Theorem gpgnbgrvtx0 49445

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 2856	. . . . 5 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
6		gpgusgra 49428	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
7	5, 6	sylan2b 606	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
8	3, 7	eqeltrid 2868	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
9		simpl 488	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
10		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2763	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	10, 11	nbusgrvtx 29986	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
13	8, 9, 12	syl2an 608	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
14		simpl 488	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
15		simpr 490	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → (1st ‘𝑋) = 0)
16	15	adantl 487	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (1st ‘𝑋) = 0)
17		simpr 490	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
18	4, 3, 10, 11	gpgvtxedg0 49434	. . . . . . 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 1450	. . . . . 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 418	. . . . 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 49424	. . . . . . . . . 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 1160	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
23	22	adantrr 730	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉)
24	4, 3, 10, 11	gpgedgvtx0 49432	. . . . . . . . 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 1160	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))
26	23, 25	jca 521	. . . . . . 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 2852	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ 𝑉))
28		preq2 4697	. . . . . . . . 9 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
29	28	eleq1d 2849	. . . . . . . 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 644	. . . . . . 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 250	. . . . . 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 49425	. . . . . . . . . 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 1161	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
34	33	adantrr 730	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉)
35	24	simp2d 1161	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))
36	34, 35	jca 521	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
37		eleq1 2852	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (2nd ‘𝑋)⟩ ∈ 𝑉))
38		preq2 4697	. . . . . . . . 9 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (2nd ‘𝑋)⟩})
39	38	eleq1d 2849	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
40	37, 39	anbi12d 644	. . . . . . 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 250	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (𝑣 = ⟨1, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
42	21	simp3d 1162	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
43	42	adantrr 730	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉)
44	43	adantr 486	. . . . . . . . 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 2852	. . . . . . . . . 10 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ 𝑉))
46	45	adantl 487	. . . . . . . . 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 260	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) ∧ 𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩) → 𝑣 ∈ 𝑉)
48	24	simp3d 1162	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺))
49	48	adantr 486	. . . . . . . . 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 4697	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
51	50	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑣 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
52	51	adantl 487	. . . . . . . . 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 260	. . . . . . . 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 521	. . . . . . 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 418	. . . . . 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 1457	. . . . 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 215	. . . 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 4697	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑋, 𝑦} = {𝑋, 𝑣})
59	58	eleq1d 2849	. . . . 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 3458	. . . . 5 ⊢ 𝑣 ∈ V
62	61	eltp 4651	. . . 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 317	. . 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 2761	. 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 2802	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)⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  {crab 3416  {cpr 4590  {ctp 4592  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  0cc0 11239  1c1 11240   + caddc 11242   − cmin 11580   / cdiv 12013  2c2 12437  3c3 12438  ℤ_≥cuz 13005  ..^cfzo 13828  ⌈cceil 13971   mod cmo 14049  Vtxcvtx 29631  Edgcedg 29682  USGraphcusgr 29787   NeighbVtx cnbgr 29970   gPetersenGr cgpg 49411

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-dec 12855  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-fl 13972  df-ceil 13973  df-mod 14050  df-hash 14516  df-dvds 16467  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-edgf 29624  df-vtx 29633  df-iedg 29634  df-edg 29683  df-upgr 29717  df-umgr 29718  df-usgr 29789  df-nbgr 29971  df-gpg 49412

This theorem is used by:  gpg3nbgrvtx0  49447  gpg3nbgrvtx0ALT  49448  gpg5nbgrvtx03star  49451  pgnbgreunbgrlem3  49489  pgnbgreunbgrlem6  49495