Theorem gpgnbgrvtx0 48003

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 2825	. . . . 5 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
6		gpgusgra 47985	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	5, 6	sylan2b 594	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
8	3, 7	eqeltrid 2837	. . 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 2734	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	10, 11	nbusgrvtx 29294	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
13	8, 9, 12	syl2an 596	. 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 47994	. . . . . . 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 1423	. . . . . 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 47981	. . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
23	22	adantrr 717	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
24	4, 3, 10, 11	gpgedgvtx0 47992	. . . . . . . . 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 1142	. . . . . . . 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 2821	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉))
28		preq2 4714	. . . . . . . . 9 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
29	28	eleq1d 2818	. . . . . . . 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 632	. . . . . . 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 47982	. . . . . . . . . 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 1143	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
34	33	adantrr 717	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
35	24	simp2d 1143	. . . . . . . 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 2821	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉))
38		preq2 4714	. . . . . . . . 9 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (2nd ‘𝑋)⟩})
39	38	eleq1d 2818	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
40	37, 39	anbi12d 632	. . . . . . 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 1144	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
43	42	adantrr 717	. . . . . . . . . 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 2821	. . . . . . . . . 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 1144	. . . . . . . . . 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 4714	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
51	50	eleq1d 2818	. . . . . . . . . 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 1430	. . . . 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 4714	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑋, 𝑦} = {𝑋, 𝑣})
59	58	eleq1d 2818	. . . . 5 ⊢ (𝑦 = 𝑣 → ({𝑋, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
60	59	elrab 3675	. . . 4 ⊢ (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
61		vex 3467	. . . . 5 ⊢ 𝑣 ∈ V
62	61	eltp 4669	. . . 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 2732	. 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 2773	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 1085   = wceq 1539   ∈ wcel 2107  {crab 3419  {cpr 4608  {ctp 4610  ⟨cop 4612  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  0cc0 11137  1c1 11138   + caddc 11140   − cmin 11474   / cdiv 11902  2c2 12303  3c3 12304  ℤ_≥cuz 12860  ..^cfzo 13676  ⌈cceil 13813   mod cmo 13891  Vtxcvtx 28942  Edgcedg 28993  USGraphcusgr 29095   NeighbVtx cnbgr 29278   gPetersenGr cgpg 47972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-sup 9464  df-inf 9465  df-dju 9923  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-xnn0 12583  df-z 12597  df-dec 12717  df-uz 12861  df-rp 13017  df-fz 13530  df-fzo 13677  df-fl 13814  df-ceil 13815  df-mod 13892  df-hash 14353  df-dvds 16274  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-edgf 28935  df-vtx 28944  df-iedg 28945  df-edg 28994  df-upgr 29028  df-umgr 29029  df-usgr 29097  df-nbgr 29279  df-gpg 47973

This theorem is referenced by:  gpg3nbgrvtx0  48005  gpg3nbgrvtx0ALT  48006  gpg5nbgrvtx03star  48009