Theorem gpgedgiov 49257

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 779	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
2		c0ex 11304	. . . . . . . . . 10 ⊢ 0 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝐼 → 0 ∈ V)
4	3	anim1i 627	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
5	4	ancoms 464	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (0 ∈ V ∧ 𝑋 ∈ 𝐼))
6		op1stg 8014	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (1st ‘⟨0, 𝑋⟩) = 0)
7	5, 6	syl 18	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1st ‘⟨0, 𝑋⟩) = 0)
8	7	ad2antlr 740	. . . . 5 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸) → (1st ‘⟨0, 𝑋⟩) = 0)
9		simpr 490	. . . . 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 2760	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13		gpgedgiov.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
14	10, 11, 12, 13	gpgvtxedg0 49255	. . . . 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 1398	. . . 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 418	. . 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 7454	. . . . . . . . 9 ⊢ ((𝑋 + 1) mod 𝑁) ∈ V
18	2, 17	pm3.2i 476	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 + 1) mod 𝑁) ∈ V)
19		opthg2 5447	. . . . . . . 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 14	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 + 1) mod 𝑁))))
21		ax-1ne0 11273	. . . . . . . . 9 ⊢ 1 ≠ 0
22		eqneqall 2966	. . . . . . . . 9 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌)))
23	21, 22	mpi 21	. . . . . . . 8 ⊢ (1 = 0 → (𝑌 = ((𝑋 + 1) mod 𝑁) → 𝑋 = 𝑌))
24	23	imp 412	. . . . . . 7 ⊢ ((1 = 0 ∧ 𝑌 = ((𝑋 + 1) mod 𝑁)) → 𝑋 = 𝑌)
25	20, 24	biimtrdi 256	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
26		1ex 11307	. . . . . . . . . . 11 ⊢ 1 ∈ V
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝐼 → 1 ∈ V)
28	27	anim1i 627	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
29	28	ancoms 464	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (1 ∈ V ∧ 𝑋 ∈ 𝐼))
30		opthg2 5447	. . . . . . . 8 ⊢ ((1 ∈ V ∧ 𝑋 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ↔ (1 = 1 ∧ 𝑌 = 𝑋)))
31	29, 30	syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ ↔ (1 = 1 ∧ 𝑌 = 𝑋)))
32		simpr 490	. . . . . . . 8 ⊢ ((1 = 1 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
33	32	eqcomd 2766	. . . . . . 7 ⊢ ((1 = 1 ∧ 𝑌 = 𝑋) → 𝑋 = 𝑌)
34	31, 33	biimtrdi 256	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨1, 𝑋⟩ → 𝑋 = 𝑌))
35		ovex 7454	. . . . . . . . 9 ⊢ ((𝑋 − 1) mod 𝑁) ∈ V
36	2, 35	pm3.2i 476	. . . . . . . 8 ⊢ (0 ∈ V ∧ ((𝑋 − 1) mod 𝑁) ∈ V)
37		opthg2 5447	. . . . . . . 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 14	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ ↔ (1 = 0 ∧ 𝑌 = ((𝑋 − 1) mod 𝑁))))
39		eqneqall 2966	. . . . . . . . 9 ⊢ (1 = 0 → (1 ≠ 0 → (𝑌 = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌)))
40	21, 39	mpi 21	. . . . . . . 8 ⊢ (1 = 0 → (𝑌 = ((𝑋 − 1) mod 𝑁) → 𝑋 = 𝑌))
41	40	imp 412	. . . . . . 7 ⊢ ((1 = 0 ∧ 𝑌 = ((𝑋 − 1) mod 𝑁)) → 𝑋 = 𝑌)
42	38, 41	biimtrdi 256	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (⟨1, 𝑌⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩ → 𝑋 = 𝑌))
43	25, 34, 42	3jaod 1456	. . . . 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 8015	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝐼) → (2nd ‘⟨0, 𝑋⟩) = 𝑋)
45	5, 44	syl 18	. . . . . 6 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (2nd ‘⟨0, 𝑋⟩) = 𝑋)
46		oveq1 7428	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) + 1) = (𝑋 + 1))
47	46	oveq1d 7436	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁))
48	47	opeq2d 4846	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩)
49	48	eqeq2d 2771	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨0, (((2nd ‘⟨0, 𝑋⟩) + 1) mod 𝑁)⟩ ↔ ⟨1, 𝑌⟩ = ⟨0, ((𝑋 + 1) mod 𝑁)⟩))
50		opeq2 4840	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ = ⟨1, 𝑋⟩)
51	50	eqeq2d 2771	. . . . . . . 8 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (⟨1, 𝑌⟩ = ⟨1, (2nd ‘⟨0, 𝑋⟩)⟩ ↔ ⟨1, 𝑌⟩ = ⟨1, 𝑋⟩))
52		oveq1 7428	. . . . . . . . . . 11 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ((2nd ‘⟨0, 𝑋⟩) − 1) = (𝑋 − 1))
53	52	oveq1d 7436	. . . . . . . . . 10 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
54	53	opeq2d 4846	. . . . . . . . 9 ⊢ ((2nd ‘⟨0, 𝑋⟩) = 𝑋 → ⟨0, (((2nd ‘⟨0, 𝑋⟩) − 1) mod 𝑁)⟩ = ⟨0, ((𝑋 − 1) mod 𝑁)⟩)
55	54	eqeq2d 2771	. . . . . . . 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 1463	. . . . . . 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 344	. . . . . 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 18	. . . . 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 260	. . . 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 487	. . 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 48	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 → 𝑋 = 𝑌))
62		simpr 490	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
63	62	ad2antlr 740	. . . . . 6 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐼)
64		opeq2 4840	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, 𝑥⟩ = ⟨0, 𝑌⟩)
65		oveq1 7428	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 1) = (𝑌 + 1))
66	65	oveq1d 7436	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 1) mod 𝑁) = ((𝑌 + 1) mod 𝑁))
67	66	opeq2d 4846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, ((𝑌 + 1) mod 𝑁)⟩)
68	64, 67	preq12d 4698	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 𝑌⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩})
69	68	eqeq2d 2771	. . . . . . . 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 4840	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, 𝑥⟩ = ⟨1, 𝑌⟩)
71	64, 70	preq12d 4698	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
72	71	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ({⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩}))
73		oveq1 7428	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑥 + 𝐾) = (𝑌 + 𝐾))
74	73	oveq1d 7436	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → ((𝑥 + 𝐾) mod 𝑁) = ((𝑌 + 𝐾) mod 𝑁))
75	74	opeq2d 4846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩ = ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩)
76	70, 75	preq12d 4698	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩} = {⟨1, 𝑌⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩})
77	76	eqeq2d 2771	. . . . . . . 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 1463	. . . . . . 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 487	. . . . . 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 2761	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
81	80	3mix2d 1356	. . . . . 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 3578	. . . . 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 49242	. . . . . 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 739	. . . . 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 260	. . . 4 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸)
87		opeq2 4840	. . . . . . 7 ⊢ (𝑋 = 𝑌 → ⟨0, 𝑋⟩ = ⟨0, 𝑌⟩)
88	87	preq1d 4696	. . . . . 6 ⊢ (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} = {⟨0, 𝑌⟩, ⟨1, 𝑌⟩})
89	88	eleq1d 2845	. . . . 5 ⊢ (𝑋 = 𝑌 → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ {⟨0, 𝑌⟩, ⟨1, 𝑌⟩} ∈ 𝐸))
90	89	adantl 487	. . . 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 260	. . 3 ⊢ ((((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) ∧ 𝑋 = 𝑌) → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸)
92	91	ex 418	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 = 𝑌 → {⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸))
93	61, 92	impbid 215	1 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  Vcvv 3450  {cpr 4586  ⟨cop 4834  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  0cc0 11204  1c1 11205   + caddc 11207   − cmin 11545   / cdiv 11977  2c2 12401  3c3 12402  ℤ_≥cuz 12969  ..^cfzo 13792  ⌈cceil 13935   mod cmo 14013  Vtxcvtx 29580  Edgcedg 29631   gPetersenGr cgpg 49232

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-oadd 8484  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-sup 9438  df-inf 9439  df-dju 9986  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-div 11978  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-xnn0 12684  df-z 12698  df-dec 12819  df-uz 12970  df-rp 13125  df-fz 13644  df-fzo 13793  df-fl 13936  df-ceil 13937  df-mod 14014  df-hash 14479  df-dvds 16429  df-struct 17331  df-slot 17366  df-ndx 17378  df-base 17394  df-edgf 29573  df-vtx 29582  df-iedg 29583  df-edg 29632  df-umgr 29667  df-usgr 29738  df-gpg 49233

This theorem is used by:  pgnbgreunbgrlem1  49305  pgnbgreunbgrlem4  49311