Theorem gpgedgiov 48096

Description: The edges of the generalized Petersen graph GPG(N,K) between an inside and an outside vertex. (Contributed by AV, 11-Nov-2025.)

Hypotheses

Ref	Expression
gpgedgiov.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgedgiov.i	⊢ 𝐼 = (0..^𝑁)
gpgedgiov.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgedgiov.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
gpgedgiov	⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))

Proof of Theorem gpgedgiov

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
2		c0ex 11101	. . . . . . . . . 10 ⊢ 0 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝐼 → 0 ∈ V)
4	3	anim1i 615	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
5	4	ancoms 458	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
6		op1stg 7928	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (1st ‘⟨0, 𝑋⟩) = 0)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1st ‘⟨0, 𝑋⟩) = 0)
8	7	ad2antlr 727	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (1st ‘⟨0, 𝑋⟩) = 0)
9		simpr 484	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸)
10		gpgedgiov.j	. . . . . 6 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
11		gpgedgiov.g	. . . . . 6 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
12		eqid 2731	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13		gpgedgiov.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
14	10, 11, 12, 13	gpgvtxedg0 48094	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘⟨0, 𝑋⟩) = 0 ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
15	1, 8, 9, 14	syl3anc 1373	. . . 4 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩))
16	15	ex 412	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩)))
17		ovex 7374	. . . . . . . . 9 ⊢ ((𝑋 + 1) mod 𝑁) ∈ V
18	2, 17	pm3.2i 470	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 + 1) mod 𝑁) ∈ V)
19		opthg2 5414	. . . . . . . 8 ⊢ ((0 ∈ V ∧ ((𝑋 + 1) mod 𝑁) ∈ V) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 + 1) mod 𝑁))))
20	18, 19	mp1i 13	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 + 1) mod 𝑁))))
21		ax-1ne0 11070	. . . . . . . . 9 ⊢ 1 ≠ 0
22		eqneqall 2939	. . . . . . . . 9 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌)))
23	21, 22	mpi 20	. . . . . . . 8 ⊢ (1 = 0 → (𝑌 = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌))
24	23	imp 406	. . . . . . 7 ⊢ ((1 = 0 ∧ 𝑌 = ((𝑋 + 1) mod 𝑁)) → 𝑋 = 𝑌)
25	20, 24	biimtrdi 253	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
26		1ex 11103	. . . . . . . . . . 11 ⊢ 1 ∈ V
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐼 → 1 ∈ V)
28	27	anim1i 615	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
29	28	ancoms 458	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
30		opthg2 5414	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 𝑋 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ↔ (1 = 1 ∧ 𝑌 = 𝑋)))
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ↔ (1 = 1 ∧ 𝑌 = 𝑋)))
32		simpr 484	. . . . . . . 8 ⊢ ((1 = 1 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
33	32	eqcomd 2737	. . . . . . 7 ⊢ ((1 = 1 ∧ 𝑌 = 𝑋) → 𝑋 = 𝑌)
34	31, 33	biimtrdi 253	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
35		ovex 7374	. . . . . . . . 9 ⊢ ((𝑋 − 1) mod 𝑁) ∈ V
36	2, 35	pm3.2i 470	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 − 1) mod 𝑁) ∈ V)
37		opthg2 5414	. . . . . . . 8 ⊢ ((0 ∈ V ∧ ((𝑋 − 1) mod 𝑁) ∈ V) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 − 1) mod 𝑁))))
38	36, 37	mp1i 13	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 − 1) mod 𝑁))))
39		eqneqall 2939	. . . . . . . . 9 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
40	21, 39	mpi 20	. . . . . . . 8 ⊢ (1 = 0 → (𝑌 = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌))
41	40	imp 406	. . . . . . 7 ⊢ ((1 = 0 ∧ 𝑌 = ((𝑋 − 1) mod 𝑁)) → 𝑋 = 𝑌)
42	38, 41	biimtrdi 253	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
43	25, 34, 42	3jaod 1431	. . . . 5 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ((⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
44		op2ndg 7929	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (2nd ‘⟨0, 𝑋⟩) = 𝑋)
45	5, 44	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (2nd ‘⟨0, 𝑋⟩) = 𝑋)
46		oveq1 7348	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) + 1) = (𝑋 + 1))
47	46	oveq1d 7356	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁))
48	47	opeq2d 4827	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩)
49	48	eqeq2d 2742	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ↔ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩))
50		opeq2 4821	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ = ⟨1, 𝑋⟩)
51	50	eqeq2d 2742	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ↔ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩))
52		oveq1 7348	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) − 1) = (𝑋 − 1))
53	52	oveq1d 7356	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
54	53	opeq2d 4827	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)
55	54	eqeq2d 2742	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ ↔ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩))
56	49, 51, 55	3orbi123d 1437	. . . . . . 7 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) ↔ (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)))
57	56	imbi1d 341	. . . . . 6 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌) ↔ ((⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
58	45, 57	syl 17	. . . . 5 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌) ↔ ((⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩) → 𝑋 = 𝑌)))
59	43, 58	mpbird 257	. . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ((⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
60	59	adantl 481	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ((⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ∨ ⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ∨ ⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩) → 𝑋 = 𝑌))
61	16, 60	syld 47	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 → 𝑋 = 𝑌))
62		simpr 484	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
63	62	ad2antlr 727	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐼)
64		opeq2 4821	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, 𝑥⟩ = ⟨0, 𝑌⟩)
65		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 1) = (𝑌 + 1))
66	65	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁))
67	66	opeq2d 4827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩)
68	64, 67	preq12d 4689	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
69	68	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩}))
70		opeq2 4821	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, 𝑥⟩ = ⟨1, 𝑌⟩)
71	64, 70	preq12d 4689	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
72	71	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩}))
73		oveq1 7348	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 𝐾) = (𝑌 + 𝐾))
74	73	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 𝐾) mod 𝑁) = ((𝑌 + 𝐾) mod 𝑁))
75	74	opeq2d 4827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ = ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩)
76	70, 75	preq12d 4689	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩})
77	76	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩} ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩}))
78	69, 72, 77	3orbi123d 1437	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩})))
79	78	adantl 481	. . . . . 6 ⊢ (((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) ∧ 𝑥 = 𝑌) → (({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) ↔ ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩})))
80		eqidd 2732	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
81	80	3mix2d 1338	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩}))
82	63, 79, 81	rspcedvd 3574	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → ∃𝑥 ∈ 𝐼 ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}))
83		gpgedgiov.i	. . . . . . 7 ⊢ 𝐼 = (0..^𝑁)
84	83, 10, 11, 13	gpgedgel 48081	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ ∃𝑥 ∈ 𝐼 ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
85	84	ad2antrr 726	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ ∃𝑥 ∈ 𝐼 ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
86	82, 85	mpbird 257	. . . 4 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸)
87		opeq2 4821	. . . . . . 7 ⊢ (𝑋 = 𝑌 → ⟨0, 𝑋⟩ = ⟨0, 𝑌⟩)
88	87	preq1d 4687	. . . . . 6 ⊢ (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
89	88	eleq1d 2816	. . . . 5 ⊢ (𝑋 = 𝑌 → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸))
90	89	adantl 481	. . . 4 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸))
91	86, 90	mpbird 257	. . 3 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸)
92	91	ex 412	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸))
93	61, 92	impbid 212	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436  {cpr 4573  ⟨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   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-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-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-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-umgr 29056  df-usgr 29124  df-gpg 48072

This theorem is referenced by:  pgnbgreunbgrlem1  48144  pgnbgreunbgrlem4  48150