pgnbgreunbgrlem2lem3 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2lem3 48881

Description: Lemma 3 for pgnbgreunbgrlem2 48882. (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 4698	. . . . . . . 8 ⊢ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} = {⟨0, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩}
2	1	eleq1i 2854	. . . . . . 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 12902	. . . . . . . 8 ⊢ 5 ∈ (ℤ_≥‘3)
5		pglem 48856	. . . . . . . 8 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
6	4, 5	pm3.2i 475	. . . . . . 7 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
7		c0ex 11195	. . . . . . . 8 ⊢ 0 ∈ V
8		vex 3459	. . . . . . . 8 ⊢ 𝑏 ∈ V
9	7, 8	op1st 7990	. . . . . . 7 ⊢ (1^st ‘⟨0, 𝑏⟩) = 0
10		eqid 2763	. . . . . . . 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 48828	. . . . . . 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 1480	. . . . . 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 256	. . . . 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 48828	. . . . . . . . 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 1480	. . . . . . . 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 11198	. . . . . . . . . . 11 ⊢ 1 ∈ V
20		ovex 7443	. . . . . . . . . . 11 ⊢ ((𝑦 + 2) mod 5) ∈ V
21	19, 20	opth 5458	. . . . . . . . . 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 11164	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
23		eqneqall 2969	. . . . . . . . . . . 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 21	. . . . . . . . . . 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 485	. . . . . . . . . 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 220	. . . . . . . . 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 5458	. . . . . . . . . 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 7991	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 𝑏⟩) = 𝑏
29	28	eqeq2i 2776	. . . . . . . . . . 11 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ ((𝑦 + 2) mod 5) = 𝑏)
30		ovex 7443	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 − 2) mod 5) ∈ V
31	19, 30	opth 5458	. . . . . . . . . . . . . . 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 2969	. . . . . . . . . . . . . . . . 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 21	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . 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 5458	. . . . . . . . . . . . . . 15 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ↔ (1 = 1 ∧ ((𝑦 − 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩)))
37	28	eqeq2i 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 − 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ ((𝑦 − 2) mod 5) = 𝑏)
38		eqeq2 2775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ((𝑦 − 2) mod 5) → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
39	38	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 − 2) mod 5) = 𝑏 → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
40	39	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^5) ∧ ((𝑦 − 2) mod 5) = 𝑏) → (((𝑦 + 2) mod 5) = 𝑏 ↔ ((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5)))
41		elfzoelz 13683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℤ)
42		2z 12621	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℤ
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℤ)
44	41, 43	zaddcld 12699	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → (𝑦 + 2) ∈ ℤ)
45	41, 43	zsubcld 12700	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → (𝑦 − 2) ∈ ℤ)
46		5nn 12322	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 5 ∈ ℕ
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → 5 ∈ ℕ)
48		difmod0 16340	. . . . . . . . . . . . . . . . . . . . . . . 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 226	. . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) ↔ (((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0))
51	41	zcnd 12696	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℂ)
52		2cnd 12314	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℂ)
53	51, 52, 52	pnncand 11603	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 − 2)) = (2 + 2))
54		2p2e4 12370	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 + 2) = 4
55	53, 54	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 − 2)) = 4)
56	55	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) − (𝑦 − 2)) mod 5) = (4 mod 5))
57	56	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0 ↔ (4 mod 5) = 0))
58		4re 12320	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℝ
59		5rp 13018	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 5 ∈ ℝ⁺
60		0re 11205	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
61		4pos 12346	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 < 4
62	60, 58, 61	ltleii 11328	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 4
63		4lt5 12415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 < 5
64		modid 13925	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((4 ∈ ℝ ∧ 5 ∈ ℝ⁺) ∧ (0 ≤ 4 ∧ 4 < 5)) → (4 mod 5) = 4)
65	58, 59, 62, 63, 64	mp4an 705	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (4 mod 5) = 4
66	65	eqeq1i 2768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 mod 5) = 0 ↔ 4 = 0)
67		4ne0 12347	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ≠ 0
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → 4 ≠ 0)
69	68	necon2bi 2988	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (4 = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
70	66, 69	sylbi 220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((4 mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
71	57, 70	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 − 2)) mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
72	50, 71	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) mod 5) = ((𝑦 − 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
73	72	adantr 485	. . . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . 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 5458	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . 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 89	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 220	. . . . . . . . . 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 513	. . . . . . . . 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 5458	. . . . . . . . . 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 485	. . . . . . . . . 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 220	. . . . . . . . 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 1454	. . . . . . . 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 18	. . . . . . 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 184	. . . . . 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 417	. . . . 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 48	. . . 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 486	. . 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 4699	. . . . . . 7 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩})
99	98	eleq1d 2848	. . . . . 6 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
100	99	adantl 486	. . . . 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 4700	. . . . . . . 8 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩})
102	101	eleq1d 2848	. . . . . . 7 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
103	102	notbid 321	. . . . . 6 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
104	103	adantr 485	. . . . 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 347	. . . 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 485	. . 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 260	. 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 411	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 209 ∧ wa 400 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 {cpr 4591 ⟨cop 4595 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 2c2 12290 3c3 12291 4c4 12292 5c5 12293 ℤcz 12586 ℤ_≥cuz 12857 ℝ⁺crp 13011 ..^cfzo 13678 ⌈cceil 13820 mod cmo 13898 Vtxcvtx 29346 Edgcedg 29397 NeighbVtx cnbgr 29682 gPetersenGr cgpg 48805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-fl 13821 df-ceil 13822 df-mod 13899 df-hash 14363 df-dvds 16306 df-struct 17202 df-slot 17237 df-ndx 17249 df-base 17265 df-edgf 29339 df-vtx 29348 df-iedg 29349 df-edg 29398 df-umgr 29433 df-usgr 29501 df-gpg 48806

This theorem is referenced by: pgnbgreunbgrlem2 48882

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