Theorem gpg3kgrtriex 48371

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 11132	. . . . . . . 8 ⊢ 1 ∈ V
2	1	prid2 4721	. . . . . . 7 ⊢ 1 ∈ {0, 1}
3	2	a1i 11	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 1 ∈ {0, 1})
4		gpg3kgrtriex.n	. . . . . . . 8 ⊢ 𝑁 = (3 · 𝐾)
5		3nn 12228	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 3 ∈ ℕ)
7		id 22	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
8	6, 7	nnmulcld 12202	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (3 · 𝐾) ∈ ℕ)
9	4, 8	eqeltrid 2841	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ ℕ)
10		lbfzo0 13619	. . . . . . 7 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
11	9, 10	sylibr 234	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 0 ∈ (0..^𝑁))
12	3, 11	opelxpd 5664	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁)))
13	4	gpg3kgrtriexlem4 48368	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
14	9, 13	jca 511	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
15		eqid 2737	. . . . . . . 8 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
16		eqid 2737	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
17	15, 16	gpgvtx 48325	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
18	17	eleq2d 2823	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
19	14, 18	syl 17	. . . . 5 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
20	12, 19	mpbird 257	. . . 4 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)))
21		gpg3kgrtriex.g	. . . . 5 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
22	21	fveq2i 6838	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
23	20, 22	eleqtrrdi 2848	. . 3 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
24		oveq2 7368	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx ⟨1, 0⟩))
25		biidd 262	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → ((𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
26	24, 25	rexeqbidv 3318	. . . . 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 3318	. . . 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 481	. . 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 48367	. . . . 5 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
30		eqid 2737	. . . . . . . . 9 ⊢ 1 = 1
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 = 1)
32	31	olcd 875	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (1 = 0 ∨ 1 = 1))
33	32, 11	jca 511	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁)))
34	29, 13	jca 511	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
35		eqid 2737	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
36	16, 15, 21, 35	opgpgvtx 48337	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
37	34, 36	syl 17	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
38	33, 37	mpbird 257	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
39		c0ex 11130	. . . . . . 7 ⊢ 0 ∈ V
40	1, 39	op1st 7943	. . . . . 6 ⊢ (1st ‘⟨1, 0⟩) = 1
41	40	a1i 11	. . . . 5 ⊢ (𝐾 ∈ ℕ → (1st ‘⟨1, 0⟩) = 1)
42		eqid 2737	. . . . . 6 ⊢ (𝐺 NeighbVtx ⟨1, 0⟩) = (𝐺 NeighbVtx ⟨1, 0⟩)
43	15, 21, 35, 42	gpgnbgrvtx1 48357	. . . . 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 839	. . . 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 2995	. . . . . 6 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → (𝑏 ≠ 𝑐 ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐))
46		preq1 4691	. . . . . . 7 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → {𝑏, 𝑐} = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐})
47	46	eleq1d 2822	. . . . . 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 633	. . . . 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 2996	. . . . . 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 4692	. . . . . . 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 2822	. . . . . 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 633	. . . . 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 5413	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ V
54	53	tpid1 4726	. . . . . 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 2826	. . . . . . 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 481	. . . . . 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 258	. . . . 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 5413	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ V
59	58	tpid3 4731	. . . . . 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 2826	. . . . . . 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 481	. . . . . 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 258	. . . . 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 48369	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
64	1, 39	op2nd 7944	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨1, 0⟩) = 0
65	64	oveq1i 7370	. . . . . . . . . . . 12 ⊢ ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾)
66		nncn 12157	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
67	66	addlidd 11338	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (0 + 𝐾) = 𝐾)
68	65, 67	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
69	68	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
70	64	oveq1i 7370	. . . . . . . . . . . . 13 ⊢ ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾)
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
72		df-neg 11371	. . . . . . . . . . . 12 ⊢ -𝐾 = (0 − 𝐾)
73	71, 72	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
74	73	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
75	63, 69, 74	3netr4d 3010	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁))
76	75	olcd 875	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (1 ≠ 1 ∨ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)))
77		ovex 7393	. . . . . . . . 9 ⊢ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ∈ V
78	1, 77	opthne 5431	. . . . . . . 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 234	. . . . . . 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 7375	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾))
82	81, 67	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
83	82	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
84	83	opeq2d 4837	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ = ⟨1, (𝐾 mod 𝑁)⟩)
85	80	oveq1d 7375	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
86	85, 72	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
87	86	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
88	87	opeq2d 4837	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ = ⟨1, (-𝐾 mod 𝑁)⟩)
89	84, 88	preq12d 4699	. . . . . . . 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 2737	. . . . . . . . 9 ⊢ {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}
91	4, 21, 90	gpg3kgrtriexlem6 48370	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} ∈ (Edg‘𝐺))
92	89, 91	eqeltrd 2837	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩} ∈ (Edg‘𝐺))
93	79, 92	jca 511	. . . . . 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 480	. . . . 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 3591	. . . 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 688	. . 3 ⊢ (𝐾 ∈ ℕ → ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
97	23, 28, 96	rspcedvd 3579	. 2 ⊢ (𝐾 ∈ ℕ → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
98		gpgusgra 48339	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
99	21, 98	eqeltrid 2841	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
100		eqid 2737	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
101		eqid 2737	. . . 4 ⊢ (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
102	35, 100, 101	usgrgrtrirex 48232	. . 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 18	. 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 257	1 ⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061  {cpr 4583  {ctp 4585  ⟨cop 4587   × cxp 5623  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  0cc0 11030  1c1 11031   + caddc 11033   · cmul 11035   − cmin 11368  -cneg 11369   / cdiv 11798  ℕcn 12149  2c2 12204  3c3 12205  ℤ_≥cuz 12755  ..^cfzo 13574  ⌈cceil 13715   mod cmo 13793  Vtxcvtx 29052  Edgcedg 29103  USGraphcusgr 29205   NeighbVtx cnbgr 29388  GrTrianglescgrtri 48219   gPetersenGr cgpg 48322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-3o 8401  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-xnn0 12479  df-z 12493  df-dec 12612  df-uz 12756  df-rp 12910  df-ico 13271  df-fz 13428  df-fzo 13575  df-fl 13716  df-ceil 13717  df-mod 13794  df-hash 14258  df-dvds 16184  df-struct 17078  df-slot 17113  df-ndx 17125  df-base 17141  df-edgf 29045  df-vtx 29054  df-iedg 29055  df-edg 29104  df-uhgr 29114  df-upgr 29138  df-umgr 29139  df-uspgr 29206  df-usgr 29207  df-nbgr 29389  df-grtri 48220  df-gpg 48323

This theorem is referenced by: (None)