pgnbgreunbgrlem2 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2 48593

Description: Lemma 2 for pgnbgreunbgr 48601. 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 11140	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3433	. . . . . . . 8 ⊢ 𝑦 ∈ V
10	8, 9	op2ndd 7953	. . . . . . 7 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
11		oveq1 7374	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
12	11	oveq1d 7382	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
13	12	opeq2d 4823	. . . . . . . . 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 4817	. . . . . . . . 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 633	. . . . . . 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 48590	. . . . . . . . . 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 7374	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
32	31	oveq1d 7382	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
33	32	opeq2d 4823	. . . . . . . . 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 633	. . . . . . 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 48592	. . . . . . . . . 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 1432	. 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 633	. . . . . 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 4676	. . . . . . . . . . 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 48590	. . . . . . . . . . . 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 4676	. . . . . . . . . . . . . 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 633	. . . . . 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 48591	. . . . . . . . . . . 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 1432	. 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 633	. . . . . 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 48592	. . . . . . . . . . . 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 633	. . . . . 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 48591	. . . . . . . . 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 1432	. 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 1431	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 1086 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 {cpr 4569 ⟨cop 4573 ‘cfv 6498 (class class class)co 7367 2^nd c2nd 7941 0cc0 11038 1c1 11039 + caddc 11041 − cmin 11377 2c2 12236 5c5 12239 ..^cfzo 13608 mod cmo 13828 Vtxcvtx 29065 Edgcedg 29116 NeighbVtx cnbgr 29401 gPetersenGr cgpg 48516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-oadd 8409 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-ceil 13752 df-mod 13829 df-hash 14293 df-dvds 16222 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-edgf 29058 df-vtx 29067 df-iedg 29068 df-edg 29117 df-umgr 29152 df-usgr 29220 df-gpg 48517

This theorem is referenced by: pgnbgreunbgrlem3 48594

W3C validator