Theorem gpgnbgrvtx1 48106

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 2823	. . . . 5 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
6		gpgusgra 48088	. . . . 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 2835	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
9		simpl 482	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → 𝑋 ∈ 𝑉)
10		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2731	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	10, 11	nbusgrvtx 29321	. . 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) = 1)) → (𝐺 NeighbVtx 𝑋) = {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)})
14		simpl 482	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
15		simpr 484	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → (1st ‘𝑋) = 1)
16	15	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (1st ‘𝑋) = 1)
17		simpr 484	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)) → {𝑋, 𝑣} ∈ (Edg‘𝐺))
18	4, 3, 10, 11	gpgvtxedg1 48095	. . . . . . 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 1424	. . . . . 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 412	. . . . 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 48085	. . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉)
23	22	adantrr 717	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉)
24	4, 3, 10, 11	gpgedgvtx1 48093	. . . . . . . . 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 1142	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
26	23, 25	jca 511	. . . . . . 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 2819	. . . . . . . 8 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ 𝑉))
28		preq2 4682	. . . . . . . . 9 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩})
29	28	eleq1d 2816	. . . . . . . 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 632	. . . . . . 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 247	. . . . . 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 48084	. . . . . . . . . 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 1143	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉)
34	33	adantrr 717	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉)
35	24	simp2d 1143	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺))
36	34, 35	jca 511	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉 ∧ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
37		eleq1 2819	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨0, (2nd ‘𝑋)⟩ ∈ 𝑉))
38		preq2 4682	. . . . . . . . 9 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → {𝑋, 𝑣} = {𝑋, ⟨0, (2nd ‘𝑋)⟩})
39	38	eleq1d 2816	. . . . . . . 8 ⊢ (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨0, (2nd ‘𝑋)⟩} ∈ (Edg‘𝐺)))
40	37, 39	anbi12d 632	. . . . . . 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 247	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (𝑣 = ⟨0, (2nd ‘𝑋)⟩ → (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺))))
42	21	simp3d 1144	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉)
43	42	adantrr 717	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉)
44	43	adantr 480	. . . . . . . . 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 2819	. . . . . . . . . 10 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → (𝑣 ∈ 𝑉 ↔ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ 𝑉))
46	45	adantl 481	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) ∧ 𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩) → 𝑣 ∈ 𝑉)
48	24	simp3d 1144	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
49	48	adantr 480	. . . . . . . . 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 4682	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {𝑋, 𝑣} = {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
51	50	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑣 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({𝑋, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑋, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
52	51	adantl 481	. . . . . . . . 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 257	. . . . . . . 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 511	. . . . . . 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 412	. . . . . 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 1431	. . . . 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 212	. . . 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 4682	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑋, 𝑦} = {𝑋, 𝑣})
59	58	eleq1d 2816	. . . . 5 ⊢ (𝑦 = 𝑣 → ({𝑋, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
60	59	elrab 3642	. . . 4 ⊢ (𝑣 ∈ {𝑦 ∈ 𝑉 ∣ {𝑋, 𝑦} ∈ (Edg‘𝐺)} ↔ (𝑣 ∈ 𝑉 ∧ {𝑋, 𝑣} ∈ (Edg‘𝐺)))
61		vex 3440	. . . . 5 ⊢ 𝑣 ∈ V
62	61	eltp 4637	. . . 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 314	. . 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 2729	. 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 2770	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)⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111  {crab 3395  {cpr 4573  {ctp 4575  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  0cc0 11001  1c1 11002   + caddc 11004   − cmin 11339   / cdiv 11769  2c2 12175  3c3 12176  ℤ_≥cuz 12727  ..^cfzo 13549  ⌈cceil 13690   mod cmo 13768  Vtxcvtx 28969  Edgcedg 29020  USGraphcusgr 29122   NeighbVtx cnbgr 29305   gPetersenGr cgpg 48071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-dju 9789  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-xnn0 12450  df-z 12464  df-dec 12584  df-uz 12728  df-rp 12886  df-ico 13246  df-fz 13403  df-fzo 13550  df-fl 13691  df-ceil 13692  df-mod 13769  df-hash 14233  df-dvds 16159  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-edgf 28962  df-vtx 28971  df-iedg 28972  df-edg 29021  df-upgr 29055  df-umgr 29056  df-usgr 29124  df-nbgr 29306  df-gpg 48072

This theorem is referenced by:  gpg3nbgrvtx1  48109  gpg5nbgr3star  48112  gpg3kgrtriex  48120  pgnbgreunbgrlem3  48149  pgnbgreunbgrlem6  48155