Theorem gpgnbgrvtx1 49142

Description: The (open) neighborhood of an inside vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-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
gpgnbgrvtx1	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})

Proof of Theorem gpgnbgrvtx1

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) = 1)) → 𝑈 = (𝐺 NeighbVtx 𝑋))
3		gpgnbgr.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
4		gpgnbgr.j	. . . . . 6 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
5	4	eleq2i 2853	. . . . 5 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
6		gpgusgra 49124	. . . . 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 2865	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
9		simpl 488	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → 𝑋 ∈ 𝑉)
10		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2761	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	10, 11	nbusgrvtx 29922	. . 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) = 1)) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
14		simpl 488	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
15		simpr 490	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → (1st ‘𝑋) = 1)
16	15	adantl 487	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1st ‘𝑋) = 1)
17		simpr 490	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
18	4, 3, 10, 11	gpgvtxedg1 49131	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
19	14, 16, 17, 18	syl2an3an 1449	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))) → (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
20	19	ex 418	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
21	4, 3, 10	gpgvtx1 49121	. . . . . . . . . 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))
22	21	simp1d 1160	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉)
23	22	adantrr 730	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉)
24	4, 3, 10, 11	gpgedgvtx1 49129	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺) ∧ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺) ∧ {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
25	24	simp1d 1160	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
26	23, 25	jca 521	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉 ∧ {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
27		eleq1 2849	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉))
28		preq2 4695	. . . . . . . . 9 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩})
29	28	eleq1d 2846	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
30	27, 29	anbi12d 644	. . . . . . 7 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉 ∧ {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))))
31	26, 30	syl5ibrcom 250	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
32	4, 3, 10	gpgvtx0 49120	. . . . . . . . . 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))
33	32	simp2d 1161	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉)
34	33	adantrr 730	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉)
35	24	simp2d 1161	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))
36	34, 35	jca 521	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
37		eleq1 2849	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉))
38		preq2 4695	. . . . . . . . 9 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (2nd ‘𝑋)⟩})
39	38	eleq1d 2846	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
40	37, 39	anbi12d 644	. . . . . . 7 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))))
41	36, 40	syl5ibrcom 250	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
42	21	simp3d 1162	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉)
43	42	adantrr 730	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉)
44	43	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉)
45		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉))
46	45	adantl 487	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ↔ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉))
47	44, 46	mpbird 260	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → 𝑣 ∈ 𝑉)
48	24	simp3d 1162	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
49	48	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
50		preq2 4695	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
51	50	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
52	51	adantl 487	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
53	49, 52	mpbird 260	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
54	47, 53	jca 521	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
55	54	ex 418	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
56	31, 41, 55	3jaod 1456	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ((𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
57	20, 56	impbid 215	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) ↔ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)))
58		preq2 4695	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑋, 𝑦} = {𝑋, 𝑣})
59	58	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝑣 → ({𝑋, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
60	59	elrab 3645	. . . 4 ⊢ (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
61		vex 3455	. . . . 5 ⊢ 𝑣 ∈ V
62	61	eltp 4650	. . . 4 ⊢ (𝑣 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ↔ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∨ 𝑣 = ⟨0, (2nd ‘𝑋)⟩ ∨ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
63	57, 60, 62	3bitr4g 317	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ 𝑣 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}))
64	63	eqrdv 2759	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
65	2, 13, 64	3eqtrd 2800	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  {crab 3413  {cpr 4586  {ctp 4588  ⟨cop 4590  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  0cc0 11193  1c1 11194   + caddc 11196   − cmin 11534   / cdiv 11966  2c2 12390  3c3 12391  ℤ_≥cuz 12958  ..^cfzo 13781  ⌈cceil 13924   mod cmo 14002  Vtxcvtx 29567  Edgcedg 29618  USGraphcusgr 29723   NeighbVtx cnbgr 29906   gPetersenGr cgpg 49107

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-rp 13114  df-ico 13475  df-fz 13633  df-fzo 13782  df-fl 13925  df-ceil 13926  df-mod 14003  df-hash 14468  df-dvds 16416  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-edgf 29560  df-vtx 29569  df-iedg 29570  df-edg 29619  df-upgr 29653  df-umgr 29654  df-usgr 29725  df-nbgr 29907  df-gpg 49108

This theorem is used by:  gpg3nbgrvtx1  49145  gpg5nbgr3star  49148  gpg3kgrtriex  49156  pgnbgreunbgrlem3  49185  pgnbgreunbgrlem6  49191