pgnbgreunbgrlem2lem3 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2lem3 48079

Description: Lemma 3 for pgnbgreunbgrlem2 48080. (Contributed by AV, 17-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
pgnbgreunbgrlem2lem3	⊢ ((((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)

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 pgnbgreunbgrlem2lem3**
Step	Hyp	Ref	Expression
1		prcom 4692	. . . . . . . 8 ⊢ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩}
2	1	eleq1i 2819	. . . . . . 7 ⊢ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸)
3	2	a1i 11	. . . . . 6 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸))
4		5eluz3 12818	. . . . . . . 8 ⊢ 5 ∈ (ℤ_≥‘3)
5		pglem 48055	. . . . . . . 8 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
6	4, 5	pm3.2i 470	. . . . . . 7 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
7		c0ex 11144	. . . . . . . 8 ⊢ 0 ∈ V
8		vex 3448	. . . . . . . 8 ⊢ 𝑏 ∈ V
9	7, 8	op1st 7955	. . . . . . 7 ⊢ (1^st ‘⟨0, 𝑏⟩) = 0
10		eqid 2729	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
11		pgnbgreunbgr.g	. . . . . . . 8 ⊢ 𝐺 = (5 gPetersenGr 2)
12		pgnbgreunbgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
13		pgnbgreunbgr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
14	10, 11, 12, 13	gpgvtxedg0 48027	. . . . . . 7 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (1^st ‘⟨0, 𝑏⟩) = 0 ∧ {⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) → (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
15	6, 9, 14	mp3an12 1453	. . . . . 6 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸 → (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
16	3, 15	biimtrdi 253	. . . . 5 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)))
17	10, 11, 12, 13	gpgvtxedg0 48027	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (1^st ‘⟨0, 𝑏⟩) = 0 ∧ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
18	6, 9, 17	mp3an12 1453	. . . . . . . 8 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
19		1ex 11146	. . . . . . . . . . 11 ⊢ 1 ∈ V
20		ovex 7402	. . . . . . . . . . 11 ⊢ ((𝑦 + 2) mod 5) ∈ V
21	19, 20	opth 5431	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)))
22		ax-1ne0 11113	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
23		eqneqall 2936	. . . . . . . . . . . 12 ⊢ (1 = 0 → (1 ≠ 0 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
24	22, 23	mpi 20	. . . . . . . . . . 11 ⊢ (1 = 0 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
26	21, 25	sylbi 217	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
27	19, 20	opth 5431	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ↔ (1 = 1 ∧ ((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩)))
28	7, 8	op2nd 7956	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 𝑏⟩) = 𝑏
29	28	eqeq2i 2742	. . . . . . . . . . 11 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ ((𝑦 + 2) mod 5) = 𝑏)
30		ovex 7402	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 − 2) mod 5) ∈ V
31	19, 30	opth 5431	. . . . . . . . . . . . . . 15 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 − 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)))
32		eqneqall 2936	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 0 → (1 ≠ 0 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))))
33	22, 32	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ (1 = 0 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
34	33	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((1 = 0 ∧ ((𝑦 − 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
35	31, 34	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
36	19, 30	opth 5431	. . . . . . . . . . . . . . 15 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ↔ (1 = 1 ∧ ((𝑦 − 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩)))
37	28	eqeq2i 2742	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 − 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ ((𝑦 − 2) mod 5) = 𝑏)
38		eqeq2 2741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ((𝑦 − 2) mod 5) → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
39	38	eqcoms 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 − 2) mod 5) = 𝑏 → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
40	39	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^5) ∧ ((𝑦 − 2) mod 5) = 𝑏) → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
41		elfzoelz 13596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℤ)
42		2z 12541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℤ
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℤ)
44	41, 43	zaddcld 12618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → (𝑦 + 2) ∈ ℤ)
45	41, 43	zsubcld 12619	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → (𝑦 − 2) ∈ ℤ)
46		5nn 12248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 5 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → 5 ∈ ℕ)
48		difmod0 16233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 + 2) ∈ ℤ ∧ (𝑦 − 2) ∈ ℤ ∧ 5 ∈ ℕ) → ((((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
49	48	bicomd 223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 + 2) ∈ ℤ ∧ (𝑦 − 2) ∈ ℤ ∧ 5 ∈ ℕ) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) ↔ (((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0))
50	44, 45, 47, 49	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) ↔ (((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0))
51	41	zcnd 12615	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℂ)
52		2cnd 12240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℂ)
53	51, 52, 52	pnncand 11548	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 − 2)) = (2 + 2))
54		2p2e4 12292	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 + 2) = 4
55	53, 54	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 − 2)) = 4)
56	55	oveq1d 7384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) − (𝑦 − 2)) mod 5) = (4 mod 5))
57	56	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0 ↔ (4 mod 5) = 0))
58		4re 12246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℝ
59		5rp 12934	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 5 ∈ ℝ⁺
60		0re 11152	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
61		4pos 12269	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 < 4
62	60, 58, 61	ltleii 11273	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 4
63		4lt5 12334	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 < 5
64		modid 13834	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((4 ∈ ℝ ∧ 5 ∈ ℝ⁺) ∧ (0 ≤ 4 ∧ 4 < 5)) → (4 mod 5) = 4)
65	58, 59, 62, 63, 64	mp4an 693	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (4 mod 5) = 4
66	65	eqeq1i 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 mod 5) = 0 ↔ 4 = 0)
67		4ne0 12270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ≠ 0
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → 4 ≠ 0)
69	68	necon2bi 2955	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (4 = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
70	66, 69	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((4 mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
71	57, 70	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
72	50, 71	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^5) ∧ ((𝑦 − 2) mod 5) = 𝑏) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
74	40, 73	sylbid 240	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0..^5) ∧ ((𝑦 − 2) mod 5) = 𝑏) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
75	74	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 − 2) mod 5) = 𝑏 → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
76	75	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 − 2) mod 5) = 𝑏 → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
77	76	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 − 2) mod 5) = 𝑏 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
78	37, 77	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 − 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
79	36, 78	simplbiim 504	. . . . . . . . . . . . . 14 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
80	19, 30	opth 5431	. . . . . . . . . . . . . . 15 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 − 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)))
81	33	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((1 = 0 ∧ ((𝑦 − 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
82	80, 81	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
83	35, 79, 82	3jaoi 1430	. . . . . . . . . . . . 13 ⊢ ((⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (((𝑦 + 2) mod 5) = 𝑏 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
84	83	com13 88	. . . . . . . . . . . 12 ⊢ (((𝑦 + 2) mod 5) = 𝑏 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ((⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
85	84	impd 410	. . . . . . . . . . 11 ⊢ (((𝑦 + 2) mod 5) = 𝑏 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
86	29, 85	sylbi 217	. . . . . . . . . 10 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
87	27, 86	simplbiim 504	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
88	19, 20	opth 5431	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)))
89	24	adantr 480	. . . . . . . . . 10 ⊢ ((1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
90	88, 89	sylbi 217	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
91	26, 87, 90	3jaoi 1430	. . . . . . . 8 ⊢ ((⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
92	18, 91	syl 17	. . . . . . 7 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
93		ax-1 6	. . . . . . 7 ⊢ (¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
94	92, 93	pm2.61i 182	. . . . . 6 ⊢ (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
95	94	ex 412	. . . . 5 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ((⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
96	16, 95	syld 47	. . . 4 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
97	96	adantl 481	. . 3 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
98		preq1 4693	. . . . . . 7 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩})
99	98	eleq1d 2813	. . . . . 6 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
100	99	adantl 481	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
101		preq2 4694	. . . . . . . 8 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩})
102	101	eleq1d 2813	. . . . . . 7 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
103	102	notbid 318	. . . . . 6 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
104	103	adantr 480	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
105	100, 104	imbi12d 344	. . . 4 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
106	105	adantr 480	. . 3 ⊢ (((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) ↔ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
107	97, 106	mpbird 257	. 2 ⊢ (((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))
108	107	imp 406	1 ⊢ ((((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)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {cpr 4587 ⟨cop 4591 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 4c4 12219 5c5 12220 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 ..^cfzo 13591 ⌈cceil 13729 mod cmo 13807 Vtxcvtx 28899 Edgcedg 28950 NeighbVtx cnbgr 29235 gPetersenGr cgpg 48004

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-ceil 13731 df-mod 13808 df-hash 14272 df-dvds 16199 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-edgf 28892 df-vtx 28901 df-iedg 28902 df-edg 28951 df-umgr 28986 df-usgr 29054 df-gpg 48005

This theorem is referenced by: pgnbgreunbgrlem2 48080

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