pgnbgreunbgrlem2 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2 48363

Description: Lemma 2 for pgnbgreunbgr 48371. Impossible cases. (Contributed by AV, 18-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
pgnbgreunbgrlem2	⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((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 pgnbgreunbgrlem2**
Step	Hyp	Ref	Expression
1		eqtr3 2758	. . . . . 6 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → 𝐿 = 𝐾)
2		eqneqall 2943	. . . . . . . 8 ⊢ (𝐾 = 𝐿 → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
3	2	impd 410	. . . . . . 7 ⊢ (𝐾 = 𝐿 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
4	3	eqcoms 2744	. . . . . 6 ⊢ (𝐿 = 𝐾 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
5	1, 4	syl 17	. . . . 5 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
6	5	a1d 25	. . . 4 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
7	6	ex 412	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
8		1ex 11128	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3444	. . . . . . . 8 ⊢ 𝑦 ∈ V
10	8, 9	op2ndd 7944	. . . . . . 7 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
11		oveq1 7365	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
12	11	oveq1d 7373	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
13	12	opeq2d 4836	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩)
14	13	eqeq2d 2747	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩))
15		opeq2 4830	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (2^nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
16	15	eqeq2d 2747	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨0, 𝑦⟩))
17	14, 16	anbi12d 632	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) ↔ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩)))
18	10, 17	syl 17	. . . . . 6 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) ↔ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩)))
19		pgnbgreunbgr.g	. . . . . . . . . . 11 ⊢ 𝐺 = (5 gPetersenGr 2)
20		pgnbgreunbgr.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
21		pgnbgreunbgr.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
22		pgnbgreunbgr.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
23	19, 20, 21, 22	pgnbgreunbgrlem2lem1 48360	. . . . . . . . . 10 ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
24	23	pm2.21d 121	. . . . . . . . 9 ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
25	24	expimpd 453	. . . . . . . 8 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
26	25	ex 412	. . . . . . 7 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
27	26	adantld 490	. . . . . 6 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
28	18, 27	biimtrdi 253	. . . . 5 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
29	28	adantr 480	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
30	29	expdcom 414	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
31		oveq1 7365	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
32	31	oveq1d 7373	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
33	32	opeq2d 4836	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ = ⟨1, ((𝑦 − 2) mod 5)⟩)
34	33	eqeq2d 2747	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩))
35	14, 34	anbi12d 632	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
36	10, 35	syl 17	. . . . . 6 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
37	19, 20, 21, 22	pgnbgreunbgrlem2lem3 48362	. . . . . . . . . 10 ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
38	37	pm2.21d 121	. . . . . . . . 9 ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
39	38	expimpd 453	. . . . . . . 8 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
40	39	ex 412	. . . . . . 7 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
41	40	adantld 490	. . . . . 6 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
42	36, 41	biimtrdi 253	. . . . 5 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
43	42	adantr 480	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
44	43	expdcom 414	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
45	7, 30, 44	3jaod 1431	. 2 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
46	10	adantr 480	. . . . . 6 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (2^nd ‘𝑋) = 𝑦)
47	15	eqeq2d 2747	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨0, 𝑦⟩))
48	13	eqeq2d 2747	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩))
49	47, 48	anbi12d 632	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) ↔ (𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩)))
50	46, 49	syl 17	. . . . 5 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) ↔ (𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩)))
51		prcom 4689	. . . . . . . . . . 11 ⊢ {⟨0, 𝑏⟩, 𝐿} = {𝐿, ⟨0, 𝑏⟩}
52	51	eleq1i 2827	. . . . . . . . . 10 ⊢ ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸)
53	19, 20, 21, 22	pgnbgreunbgrlem2lem1 48360	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸)
54		prcom 4689	. . . . . . . . . . . . . 14 ⊢ {𝐾, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, 𝐾}
55	54	eleq1i 2827	. . . . . . . . . . . . 13 ⊢ ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸)
56		pm2.21 123	. . . . . . . . . . . . 13 ⊢ (¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → ({⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
57	55, 56	biimtrid 242	. . . . . . . . . . . 12 ⊢ (¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
58	53, 57	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
59	58	ex 412	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨0, 𝑏⟩} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
60	52, 59	biimtrid 242	. . . . . . . . 9 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
61	60	impcomd 411	. . . . . . . 8 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
62	61	ex 412	. . . . . . 7 ⊢ ((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
63	62	ancoms 458	. . . . . 6 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
64	63	adantld 490	. . . . 5 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
65	50, 64	biimtrdi 253	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
66	65	expdcom 414	. . 3 ⊢ (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
67		eqtr3 2758	. . . . . . . 8 ⊢ ((𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩) → 𝐾 = 𝐿)
68	67	ancoms 458	. . . . . . 7 ⊢ ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → 𝐾 = 𝐿)
69	68, 2	syl 17	. . . . . 6 ⊢ ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
70	69	impd 410	. . . . 5 ⊢ ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
71	70	a1d 25	. . . 4 ⊢ ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
72	71	ex 412	. . 3 ⊢ (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
73	47, 34	anbi12d 632	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
74	46, 73	syl 17	. . . . 5 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
75	19, 20, 21, 22	pgnbgreunbgrlem2lem2 48361	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸)
76	75, 57	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
77	76	ex 412	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨0, 𝑏⟩} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
78	52, 77	biimtrid 242	. . . . . . . . 9 ⊢ (((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
79	78	impcomd 411	. . . . . . . 8 ⊢ (((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
80	79	ex 412	. . . . . . 7 ⊢ ((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
81	80	ancoms 458	. . . . . 6 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
82	81	adantld 490	. . . . 5 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
83	74, 82	biimtrdi 253	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
84	83	expdcom 414	. . 3 ⊢ (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
85	66, 72, 84	3jaod 1431	. 2 ⊢ (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
86	33	eqeq2d 2747	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩))
87	86, 48	anbi12d 632	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) ↔ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩)))
88	46, 87	syl 17	. . . . 5 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) ↔ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩)))
89	19, 20, 21, 22	pgnbgreunbgrlem2lem3 48362	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸)
90	89, 57	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
91	90	ex 412	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨0, 𝑏⟩} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
92	52, 91	biimtrid 242	. . . . . . . . 9 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩)))
93	92	impcomd 411	. . . . . . . 8 ⊢ (((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
94	93	ex 412	. . . . . . 7 ⊢ ((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
95	94	ancoms 458	. . . . . 6 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
96	95	adantld 490	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
97	88, 96	biimtrdi 253	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
98	97	expdcom 414	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
99	86, 16	anbi12d 632	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) ↔ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩)))
100	46, 99	syl 17	. . . . 5 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) ↔ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩)))
101	19, 20, 21, 22	pgnbgreunbgrlem2lem2 48361	. . . . . . . . 9 ⊢ ((((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
102	101	pm2.21d 121	. . . . . . . 8 ⊢ ((((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
103	102	expimpd 453	. . . . . . 7 ⊢ (((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
104	103	ex 412	. . . . . 6 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
105	104	adantld 490	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
106	100, 105	biimtrdi 253	. . . 4 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
107	106	expdcom 414	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
108		eqtr3 2758	. . . . . . 7 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → 𝐿 = 𝐾)
109	108	eqcomd 2742	. . . . . 6 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → 𝐾 = 𝐿)
110	109, 3	syl 17	. . . . 5 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
111	110	a1d 25	. . . 4 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
112	111	ex 412	. . 3 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
113	98, 107, 112	3jaod 1431	. 2 ⊢ (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
114	45, 85, 113	3jaoi 1430	1 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((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: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 {cpr 4582 ⟨cop 4586 ‘cfv 6492 (class class class)co 7358 2^nd c2nd 7932 0cc0 11026 1c1 11027 + caddc 11029 − cmin 11364 2c2 12200 5c5 12203 ..^cfzo 13570 mod cmo 13789 Vtxcvtx 29069 Edgcedg 29120 NeighbVtx cnbgr 29405 gPetersenGr cgpg 48286

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-ceil 13713 df-mod 13790 df-hash 14254 df-dvds 16180 df-struct 17074 df-slot 17109 df-ndx 17121 df-base 17137 df-edgf 29062 df-vtx 29071 df-iedg 29072 df-edg 29121 df-umgr 29156 df-usgr 29224 df-gpg 48287

This theorem is referenced by: pgnbgreunbgrlem3 48364

W3C validator