pgnbgreunbgrlem2 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2 49291

Description: Lemma 2 for pgnbgreunbgr 49299. 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 2782	. . . . . 6 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩) → 𝐿 = 𝐾)
2		eqneqall 2966	. . . . . . . 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 416	. . . . . . 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 2768	. . . . . 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 18	. . . . 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 26	. . . 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 418	. . 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 11304	. . . . . . . 8 ⊢ 1 ∈ V
9		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
10	8, 9	op2ndd 8012	. . . . . . 7 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
11		oveq1 7427	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
12	11	oveq1d 7435	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
13	12	opeq2d 4839	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩)
14	13	eqeq2d 2771	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩))
15		opeq2 4833	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (2^nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
16	15	eqeq2d 2771	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨0, 𝑦⟩))
17	14, 16	anbi12d 644	. . . . . . 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 18	. . . . . 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 49288	. . . . . . . . . 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 122	. . . . . . . . 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 459	. . . . . . . 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 418	. . . . . . 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 496	. . . . . 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 256	. . . . 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 486	. . . 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 420	. . 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 7427	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
32	31	oveq1d 7435	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
33	32	opeq2d 4839	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ = ⟨1, ((𝑦 − 2) mod 5)⟩)
34	33	eqeq2d 2771	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩))
35	14, 34	anbi12d 644	. . . . . . 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 18	. . . . . 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 49290	. . . . . . . . . 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 122	. . . . . . . . 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 459	. . . . . . . 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 418	. . . . . . 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 496	. . . . . 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 256	. . . . 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 486	. . . 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 420	. . 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 1456	. 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 486	. . . . . 6 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (2^nd ‘𝑋) = 𝑦)
47	15	eqeq2d 2771	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨0, 𝑦⟩))
48	13	eqeq2d 2771	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩))
49	47, 48	anbi12d 644	. . . . . 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 18	. . . . 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 4692	. . . . . . . . . . 11 ⊢ {⟨0, 𝑏⟩, 𝐿} = {𝐿, ⟨0, 𝑏⟩}
52	51	eleq1i 2851	. . . . . . . . . 10 ⊢ ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {𝐿, ⟨0, 𝑏⟩} ∈ 𝐸)
53	19, 20, 21, 22	pgnbgreunbgrlem2lem1 49288	. . . . . . . . . . . 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 4692	. . . . . . . . . . . . . 14 ⊢ {𝐾, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, 𝐾}
55	54	eleq1i 2851	. . . . . . . . . . . . 13 ⊢ ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸)
56		pm2.21 124	. . . . . . . . . . . . 13 ⊢ (¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → ({⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
57	55, 56	biimtrid 245	. . . . . . . . . . . 12 ⊢ (¬ {⟨0, 𝑏⟩, 𝐾} ∈ 𝐸 → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → 𝑋 = ⟨0, 𝑏⟩))
58	53, 57	syl 18	. . . . . . . . . . 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 418	. . . . . . . . . 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 245	. . . . . . . . 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 417	. . . . . . . 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 418	. . . . . . 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 464	. . . . . 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 496	. . . . 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 256	. . . 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 420	. . 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 2782	. . . . . . . 8 ⊢ ((𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩) → 𝐾 = 𝐿)
68	67	ancoms 464	. . . . . . 7 ⊢ ((𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∧ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩) → 𝐾 = 𝐿)
69	68, 2	syl 18	. . . . . 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 416	. . . . 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 26	. . . 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 418	. . 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 644	. . . . . 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 18	. . . . 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 49289	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 418	. . . . . . . . . 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 245	. . . . . . . . 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 417	. . . . . . . 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 418	. . . . . . 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 464	. . . . . 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 496	. . . . 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 256	. . . 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 420	. . 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 1456	. 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 2771	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩))
87	86, 48	anbi12d 644	. . . . . 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 18	. . . . 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 49290	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 418	. . . . . . . . . 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 245	. . . . . . . . 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 417	. . . . . . . 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 418	. . . . . . 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 464	. . . . . 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 496	. . . . 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 256	. . . 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 420	. . 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 644	. . . . . 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 18	. . . . 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 49289	. . . . . . . . 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 122	. . . . . . . 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 459	. . . . . . 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 418	. . . . . 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 496	. . . . 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 256	. . . 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 420	. . 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 2782	. . . . . . 7 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → 𝐿 = 𝐾)
109	108	eqcomd 2766	. . . . . 6 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → 𝐾 = 𝐿)
110	109, 3	syl 18	. . . . 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 26	. . . 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 418	. . 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 1456	. 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 1454	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 {cpr 4585 ⟨cop 4589 ‘cfv 6538 (class class class)co 7420 2^nd c2nd 8000 0cc0 11201 1c1 11202 + caddc 11204 − cmin 11542 2c2 12398 5c5 12401 ..^cfzo 13789 mod cmo 14010 Vtxcvtx 29575 Edgcedg 29626 NeighbVtx cnbgr 29914 gPetersenGr cgpg 49214

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-oadd 8481 df-er 8718 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9435 df-inf 9436 df-dju 9983 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-xnn0 12681 df-z 12695 df-dec 12816 df-uz 12967 df-rp 13122 df-fz 13641 df-fzo 13790 df-fl 13933 df-ceil 13934 df-mod 14011 df-hash 14476 df-dvds 16424 df-struct 17326 df-slot 17361 df-ndx 17373 df-base 17389 df-edgf 29568 df-vtx 29577 df-iedg 29578 df-edg 29627 df-umgr 29662 df-usgr 29733 df-gpg 49215

This theorem is used by: pgnbgreunbgrlem3 49292

W3C validator