Theorem gpgedgiov 48425

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 767	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
2		c0ex 11138	. . . . . . . . . 10 ⊢ 0 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝐼 → 0 ∈ V)
4	3	anim1i 616	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
5	4	ancoms 458	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
6		op1stg 7955	. . . . . . 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 728	. . . . 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 2737	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13		gpgedgiov.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
14	10, 11, 12, 13	gpgvtxedg0 48423	. . . . 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 1374	. . . 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 7401	. . . . . . . . 9 ⊢ ((𝑋 + 1) mod 𝑁) ∈ V
18	2, 17	pm3.2i 470	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 + 1) mod 𝑁) ∈ V)
19		opthg2 5435	. . . . . . . 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 11107	. . . . . . . . 9 ⊢ 1 ≠ 0
22		eqneqall 2944	. . . . . . . . 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 11140	. . . . . . . . . . 11 ⊢ 1 ∈ V
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐼 → 1 ∈ V)
28	27	anim1i 616	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
29	28	ancoms 458	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
30		opthg2 5435	. . . . . . . 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 2743	. . . . . . 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 7401	. . . . . . . . 9 ⊢ ((𝑋 − 1) mod 𝑁) ∈ V
36	2, 35	pm3.2i 470	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 − 1) mod 𝑁) ∈ V)
37		opthg2 5435	. . . . . . . 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 2944	. . . . . . . . 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 1432	. . . . 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 7956	. . . . . . 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 7375	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) + 1) = (𝑋 + 1))
47	46	oveq1d 7383	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁))
48	47	opeq2d 4838	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩)
49	48	eqeq2d 2748	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ↔ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩))
50		opeq2 4832	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ = ⟨1, 𝑋⟩)
51	50	eqeq2d 2748	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ↔ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩))
52		oveq1 7375	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) − 1) = (𝑋 − 1))
53	52	oveq1d 7383	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
54	53	opeq2d 4838	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)
55	54	eqeq2d 2748	. . . . . . . 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 1438	. . . . . . 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 728	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐼)
64		opeq2 4832	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, 𝑥⟩ = ⟨0, 𝑌⟩)
65		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 1) = (𝑌 + 1))
66	65	oveq1d 7383	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁))
67	66	opeq2d 4838	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩)
68	64, 67	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
69	68	eqeq2d 2748	. . . . . . . 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 4832	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, 𝑥⟩ = ⟨1, 𝑌⟩)
71	64, 70	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
72	71	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩}))
73		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 𝐾) = (𝑌 + 𝐾))
74	73	oveq1d 7383	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 𝐾) mod 𝑁) = ((𝑌 + 𝐾) mod 𝑁))
75	74	opeq2d 4838	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ = ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩)
76	70, 75	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩})
77	76	eqeq2d 2748	. . . . . . . 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 1438	. . . . . . 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 2738	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
81	80	3mix2d 1339	. . . . . 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 3580	. . . . 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 48410	. . . . . 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 727	. . . . 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 4832	. . . . . . 7 ⊢ (𝑋 = 𝑌 → ⟨0, 𝑋⟩ = ⟨0, 𝑌⟩)
88	87	preq1d 4698	. . . . . 6 ⊢ (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
89	88	eleq1d 2822	. . . . 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 1086   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3442  {cpr 4584  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  0cc0 11038  1c1 11039   + caddc 11041   − cmin 11376   / cdiv 11806  2c2 12212  3c3 12213  ℤ_≥cuz 12763  ..^cfzo 13582  ⌈cceil 13723   mod cmo 13801  Vtxcvtx 29081  Edgcedg 29132   gPetersenGr cgpg 48400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-ceil 13725  df-mod 13802  df-hash 14266  df-dvds 16192  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-edgf 29074  df-vtx 29083  df-iedg 29084  df-edg 29133  df-umgr 29168  df-usgr 29236  df-gpg 48401

This theorem is referenced by:  pgnbgreunbgrlem1  48473  pgnbgreunbgrlem4  48479