pgnbgreunbgrlem1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > pgnbgreunbgrlem1		Structured version Visualization version GIF version

Theorem pgnbgreunbgrlem1 48275

Description: Lemma 1 for pgnbgreunbgr 48287. (Contributed by AV, 15-Nov-2025.)

**Hypotheses**
Ref	Expression
pgnbgreunbgr.g	⊢ 𝐺 = (5 gPetersenGr 2)
pgnbgreunbgr.v	⊢ 𝑉 = (Vtx‘𝐺)
pgnbgreunbgr.e	⊢ 𝐸 = (Edg‘𝐺)
pgnbgreunbgr.n	⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)

**Assertion**
Ref	Expression
pgnbgreunbgrlem1	⊢ ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))

Distinct variable group: 𝑦,𝑏 Allowed substitution hints: 𝐸(𝑦,𝑏) 𝐺(𝑦,𝑏) 𝐾(𝑦,𝑏) 𝐿(𝑦,𝑏) 𝑁(𝑦,𝑏) 𝑉(𝑦,𝑏) 𝑋(𝑦,𝑏)

**Proof of Theorem pgnbgreunbgrlem1**
Step	Hyp	Ref	Expression
1		c0ex 11117	. . . . . 6 ⊢ 0 ∈ V
2		vex 3441	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7941	. . . . 5 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4		oveq1 7362	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 1) = (𝑦 + 1))
5	4	oveq1d 7370	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 1) mod 5) = ((𝑦 + 1) mod 5))
6	5	opeq2d 4833	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩)
7	6	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩))
8		opeq2 4827	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (2^nd ‘𝑋)⟩ = ⟨1, 𝑦⟩)
9	8	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨1, 𝑦⟩))
10		oveq1 7362	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 1) = (𝑦 − 1))
11	10	oveq1d 7370	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 1) mod 5) = ((𝑦 − 1) mod 5))
12	11	opeq2d 4833	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩)
13	12	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩))
14	7, 9, 13	3orbi123d 1437	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ↔ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩)))
15	6	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩))
16	8	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨1, 𝑦⟩))
17	12	eqeq2d 2744	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))
18	15, 16, 17	3orbi123d 1437	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ↔ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)))
19	14, 18	anbi12d 632	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) ↔ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))))
20		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩)
21		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩)
22	20, 21	neeq12d 2990	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩))
23		eqid 2733	. . . . . . . . . . . . 13 ⊢ ⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩
24		eqneqall 2940	. . . . . . . . . . . . 13 ⊢ (⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩ → (⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
25	23, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
26	22, 25	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
27	26	impd 410	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
28	27	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
29		5eluz3 12787	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘3)
30		pglem 48253	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
31		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
32		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (0..^5) = (0..^5)
33		pgnbgreunbgr.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (5 gPetersenGr 2)
34		pgnbgreunbgr.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
35	31, 32, 33, 34	gpgedgiov 48227	. . . . . . . . . . . . . . 15 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
36	29, 30, 35	mpanl12 702	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
37		opeq2 4827	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
38	37	eqcoms 2741	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
39	36, 38	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
40	39	adantld 490	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
41		preq1 4687	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩})
42	41	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
43		preq2 4688	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, 𝑦⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩})
44	43	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, 𝑦⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸))
45	42, 44	bi2anan9r 639	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸)))
46	45	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
47	40, 46	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
48	47	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
49	48	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
50		prcom 4686	. . . . . . . . . . . . . . 15 ⊢ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩}
51	50	eleq1i 2824	. . . . . . . . . . . . . 14 ⊢ ({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
52		prcom 4686	. . . . . . . . . . . . . . 15 ⊢ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩}
53	52	eleq1i 2824	. . . . . . . . . . . . . 14 ⊢ ({⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸)
54	51, 53	anbi12ci 629	. . . . . . . . . . . . 13 ⊢ (({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸) ↔ ({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸))
55		5nn 12222	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℕ
56	55	nnzi 12506	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℤ
57		uzid 12757	. . . . . . . . . . . . . . . 16 ⊢ (5 ∈ ℤ → 5 ∈ (ℤ_≥‘5))
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘5)
59	31, 32, 33, 34	gpgedg2ov 48228	. . . . . . . . . . . . . . 15 ⊢ (((5 ∈ (ℤ_≥‘5) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
60	58, 30, 59	mpanl12 702	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
61		equcomiv 2015	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑦 = 𝑏)
62	61	opeq2d 4833	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
63	60, 62	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
64	54, 63	biimtrid 242	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
65		preq2 4688	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩})
66	65	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸))
67	42, 66	bi2anan9r 639	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)))
68	67	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
69	64, 68	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
70	69	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
71	70	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
72	28, 49, 71	3jaoi 1430	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
73		prcom 4686	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩}
74	73	eleq1i 2824	. . . . . . . . . . . . . 14 ⊢ ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸)
75	74, 39	biimtrid 242	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
76	75	adantrd 491	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
77		preq1 4687	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨1, 𝑦⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨1, 𝑦⟩, ⟨0, 𝑏⟩})
78	77	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, 𝑦⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
79		preq2 4688	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩})
80	79	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸))
81	78, 80	bi2anan9r 639	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)))
82	81	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
83	76, 82	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
84	83	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
85	84	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
86		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐾 = ⟨1, 𝑦⟩)
87		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐿 = ⟨1, 𝑦⟩)
88	86, 87	neeq12d 2990	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩))
89		eqid 2733	. . . . . . . . . . . . 13 ⊢ ⟨1, 𝑦⟩ = ⟨1, 𝑦⟩
90		eqneqall 2940	. . . . . . . . . . . . 13 ⊢ (⟨1, 𝑦⟩ = ⟨1, 𝑦⟩ → (⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
91	89, 90	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
92	88, 91	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
93	92	impd 410	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
94	93	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
95	75	adantrd 491	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
96	77	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → {𝐾, ⟨0, 𝑏⟩} = {⟨1, 𝑦⟩, ⟨0, 𝑏⟩})
97	96	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
98	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩})
99	98	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸))
100	97, 99	anbi12d 632	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)))
101	100	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
102	95, 101	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
103	102	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
104	103	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
105	85, 94, 104	3jaoi 1430	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
106	60, 38	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
107	106	adantl 481	. . . . . . . . . . . 12 ⊢ ((⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
108	107	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
109		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)
110		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩)
111	109, 110	neeq12d 2990	. . . . . . . . . . . . 13 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩))
112	111	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩))
113	112	anbi1d 631	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ↔ (⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)))))
114		preq1 4687	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩})
115	114	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
116	115, 80	bi2anan9r 639	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)))
117	116	imbi1d 341	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
118	108, 113, 117	3imtr4d 294	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
119	118	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
120	39	adantld 490	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
121	120	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
122	114	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩})
123	122	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
124	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸))
125	123, 124	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸)))
126	125	imbi1d 341	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩) ↔ (({⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
127	121, 126	imbitrrid 246	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
128	127	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
129		eqeq2 2745	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑦 − 1) mod 5)⟩ = 𝐿 → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
130	129	eqcoms 2741	. . . . . . . . . 10 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
131		eqneqall 2940	. . . . . . . . . . 11 ⊢ (𝐾 = 𝐿 → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
132	131	impd 410	. . . . . . . . . 10 ⊢ (𝐾 = 𝐿 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
133	130, 132	biimtrdi 253	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
134	119, 128, 133	3jaoi 1430	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
135	72, 105, 134	3jaod 1431	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
136	135	imp 406	. . . . . 6 ⊢ (((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
137	19, 136	biimtrdi 253	. . . . 5 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
138	3, 137	syl 17	. . . 4 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
139		eqeq1 2737	. . . . . 6 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (𝑋 = ⟨0, 𝑏⟩ ↔ ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
140	139	imbi2d 340	. . . . 5 ⊢ (𝑋 = ⟨0, 𝑦⟩ → ((({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩) ↔ (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)))
141	140	imbi2d 340	. . . 4 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)) ↔ ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))))
142	138, 141	sylibrd 259	. . 3 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
143	142	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
144	143	expdcom 414	1 ⊢ ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 {cpr 4579 ⟨cop 4583 ‘cfv 6489 (class class class)co 7355 2^nd c2nd 7929 0cc0 11017 1c1 11018 + caddc 11020 − cmin 11355 / cdiv 11785 2c2 12191 3c3 12192 5c5 12194 ℤcz 12479 ℤ_≥cuz 12742 ..^cfzo 13561 ⌈cceil 13702 mod cmo 13780 Vtxcvtx 28995 Edgcedg 29046 NeighbVtx cnbgr 29331 gPetersenGr cgpg 48202

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-oadd 8398 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9337 df-inf 9338 df-dju 9805 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-xnn0 12466 df-z 12480 df-dec 12599 df-uz 12743 df-rp 12897 df-ico 13258 df-fz 13415 df-fzo 13562 df-fl 13703 df-ceil 13704 df-mod 13781 df-hash 14245 df-dvds 16171 df-struct 17065 df-slot 17100 df-ndx 17112 df-base 17128 df-edgf 28988 df-vtx 28997 df-iedg 28998 df-edg 29047 df-umgr 29082 df-usgr 29150 df-gpg 48203

This theorem is referenced by: pgnbgreunbgrlem3 48280

W3C validator