gpg3nbgrvtx0 - Mathbox for Alexander van der Vekens

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Theorem gpg3nbgrvtx0 48651

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 ∧ (1^st ‘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 48649	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 𝑈 = {⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩})
6	5	fveq2d 6865	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (♯‘𝑈) = (♯‘{⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩}))
7		0ne1 12284	. . . . . . 7 ⊢ 0 ≠ 1
8	7	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 0 ≠ 1)
9	8	orcd 884	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (0 ≠ 1 ∨ (((2^nd ‘𝑋) + 1) mod 𝑁) ≠ (2^nd ‘𝑋)))
10		c0ex 11168	. . . . . 6 ⊢ 0 ∈ V
11		ovex 7423	. . . . . 6 ⊢ (((2^nd ‘𝑋) + 1) mod 𝑁) ∈ V
12	10, 11	opthne 5449	. . . . 5 ⊢ (⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2^nd ‘𝑋)⟩ ↔ (0 ≠ 1 ∨ (((2^nd ‘𝑋) + 1) mod 𝑁) ≠ (2^nd ‘𝑋)))
13	9, 12	sylibr 236	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2^nd ‘𝑋)⟩)
14		ax-1ne0 11137	. . . . . . 7 ⊢ 1 ≠ 0
15	14	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 1 ≠ 0)
16	15	orcd 884	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (1 ≠ 0 ∨ (2^nd ‘𝑋) ≠ (((2^nd ‘𝑋) − 1) mod 𝑁)))
17		1ex 11171	. . . . . 6 ⊢ 1 ∈ V
18		fvex 6874	. . . . . 6 ⊢ (2^nd ‘𝑋) ∈ V
19	17, 18	opthne 5449	. . . . 5 ⊢ (⟨1, (2^nd ‘𝑋)⟩ ≠ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ↔ (1 ≠ 0 ∨ (2^nd ‘𝑋) ≠ (((2^nd ‘𝑋) − 1) mod 𝑁)))
20	16, 19	sylibr 236	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ⟨1, (2^nd ‘𝑋)⟩ ≠ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩)
21		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
22	21	anim2i 626	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
23		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^𝑁)
24	23, 1, 2, 3	gpgvtxel2 48623	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2^nd ‘𝑋) ∈ (0..^𝑁))
25		elfzoelz 13659	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑋) ∈ (0..^𝑁) → (2^nd ‘𝑋) ∈ ℤ)
26	22, 24, 25	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (2^nd ‘𝑋) ∈ ℤ)
27	26	zcnd 12673	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (2^nd ‘𝑋) ∈ ℂ)
28		1cnd 11170	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 1 ∈ ℂ)
29		2cnd 12291	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 2 ∈ ℂ)
30	27, 28, 29	subadd23d 11559	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) − 1) + 2) = ((2^nd ‘𝑋) + (2 − 1)))
31		2m1e1 12337	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
32	31	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (2 − 1) = 1)
33	32	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((2^nd ‘𝑋) + (2 − 1)) = ((2^nd ‘𝑋) + 1))
34	30, 33	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) − 1) + 2) = ((2^nd ‘𝑋) + 1))
35	34	eqcomd 2767	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((2^nd ‘𝑋) + 1) = (((2^nd ‘𝑋) − 1) + 2))
36	35	oveq1d 7405	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) + 1) mod 𝑁) = ((((2^nd ‘𝑋) − 1) + 2) mod 𝑁))
37		1zzd 12597	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 1 ∈ ℤ)
38	26, 37	zsubcld 12677	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((2^nd ‘𝑋) − 1) ∈ ℤ)
39	38	zred 12672	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((2^nd ‘𝑋) − 1) ∈ ℝ)
40		2re 12287	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 2 ∈ ℝ)
42		eluz3nn 12885	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
43	42	nnrpd 13030	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℝ⁺)
44	43	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 𝑁 ∈ ℝ⁺)
45		modaddabs 13916	. . . . . . . . . . 11 ⊢ ((((2^nd ‘𝑋) − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) = ((((2^nd ‘𝑋) − 1) + 2) mod 𝑁))
46	39, 41, 44, 45	syl3anc 1389	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) = ((((2^nd ‘𝑋) − 1) + 2) mod 𝑁))
47	46	eqcomd 2767	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((((2^nd ‘𝑋) − 1) + 2) mod 𝑁) = (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁))
48	36, 47	eqtrd 2796	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) + 1) mod 𝑁) = (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁))
49	42	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 𝑁 ∈ ℕ)
50	38, 49	zmodcld 13897	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ₀)
51		modlt 13885	. . . . . . . . . . 11 ⊢ ((((2^nd ‘𝑋) − 1) ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → (((2^nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
52	39, 44, 51	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) − 1) mod 𝑁) < 𝑁)
53	50, 52	jca 519	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((((2^nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ₀ ∧ (((2^nd ‘𝑋) − 1) mod 𝑁) < 𝑁))
54		2nn0 12493	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ₀
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 2 ∈ ℕ₀)
56		eluz2 12840	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁))
57	40	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 ∈ ℝ)
58		3re 12293	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
59	58	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ∈ ℝ)
60		zre 12567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
61	60	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 𝑁 ∈ ℝ)
62		2lt3 12386	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 3
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 3)
64		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 3 ≤ 𝑁)
65	57, 59, 61, 63, 64	ltletrd 11338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
66	65	3adant1 1142	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁)
67	56, 66	sylbi 219	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 2 < 𝑁)
68		elfzo0 13701	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (0..^𝑁) ↔ (2 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁))
69	55, 42, 67, 68	syl3anbrc 1356	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 2 ∈ (0..^𝑁))
70		zmodidfzoimp 13906	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0..^𝑁) → (2 mod 𝑁) = 2)
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (2 mod 𝑁) = 2)
72		2nn 12286	. . . . . . . . . . . 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 13885	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → (2 mod 𝑁) < 𝑁)
76	74, 43, 75	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (2 mod 𝑁) < 𝑁)
77	73, 76	jca 519	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
78	77	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁))
79		addmodne 47897	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ((((2^nd ‘𝑋) − 1) mod 𝑁) ∈ ℕ₀ ∧ (((2^nd ‘𝑋) − 1) mod 𝑁) < 𝑁) ∧ ((2 mod 𝑁) ∈ ℕ ∧ (2 mod 𝑁) < 𝑁)) → (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) ≠ (((2^nd ‘𝑋) − 1) mod 𝑁))
80	49, 53, 78, 79	syl3anc 1389	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((((2^nd ‘𝑋) − 1) mod 𝑁) + (2 mod 𝑁)) mod 𝑁) ≠ (((2^nd ‘𝑋) − 1) mod 𝑁))
81	48, 80	eqnetrd 3023	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) + 1) mod 𝑁) ≠ (((2^nd ‘𝑋) − 1) mod 𝑁))
82	81	necomd 3011	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (((2^nd ‘𝑋) − 1) mod 𝑁) ≠ (((2^nd ‘𝑋) + 1) mod 𝑁))
83	82	olcd 885	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (0 ≠ 0 ∨ (((2^nd ‘𝑋) − 1) mod 𝑁) ≠ (((2^nd ‘𝑋) + 1) mod 𝑁)))
84		ovex 7423	. . . . . 6 ⊢ (((2^nd ‘𝑋) − 1) mod 𝑁) ∈ V
85	10, 84	opthne 5449	. . . . 5 ⊢ (⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ↔ (0 ≠ 0 ∨ (((2^nd ‘𝑋) − 1) mod 𝑁) ≠ (((2^nd ‘𝑋) + 1) mod 𝑁)))
86	83, 85	sylibr 236	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩)
87	13, 20, 86	3jca 1140	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2^nd ‘𝑋)⟩ ∧ ⟨1, (2^nd ‘𝑋)⟩ ≠ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩))
88		opex 5430	. . . 4 ⊢ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
89		opex 5430	. . . 4 ⊢ ⟨1, (2^nd ‘𝑋)⟩ ∈ V
90		opex 5430	. . . 4 ⊢ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
91		hashtpg 14493	. . . 4 ⊢ ((⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V ∧ ⟨1, (2^nd ‘𝑋)⟩ ∈ V ∧ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V) → ((⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2^nd ‘𝑋)⟩ ∧ ⟨1, (2^nd ‘𝑋)⟩ ≠ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩) ↔ (♯‘{⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3))
92	88, 89, 90, 91	mp3an 1481	. . 3 ⊢ ((⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨1, (2^nd ‘𝑋)⟩ ∧ ⟨1, (2^nd ‘𝑋)⟩ ≠ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ∧ ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩) ↔ (♯‘{⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3)
93	87, 92	sylib 220	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (♯‘{⟨0, (((2^nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 𝑁)⟩}) = 3)
94	6, 93	eqtrd 2796	1 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → (♯‘𝑈) = 3)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 Vcvv 3453 {ctp 4585 ⟨cop 4587 class class class wbr 5099 ‘cfv 6515 (class class class)co 7390 1^st c1st 7962 2^nd c2nd 7963 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 < clt 11211 ≤ cle 11212 − cmin 11409 / cdiv 11839 ℕcn 12205 2c2 12267 3c3 12268 ℕ₀cn0 12476 ℤcz 12563 ℤ_≥cuz 12834 ℝ⁺crp 12988 ..^cfzo 13654 ⌈cceil 13796 mod cmo 13874 ♯chash 14338 Vtxcvtx 29141 NeighbVtx cnbgr 29477 gPetersenGr cgpg 48615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-oadd 8434 df-er 8671 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9383 df-inf 9384 df-dju 9854 df-card 9892 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-sub 11411 df-neg 11412 df-div 11840 df-nn 12206 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12477 df-xnn0 12550 df-z 12564 df-dec 12684 df-uz 12835 df-rp 12989 df-fz 13508 df-fzo 13655 df-fl 13797 df-ceil 13798 df-mod 13875 df-hash 14339 df-dvds 16268 df-struct 17164 df-slot 17199 df-ndx 17211 df-base 17227 df-edgf 29134 df-vtx 29143 df-iedg 29144 df-edg 29193 df-upgr 29227 df-umgr 29228 df-usgr 29296 df-nbgr 29478 df-gpg 48616

This theorem is referenced by: gpgcubic 48654 gpg5nbgrvtx03star 48655

W3C validator