pgnbgreunbgrlem1 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem1 48093

Description: Lemma 1 for pgnbgreunbgr 48105. (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 11174	. . . . . 6 ⊢ 0 ∈ V
2		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7981	. . . . 5 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4		oveq1 7396	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 1) = (𝑦 + 1))
5	4	oveq1d 7404	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 1) mod 5) = ((𝑦 + 1) mod 5))
6	5	opeq2d 4846	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩)
7	6	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩))
8		opeq2 4840	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (2^nd ‘𝑋)⟩ = ⟨1, 𝑦⟩)
9	8	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨1, 𝑦⟩))
10		oveq1 7396	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 1) = (𝑦 − 1))
11	10	oveq1d 7404	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 1) mod 5) = ((𝑦 − 1) mod 5))
12	11	opeq2d 4846	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩)
13	12	eqeq2d 2741	. . . . . . . 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 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩))
16	8	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨1, 𝑦⟩))
17	12	eqeq2d 2741	. . . . . . . 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 2987	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ ⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩
24		eqneqall 2937	. . . . . . . . . . . . 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 12848	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘3)
30		pglem 48072	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
31		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
32		eqid 2730	. . . . . . . . . . . . . . . 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 48046	. . . . . . . . . . . . . . 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 4840	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ⟨0, 𝑦⟩ = ⟨0, 𝑏⟩)
38	37	eqcoms 2738	. . . . . . . . . . . . . 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 4699	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩})
42	41	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
43		preq2 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, 𝑦⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩})
44	43	eleq1d 2814	. . . . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . 15 ⊢ {⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩}
51	50	eleq1i 2820	. . . . . . . . . . . . . 14 ⊢ ({⟨0, ((𝑦 + 1) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
52		prcom 4698	. . . . . . . . . . . . . . 15 ⊢ {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩}
53	52	eleq1i 2820	. . . . . . . . . . . . . 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 12273	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℕ
56	55	nnzi 12563	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℤ
57		uzid 12814	. . . . . . . . . . . . . . . 16 ⊢ (5 ∈ ℤ → 5 ∈ (ℤ_≥‘5))
58	56, 57	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘5)
59	31, 32, 33, 34	gpgedg2ov 48047	. . . . . . . . . . . . . . 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 2014	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑦 = 𝑏)
62	61	opeq2d 4846	. . . . . . . . . . . . . 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 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 − 1) mod 5)⟩})
66	65	eleq1d 2814	. . . . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨1, 𝑦⟩}
74	73	eleq1i 2820	. . . . . . . . . . . . . 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 4699	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨1, 𝑦⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨1, 𝑦⟩, ⟨0, 𝑏⟩})
78	77	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, 𝑦⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
79		preq2 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨0, ((𝑦 + 1) mod 5)⟩})
80	79	eleq1d 2814	. . . . . . . . . . . . . 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 2987	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩))
89		eqid 2730	. . . . . . . . . . . . 13 ⊢ ⟨1, 𝑦⟩ = ⟨1, 𝑦⟩
90		eqneqall 2937	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . 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 4699	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, ((𝑦 − 1) mod 5)⟩, ⟨0, 𝑏⟩})
115	114	eleq1d 2814	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . . 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 2742	. . . . . . . . . . 11 ⊢ (⟨0, ((𝑦 − 1) mod 5)⟩ = 𝐿 → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
130	129	eqcoms 2738	. . . . . . . . . 10 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ 𝐾 = 𝐿))
131		eqneqall 2937	. . . . . . . . . . 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 2734	. . . . . 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 1540 ∈ wcel 2109 ≠ wne 2926 {cpr 4593 ⟨cop 4597 ‘cfv 6513 (class class class)co 7389 2^nd c2nd 7969 0cc0 11074 1c1 11075 + caddc 11077 − cmin 11411 / cdiv 11841 2c2 12242 3c3 12243 5c5 12245 ℤcz 12535 ℤ_≥cuz 12799 ..^cfzo 13621 ⌈cceil 13759 mod cmo 13837 Vtxcvtx 28929 Edgcedg 28980 NeighbVtx cnbgr 29265 gPetersenGr cgpg 48021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-oadd 8440 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-ico 13318 df-fz 13475 df-fzo 13622 df-fl 13760 df-ceil 13761 df-mod 13838 df-hash 14302 df-dvds 16229 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-edgf 28922 df-vtx 28931 df-iedg 28932 df-edg 28981 df-umgr 29016 df-usgr 29084 df-gpg 48022

This theorem is referenced by: pgnbgreunbgrlem3 48098

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