . . 4 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
91		hashtpg 14669	. . . 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 1491	. . 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 2798	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 2147   ≠ wne 2959  Vcvv 3454  {ctp 4592  ⟨cop 4838   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  1^st c1st 8031  2^nd c2nd 8032  ℝcr 11236  0cc0 11237  1c1 11238   + caddc 11240   < clt 11380   ≤ cle 11381   − cmin 11578   / cdiv 12011  ℕcn 12373  2c2 12435  3c3 12436  ℕ₀cn0 12644  ℤcz 12731  ℤ_≥cuz 13003  ℝ⁺crp 13158  ..^cfzo 13826  ⌈cceil 13969   mod cmo 14047  ♯chash 14513  Vtxcvtx 29620   NeighbVtx cnbgr 29959   gPetersenGr cgpg 49405

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-2o 8513  df-oadd 8516  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-inf 9470  df-dju 10017  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-div 12012  df-nn 12374  df-2 12443  df-3 12444  df-4 12445  df-5 12446  df-6 12447  df-7 12448  df-8 12449  df-9 12450  df-n0 12645  df-xnn0 12718  df-z 12732  df-dec 12853  df-uz 13004  df-rp 13159  df-fz 13678  df-fzo 13827  df-fl 13970  df-ceil 13971  df-mod 14048  df-hash 14514  df-dvds 16464  df-struct 17366  df-slot 17401  df-ndx 17413  df-base 17429  df-edgf 29613  df-vtx 29622  df-iedg 29623  df-edg 29672  df-upgr 29706  df-umgr 29707  df-usgr 29778  df-nbgr 29960  df-gpg 49406

This theorem is used by:  gpgcubic  49444  gpg5nbgrvtx03star  49445