Theorem gpgnbgrvtx0 48434

Description: The (open) neighborhood of an outside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 28-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
gpgnbgrvtx0	⊢ (((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)⟩})

Proof of Theorem gpgnbgrvtx0

Dummy variables 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gpgnbgr.u	. . 3 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
2	1	a1i 11	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = (𝐺 NeighbVtx 𝑋))
3		gpgnbgr.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
4		gpgnbgr.j	. . . . . 6 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
5	4	eleq2i 2829	. . . . 5 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
6		gpgusgra 48417	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	5, 6	sylan2b 595	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
8	3, 7	eqeltrid 2841	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
9		simpl 482	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
10		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2737	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	10, 11	nbusgrvtx 29433	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
13	8, 9, 12	syl2an 597	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
14		simpl 482	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
15		simpr 484	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → (1st ‘𝑋) = 0)
16	15	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (1st ‘𝑋) = 0)
17		simpr 484	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
18	4, 3, 10, 11	gpgvtxedg0 48423	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
19	14, 16, 17, 18	syl2an3an 1425	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))) → (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
20	19	ex 412	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
21	4, 3, 10	gpgvtx0 48413	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))
22	21	simp1d 1143	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
23	22	adantrr 718	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
24	4, 3, 10, 11	gpgedgvtx0 48421	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺) ∧ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺) ∧ {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
25	24	simp1d 1143	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))
26	23, 25	jca 511	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
27		eleq1 2825	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉))
28		preq2 4693	. . . . . . . . 9 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
29	28	eleq1d 2822	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
30	27, 29	anbi12d 633	. . . . . . 7 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉 ∧ {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))))
31	26, 30	syl5ibrcom 247	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
32	4, 3, 10	gpgvtx1 48414	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉 ∧ ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉))
33	32	simp2d 1144	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
34	33	adantrr 718	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
35	24	simp2d 1144	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))
36	34, 35	jca 511	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
37		eleq1 2825	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉))
38		preq2 4693	. . . . . . . . 9 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (2nd ‘𝑋)⟩})
39	38	eleq1d 2822	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
40	37, 39	anbi12d 633	. . . . . . 7 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))))
41	36, 40	syl5ibrcom 247	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
42	21	simp3d 1145	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
43	42	adantrr 718	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
44	43	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
45		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))
46	45	adantl 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))
47	44, 46	mpbird 257	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → 𝑣 ∈ 𝑉)
48	24	simp3d 1145	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))
49	48	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))
50		preq2 4693	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
51	50	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
52	51	adantl 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
53	49, 52	mpbird 257	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
54	47, 53	jca 511	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
55	54	ex 412	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
56	31, 41, 55	3jaod 1432	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
57	20, 56	impbid 212	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)))
58		preq2 4693	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑋, 𝑦} = {𝑋, 𝑣})
59	58	eleq1d 2822	. . . . 5 ⊢ (𝑦 = 𝑣 → ({𝑋, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
60	59	elrab 3648	. . . 4 ⊢ (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
61		vex 3446	. . . . 5 ⊢ 𝑣 ∈ V
62	61	eltp 4648	. . . 4 ⊢ (𝑣 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ↔ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∨ 𝑣 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
63	57, 60, 62	3bitr4g 314	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ 𝑣 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}))
64	63	eqrdv 2735	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
65	2, 13, 64	3eqtrd 2776	1 ⊢ (((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)⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  {crab 3401  {cpr 4584  {ctp 4586  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  0cc0 11038  1c1 11039   + caddc 11041   − cmin 11376   / cdiv 11806  2c2 12212  3c3 12213  ℤ_≥cuz 12763  ..^cfzo 13582  ⌈cceil 13723   mod cmo 13801  Vtxcvtx 29081  Edgcedg 29132  USGraphcusgr 29234   NeighbVtx cnbgr 29417   gPetersenGr cgpg 48400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-ceil 13725  df-mod 13802  df-hash 14266  df-dvds 16192  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-edgf 29074  df-vtx 29083  df-iedg 29084  df-edg 29133  df-upgr 29167  df-umgr 29168  df-usgr 29236  df-nbgr 29418  df-gpg 48401

This theorem is referenced by:  gpg3nbgrvtx0  48436  gpg3nbgrvtx0ALT  48437  gpg5nbgrvtx03star  48440  pgnbgreunbgrlem3  48478  pgnbgreunbgrlem6  48484