Theorem gpg3kgrtriex 48099

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 11100	. . . . . . . 8 ⊢ 1 ∈ V
2	1	prid2 4714	. . . . . . 7 ⊢ 1 ∈ {0, 1}
3	2	a1i 11	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 1 ∈ {0, 1})
4		gpg3kgrtriex.n	. . . . . . . 8 ⊢ 𝑁 = (3 · 𝐾)
5		3nn 12196	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 3 ∈ ℕ)
7		id 22	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
8	6, 7	nnmulcld 12170	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (3 · 𝐾) ∈ ℕ)
9	4, 8	eqeltrid 2833	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ ℕ)
10		lbfzo0 13591	. . . . . . 7 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
11	9, 10	sylibr 234	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 0 ∈ (0..^𝑁))
12	3, 11	opelxpd 5653	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁)))
13	4	gpg3kgrtriexlem4 48096	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
14	9, 13	jca 511	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
15		eqid 2730	. . . . . . . 8 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
16		eqid 2730	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
17	15, 16	gpgvtx 48053	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
18	17	eleq2d 2815	. . . . . 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 6820	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
23	20, 22	eleqtrrdi 2840	. . 3 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
24		oveq2 7349	. . . . 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 3311	. . . . 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 3311	. . . 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 48095	. . . . 5 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
30		eqid 2730	. . . . . . . . 9 ⊢ 1 = 1
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 = 1)
32	31	olcd 874	. . . . . . 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 2730	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
36	16, 15, 21, 35	opgpgvtx 48065	. . . . . . 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 11098	. . . . . . 7 ⊢ 0 ∈ V
40	1, 39	op1st 7924	. . . . . 6 ⊢ (1st ‘⟨1, 0⟩) = 1
41	40	a1i 11	. . . . 5 ⊢ (𝐾 ∈ ℕ → (1st ‘⟨1, 0⟩) = 1)
42		eqid 2730	. . . . . 6 ⊢ (𝐺 NeighbVtx ⟨1, 0⟩) = (𝐺 NeighbVtx ⟨1, 0⟩)
43	15, 21, 35, 42	gpgnbgrvtx1 48085	. . . . 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 838	. . . 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 2988	. . . . . 6 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → (𝑏 ≠ 𝑐 ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐))
46		preq1 4684	. . . . . . 7 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → {𝑏, 𝑐} = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐})
47	46	eleq1d 2814	. . . . . 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 632	. . . . 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 2989	. . . . . 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 4685	. . . . . . 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 2814	. . . . . 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 632	. . . . 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 5402	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ V
54	53	tpid1 4719	. . . . . 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 2818	. . . . . . 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 5402	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ V
59	58	tpid3 4724	. . . . . 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 2818	. . . . . . 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 48097	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
64	1, 39	op2nd 7925	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨1, 0⟩) = 0
65	64	oveq1i 7351	. . . . . . . . . . . 12 ⊢ ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾)
66		nncn 12125	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
67	66	addlidd 11306	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (0 + 𝐾) = 𝐾)
68	65, 67	eqtrid 2777	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
69	68	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
70	64	oveq1i 7351	. . . . . . . . . . . . 13 ⊢ ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾)
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
72		df-neg 11339	. . . . . . . . . . . 12 ⊢ -𝐾 = (0 − 𝐾)
73	71, 72	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
74	73	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
75	63, 69, 74	3netr4d 3003	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁))
76	75	olcd 874	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (1 ≠ 1 ∨ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ≠ (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)))
77		ovex 7374	. . . . . . . . 9 ⊢ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ∈ V
78	1, 77	opthne 5420	. . . . . . . 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 7356	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾))
82	81, 67	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
83	82	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
84	83	opeq2d 4830	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ = ⟨1, (𝐾 mod 𝑁)⟩)
85	80	oveq1d 7356	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
86	85, 72	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
87	86	oveq1d 7356	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
88	87	opeq2d 4830	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ = ⟨1, (-𝐾 mod 𝑁)⟩)
89	84, 88	preq12d 4692	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}
91	4, 21, 90	gpg3kgrtriexlem6 48098	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} ∈ (Edg‘𝐺))
92	89, 91	eqeltrd 2829	. . . . . . 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 3589	. . . 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 687	. . 3 ⊢ (𝐾 ∈ ℕ → ∃𝑏 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)∃𝑐 ∈ (𝐺 NeighbVtx ⟨1, 0⟩)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
97	23, 28, 96	rspcedvd 3577	. 2 ⊢ (𝐾 ∈ ℕ → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
98		gpgusgra 48067	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
99	21, 98	eqeltrid 2833	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
100		eqid 2730	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
101		eqid 2730	. . . 4 ⊢ (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
102	35, 100, 101	usgrgrtrirex 47960	. . 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 847   = wceq 1541  ∃wex 1780   ∈ wcel 2110   ≠ wne 2926  ∃wrex 3054  {cpr 4576  {ctp 4578  ⟨cop 4580   × cxp 5612  ‘cfv 6477  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  0cc0 10998  1c1 10999   + caddc 11001   · cmul 11003   − cmin 11336  -cneg 11337   / cdiv 11766  ℕcn 12117  2c2 12172  3c3 12173  ℤ_≥cuz 12724  ..^cfzo 13546  ⌈cceil 13687   mod cmo 13765  Vtxcvtx 28967  Edgcedg 29018  USGraphcusgr 29120   NeighbVtx cnbgr 29303  GrTrianglescgrtri 47947   gPetersenGr cgpg 48050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-3o 8382  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-dju 9786  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-div 11767  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-xnn0 12447  df-z 12461  df-dec 12581  df-uz 12725  df-rp 12883  df-ico 13243  df-fz 13400  df-fzo 13547  df-fl 13688  df-ceil 13689  df-mod 13766  df-hash 14230  df-dvds 16156  df-struct 17050  df-slot 17085  df-ndx 17097  df-base 17113  df-edgf 28960  df-vtx 28969  df-iedg 28970  df-edg 29019  df-uhgr 29029  df-upgr 29053  df-umgr 29054  df-uspgr 29121  df-usgr 29122  df-nbgr 29304  df-grtri 47948  df-gpg 48051

This theorem is referenced by: (None)