Theorem gpg3kgrtriex 48709

Description: All generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles. (Contributed by AV, 1-Oct-2025.)

Hypotheses

Ref	Expression
gpg3kgrtriex.n	⊢ 𝑁 = (3 · 𝐾)
gpg3kgrtriex.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)

Assertion

Ref	Expression
gpg3kgrtriex	⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺))

Distinct variable group:   𝑡,𝐺

Allowed substitution hints:   𝐾(𝑡)   𝑁(𝑡)

Proof of Theorem gpg3kgrtriex

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1ex 11191	. . . . . . . 8 ⊢ 1 ∈ V
2	1	prid2 4725	. . . . . . 7 ⊢ 1 ∈ {0, 1}
3	2	a1i 11	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 1 ∈ {0, 1})
4		gpg3kgrtriex.n	. . . . . . . 8 ⊢ 𝑁 = (3 · 𝐾)
5		3nn 12311	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 3 ∈ ℕ)
7		id 23	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
8	6, 7	nnmulcld 12280	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (3 · 𝐾) ∈ ℕ)
9	4, 8	eqeltrid 2869	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ ℕ)
10		lbfzo0 13719	. . . . . . 7 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
11	9, 10	sylibr 237	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 0 ∈ (0..^𝑁))
12	3, 11	opelxpd 5691	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁)))
13	4	gpg3kgrtriexlem4 48706	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
14	9, 13	jca 520	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
15		eqid 2765	. . . . . . . 8 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
16		eqid 2765	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
17	15, 16	gpgvtx 48663	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
18	17	eleq2d 2851	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
19	14, 18	syl 18	. . . . 5 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
20	12, 19	mpbird 260	. . . 4 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)))
21		gpg3kgrtriex.g	. . . . 5 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
22	21	fveq2i 6874	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
23	20, 22	eleqtrrdi 2876	. . 3 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
24		oveq2 7408	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx ⟨1, 0⟩))
25		biidd 265	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → ((𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
26	24, 25	rexeqbidv 3340	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → (∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ ∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
27	24, 26	rexeqbidv 3340	. . . 4 ⊢ (𝑎 = ⟨1, 0⟩ → (∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
28	27	adantl 486	. . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑎 = ⟨1, 0⟩) → (∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
29	4	gpg3kgrtriexlem3 48705	. . . . 5 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
30		eqid 2765	. . . . . . . . 9 ⊢ 1 = 1
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 = 1)
32	31	olcd 887	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (1 = 0 ∨ 1 = 1))
33	32, 11	jca 520	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁)))
34	29, 13	jca 520	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
35		eqid 2765	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
36	16, 15, 21, 35	opgpgvtx 48675	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
37	34, 36	syl 18	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
38	33, 37	mpbird 260	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
39		c0ex 11188	. . . . . . 7 ⊢ 0 ∈ V
40	1, 39	op1st 7982	. . . . . 6 ⊢ (1st ‘⟨1, 0⟩) = 1
41	40	a1i 11	. . . . 5 ⊢ (𝐾 ∈ ℕ → (1st ‘⟨1, 0⟩) = 1)
42		eqid 2765	. . . . . 6 ⊢ (𝐺 NeighbVtx ⟨1, 0⟩) = (𝐺 NeighbVtx ⟨1, 0⟩)
43	15, 21, 35, 42	gpgnbgrvtx1 48695	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) ∧ (⟨1, 0⟩ ∈ (Vtx‘𝐺) ∧ (1st ‘⟨1, 0⟩) = 1)) → (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩})
44	29, 13, 38, 41, 43	syl22anc 851	. . . 4 ⊢ (𝐾 ∈ ℕ → (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩})
45		neeq1 3022	. . . . . 6 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → (𝑏 ≠ 𝑐 ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐))
46		preq1 4695	. . . . . . 7 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → {𝑏, 𝑐} = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐})
47	46	eleq1d 2850	. . . . . 6 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → ({𝑏, 𝑐} ∈ (Edg‘𝐺) ↔ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐} ∈ (Edg‘𝐺)))
48	45, 47	anbi12d 643	. . . . 5 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → ((𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐 ∧ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐} ∈ (Edg‘𝐺))))
49		neeq2 3023	. . . . . 6 ⊢ (𝑐 = ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ → (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐 ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩))
50		preq2 4696	. . . . . . 7 ⊢ (𝑐 = ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ → {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐} = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩})
51	50	eleq1d 2850	. . . . . 6 ⊢ (𝑐 = ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ → ({⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐} ∈ (Edg‘𝐺) ↔ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
52	49, 51	anbi12d 643	. . . . 5 ⊢ (𝑐 = ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ → ((⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐 ∧ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐} ∈ (Edg‘𝐺)) ↔ (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∧ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))))
53		opex 5436	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ V
54	53	tpid1 4730	. . . . . 6 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}
55		eleq2 2854	. . . . . . 7 ⊢ ((𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} → (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩) ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}))
56	55	adantl 486	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩) ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}))
57	54, 56	mpbiri 261	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩))
58		opex 5436	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ V
59	58	tpid3 4735	. . . . . 6 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}
60		eleq2 2854	. . . . . . 7 ⊢ ((𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} → (⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩) ↔ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}))
61	60	adantl 486	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → (⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩) ↔ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}))
62	59, 61	mpbiri 261	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ (𝐺 NeighbVtx ⟨1, 0⟩))
63	4	gpg3kgrtriexlem5 48707	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
64	1, 39	op2nd 7983	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨1, 0⟩) = 0
65	64	oveq1i 7410	. . . . . . . . . . . 12 ⊢ ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾)
66		nncn 12232	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
67	66	addlidd 11399	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (0 + 𝐾) = 𝐾)
68	65, 67	eqtrid 2812	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
69	68	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
70	64	oveq1i 7410	. . . . . . . . . . . . 13 ⊢ ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾)
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
72		df-neg 11432	. . . . . . . . . . . 12 ⊢ -𝐾 = (0 − 𝐾)
73	71, 72	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
74	73	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
75	63, 69, 74	3netr4d 3037	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁))
76	75	olcd 887	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (1 ≠ 1 ∨ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)))
77		ovex 7433	. . . . . . . . 9 ⊢ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ∈ V
78	1, 77	opthne 5455	. . . . . . . 8 ⊢ (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ↔ (1 ≠ 1 ∨ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)))
79	76, 78	sylibr 237	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩)
80	64	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → (2nd ‘⟨1, 0⟩) = 0)
81	80	oveq1d 7415	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾))
82	81, 67	eqtrd 2800	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
83	82	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
84	83	opeq2d 4841	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ = ⟨1, (𝐾 mod 𝑁)⟩)
85	80	oveq1d 7415	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
86	85, 72	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
87	86	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
88	87	opeq2d 4841	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ = ⟨1, (-𝐾 mod 𝑁)⟩)
89	84, 88	preq12d 4703	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩})
90		eqid 2765	. . . . . . . . 9 ⊢ {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}
91	4, 21, 90	gpg3kgrtriexlem6 48708	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} ∈ (Edg‘𝐺))
92	89, 91	eqeltrd 2865	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
93	79, 92	jca 520	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∧ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
94	93	adantr 485	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → (⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∧ {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺)))
95	48, 52, 57, 62, 94	2rspcedvdw 3598	. . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (𝐺 NeighbVtx ⟨1, 0⟩) = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘⟨1, 0⟩)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩}) → ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
96	44, 95	mpdan 699	. . 3 ⊢ (𝐾 ∈ ℕ → ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
97	23, 28, 96	rspcedvd 3586	. 2 ⊢ (𝐾 ∈ ℕ → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
98		gpgusgra 48677	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
99	21, 98	eqeltrid 2869	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
100		eqid 2765	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
101		eqid 2765	. . . 4 ⊢ (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
102	35, 100, 101	usgrgrtrirex 48570	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
103	34, 99, 102	3syl 19	. 2 ⊢ (𝐾 ∈ ℕ → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
104	97, 103	mpbird 260	1 ⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {cpr 4587  {ctp 4589  ⟨cop 4591   × cxp 5650  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  0cc0 11088  1c1 11089   + caddc 11091   · cmul 11093   − cmin 11429  -cneg 11430   / cdiv 11859  ℕcn 12224  2c2 12286  3c3 12287  ℤ_≥cuz 12853  ..^cfzo 13673  ⌈cceil 13815   mod cmo 13893  Vtxcvtx 29255  Edgcedg 29306  USGraphcusgr 29408   NeighbVtx cnbgr 29591  GrTrianglescgrtri 48557   gPetersenGr cgpg 48660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-3o 8443  df-oadd 8445  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-rp 13008  df-ico 13369  df-fz 13527  df-fzo 13674  df-fl 13816  df-ceil 13817  df-mod 13894  df-hash 14358  df-dvds 16301  df-struct 17197  df-slot 17232  df-ndx 17244  df-base 17260  df-edgf 29248  df-vtx 29257  df-iedg 29258  df-edg 29307  df-uhgr 29317  df-upgr 29341  df-umgr 29342  df-uspgr 29409  df-usgr 29410  df-nbgr 29592  df-grtri 48558  df-gpg 48661

This theorem is referenced by: (None)