pgnbgreunbgrlem1 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem1 48696

Description: Lemma 1 for pgnbgreunbgr 48708. (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 11167	. . . . . 6 ⊢ 0 ∈ V
2		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7976	. . . . 5 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4		oveq1 7398	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 1) = (𝑦 + 1))
5	4	oveq1d 7406	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 1) mod 5) = ((𝑦 + 1) mod 5))
6	5	opeq2d 4835	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩)
7	6	eqeq2d 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩))
8		opeq2 4829	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (2^nd ‘𝑋)⟩ = ⟨1, 𝑦⟩)
9	8	eqeq2d 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨1, 𝑦⟩))
10		oveq1 7398	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 1) = (𝑦 − 1))
11	10	oveq1d 7406	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 1) mod 5) = ((𝑦 − 1) mod 5))
12	11	opeq2d 4835	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩)
13	12	eqeq2d 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩))
14	7, 9, 13	3orbi123d 1455	. . . . . . 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 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩))
16	8	eqeq2d 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨1, 𝑦⟩))
17	12	eqeq2d 2772	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))
18	15, 16, 17	3orbi123d 1455	. . . . . . 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 641	. . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩)
21		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩)
22	20, 21	neeq12d 3017	. . . . . . . . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ ⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩
24		eqneqall 2967	. . . . . . . . . . . . 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 255	. . . . . . . . . . 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 414	. . . . . . . . . 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 416	. . . . . . . . 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 12878	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘3)
30		pglem 48674	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
31		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
32		eqid 2761	. . . . . . . . . . . . . . . 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 48648	. . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
37		opeq2 4829	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
38	37	eqcoms 2769	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
39	36, 38	biimtrdi 255	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
40	39	adantld 494	. . . . . . . . . . . 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 4689	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩})
42	41	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
43		preq2 4690	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, 𝑦⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩})
44	43	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, 𝑦⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸))
45	42, 44	bi2anan9r 648	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 494	. . . . . . . . . 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 416	. . . . . . . . 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 4688	. . . . . . . . . . . . . . 15 ⊢ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩}
51	50	eleq1i 2852	. . . . . . . . . . . . . 14 ⊢ ({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
52		prcom 4688	. . . . . . . . . . . . . . 15 ⊢ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩}
53	52	eleq1i 2852	. . . . . . . . . . . . . 14 ⊢ ({⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸)
54	51, 53	anbi12ci 638	. . . . . . . . . . . . 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 12298	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℕ
56	55	nnzi 12589	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℤ
57		uzid 12848	. . . . . . . . . . . . . . . 16 ⊢ (5 ∈ ℤ → 5 ∈ (ℤ_≥‘5))
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘5)
59	31, 32, 33, 34	gpgedg2ov 48649	. . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . 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 2033	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑦 = 𝑏)
62	61	opeq2d 4835	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
63	60, 62	biimtrdi 255	. . . . . . . . . . . . 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 244	. . . . . . . . . . . 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 4690	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩})
66	65	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸))
67	42, 66	bi2anan9r 648	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 494	. . . . . . . . . 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 416	. . . . . . . . 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 1446	. . . . . . . 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 4688	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩}
74	73	eleq1i 2852	. . . . . . . . . . . . . 14 ⊢ ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸)
75	74, 39	biimtrid 244	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
76	75	adantrd 495	. . . . . . . . . . . 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 4689	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨1, 𝑦⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨1, 𝑦⟩, ⟨0, 𝑏⟩})
78	77	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, 𝑦⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
79		preq2 4690	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩})
80	79	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸))
81	78, 80	bi2anan9r 648	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 494	. . . . . . . . . 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 416	. . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐾 = ⟨1, 𝑦⟩)
87		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐿 = ⟨1, 𝑦⟩)
88	86, 87	neeq12d 3017	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩))
89		eqid 2761	. . . . . . . . . . . . 13 ⊢ ⟨1, 𝑦⟩ = ⟨1, 𝑦⟩
90		eqneqall 2967	. . . . . . . . . . . . 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 255	. . . . . . . . . . 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 414	. . . . . . . . . 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 416	. . . . . . . . 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 495	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → {𝐾, ⟨0, 𝑏⟩} = {⟨1, 𝑦⟩, ⟨0, 𝑏⟩})
97	96	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
98	65	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩})
99	98	eleq1d 2846	. . . . . . . . . . . . . 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 641	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 494	. . . . . . . . . 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 416	. . . . . . . . 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 1446	. . . . . . . 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 255	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)
110		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩)
111	109, 110	neeq12d 3017	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 640	. . . . . . . . . . 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 4689	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩})
115	114	eleq1d 2846	. . . . . . . . . . . . 13 ⊢ (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
116	115, 80	bi2anan9r 648	. . . . . . . . . . . 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 343	. . . . . . . . . . 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 296	. . . . . . . . . 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 416	. . . . . . . . 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 494	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩})
123	122	eleq1d 2846	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, 𝑦⟩} ∈ 𝐸))
125	123, 124	anbi12d 641	. . . . . . . . . . . 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 343	. . . . . . . . . . 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 248	. . . . . . . . . 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 416	. . . . . . . . 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 2773	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑦 − 1) mod 5)⟩ = 𝐿 → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
130	129	eqcoms 2769	. . . . . . . . . 10 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
131		eqneqall 2967	. . . . . . . . . . 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 414	. . . . . . . . . 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 255	. . . . . . . . 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 1446	. . . . . . . 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 1448	. . . . . . 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 410	. . . . . 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 255	. . . . 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 2765	. . . . . 6 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (𝑋 = ⟨0, 𝑏⟩ ↔ ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩))
140	139	imbi2d 342	. . . . 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 342	. . . 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 261	. . 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 485	. 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 418	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 208 ∧ wa 399 ∨ w3o 1096 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 {cpr 4581 ⟨cop 4585 ‘cfv 6516 (class class class)co 7391 2^nd c2nd 7964 0cc0 11067 1c1 11068 + caddc 11070 − cmin 11408 / cdiv 11838 2c2 12266 3c3 12267 5c5 12269 ℤcz 12562 ℤ_≥cuz 12833 ..^cfzo 13653 ⌈cceil 13795 mod cmo 13873 Vtxcvtx 29154 Edgcedg 29205 NeighbVtx cnbgr 29490 gPetersenGr cgpg 48623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-oadd 8435 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9382 df-inf 9383 df-dju 9853 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-rp 12988 df-ico 13349 df-fz 13507 df-fzo 13654 df-fl 13796 df-ceil 13797 df-mod 13874 df-hash 14338 df-dvds 16278 df-struct 17174 df-slot 17209 df-ndx 17221 df-base 17237 df-edgf 29147 df-vtx 29156 df-iedg 29157 df-edg 29206 df-umgr 29241 df-usgr 29309 df-gpg 48624

This theorem is referenced by: pgnbgreunbgrlem3 48701

W3C validator