pgnbgreunbgrlem2 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > pgnbgreunbgrlem2		Structured version Visualization version GIF version

Theorem pgnbgreunbgrlem2 48121

Description: Lemma 2 for pgnbgreunbgr 48129. 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 2751	. . . . . 6 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → 𝐿 = 𝐾)
2		eqneqall 2936	. . . . . . . 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 2737	. . . . . 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 11130	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3442	. . . . . . . 8 ⊢ 𝑦 ∈ V
10	8, 9	op2ndd 7942	. . . . . . 7 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
11		oveq1 7360	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
12	11	oveq1d 7368	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
13	12	opeq2d 4834	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩)
14	13	eqeq2d 2740	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩))
15		opeq2 4828	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (2^nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
16	15	eqeq2d 2740	. . . . . . . 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 48118	. . . . . . . . . 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 7360	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
32	31	oveq1d 7368	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
33	32	opeq2d 4834	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ = ⟨1, ((𝑦 − 2) mod 5)⟩)
34	33	eqeq2d 2740	. . . . . . . 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 48120	. . . . . . . . . 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 2740	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨0, 𝑦⟩))
48	13	eqeq2d 2740	. . . . . . 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 4686	. . . . . . . . . . 11 ⊢ {⟨0, 𝑏⟩, 𝐿} = {𝐿, ⟨0, 𝑏⟩}
52	51	eleq1i 2819	. . . . . . . . . 10 ⊢ ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸)
53	19, 20, 21, 22	pgnbgreunbgrlem2lem1 48118	. . . . . . . . . . . 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 4686	. . . . . . . . . . . . . 14 ⊢ {𝐾, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, 𝐾}
55	54	eleq1i 2819	. . . . . . . . . . . . 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 2751	. . . . . . . 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 48119	. . . . . . . . . . . 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 2740	. . . . . . 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 48120	. . . . . . . . . . . 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 48119	. . . . . . . . 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 2751	. . . . . . 7 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → 𝐿 = 𝐾)
109	108	eqcomd 2735	. . . . . 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 1540 ∈ wcel 2109 ≠ wne 2925 {cpr 4581 ⟨cop 4585 ‘cfv 6486 (class class class)co 7353 2^nd c2nd 7930 0cc0 11028 1c1 11029 + caddc 11031 − cmin 11366 2c2 12202 5c5 12205 ..^cfzo 13576 mod cmo 13792 Vtxcvtx 28960 Edgcedg 29011 NeighbVtx cnbgr 29296 gPetersenGr cgpg 48044

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12611 df-uz 12755 df-rp 12913 df-fz 13430 df-fzo 13577 df-fl 13715 df-ceil 13716 df-mod 13793 df-hash 14257 df-dvds 16183 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17140 df-edgf 28953 df-vtx 28962 df-iedg 28963 df-edg 29012 df-umgr 29047 df-usgr 29115 df-gpg 48045

This theorem is referenced by: pgnbgreunbgrlem3 48122

W3C validator