Theorem gpg3kgrtriex 48821

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		1elpr01 11203	. . . . . . 7 ⊢ 1 ∈ {0, 1}
2	1	a1i 11	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 1 ∈ {0, 1})
3		gpg3kgrtriex.n	. . . . . . . 8 ⊢ 𝑁 = (3 · 𝐾)
4		3nn 12319	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
5	4	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 3 ∈ ℕ)
6		id 23	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
7	5, 6	nnmulcld 12288	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (3 · 𝐾) ∈ ℕ)
8	3, 7	eqeltrid 2865	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ ℕ)
9		lbfzo0 13728	. . . . . . 7 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
10	8, 9	sylibr 237	. . . . . 6 ⊢ (𝐾 ∈ ℕ → 0 ∈ (0..^𝑁))
11	2, 10	opelxpd 5700	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁)))
12	3	gpg3kgrtriexlem4 48818	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
13	8, 12	jca 520	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
14		eqid 2761	. . . . . . . 8 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
15		eqid 2761	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
16	14, 15	gpgvtx 48775	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × (0..^𝑁)))
17	16	eleq2d 2847	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
18	13, 17	syl 18	. . . . 5 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)) ↔ ⟨1, 0⟩ ∈ ({0, 1} × (0..^𝑁))))
19	11, 18	mpbird 260	. . . 4 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘(𝑁 gPetersenGr 𝐾)))
20		gpg3kgrtriex.g	. . . . 5 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
21	20	fveq2i 6884	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘(𝑁 gPetersenGr 𝐾))
22	19, 21	eleqtrrdi 2872	. . 3 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
23		oveq2 7418	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx ⟨1, 0⟩))
24		biidd 265	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → ((𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
25	23, 24	rexeqbidv 3337	. . . . 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))))
26	23, 25	rexeqbidv 3337	. . . 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))))
27	26	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))))
28	3	gpg3kgrtriexlem3 48817	. . . . 5 ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
29		eqid 2761	. . . . . . . . 9 ⊢ 1 = 1
30	29	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 = 1)
31	30	olcd 887	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (1 = 0 ∨ 1 = 1))
32	31, 10	jca 520	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁)))
33	28, 12	jca 520	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))))
34		eqid 2761	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35	15, 14, 20, 34	opgpgvtx 48787	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
36	33, 35	syl 18	. . . . . 6 ⊢ (𝐾 ∈ ℕ → (⟨1, 0⟩ ∈ (Vtx‘𝐺) ↔ ((1 = 0 ∨ 1 = 1) ∧ 0 ∈ (0..^𝑁))))
37	32, 36	mpbird 260	. . . . 5 ⊢ (𝐾 ∈ ℕ → ⟨1, 0⟩ ∈ (Vtx‘𝐺))
38		1ex 11202	. . . . . . 7 ⊢ 1 ∈ V
39		c0ex 11199	. . . . . . 7 ⊢ 0 ∈ V
40	38, 39	op1st 7993	. . . . . 6 ⊢ (1st ‘⟨1, 0⟩) = 1
41	40	a1i 11	. . . . 5 ⊢ (𝐾 ∈ ℕ → (1st ‘⟨1, 0⟩) = 1)
42		eqid 2761	. . . . . 6 ⊢ (𝐺 NeighbVtx ⟨1, 0⟩) = (𝐺 NeighbVtx ⟨1, 0⟩)
43	14, 20, 34, 42	gpgnbgrvtx1 48807	. . . . 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	28, 12, 37, 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 3018	. . . . . 6 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → (𝑏 ≠ 𝑐 ↔ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ≠ 𝑐))
46		preq1 4698	. . . . . . 7 ⊢ (𝑏 = ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ → {𝑏, 𝑐} = {⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩, 𝑐})
47	46	eleq1d 2846	. . . . . 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 3019	. . . . . 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 4699	. . . . . . 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 2846	. . . . . 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 5445	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ ∈ V
54	53	tpid1 4733	. . . . . 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 2850	. . . . . . 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 5445	. . . . . . 7 ⊢ ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ ∈ V
59	58	tpid3 4738	. . . . . 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 2850	. . . . . . 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	3	gpg3kgrtriexlem5 48819	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
64	38, 39	op2nd 7994	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨1, 0⟩) = 0
65	64	oveq1i 7420	. . . . . . . . . . . 12 ⊢ ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾)
66		nncn 12240	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
67	66	addlidd 11410	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (0 + 𝐾) = 𝐾)
68	65, 67	eqtrid 2808	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
69	68	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
70	64	oveq1i 7420	. . . . . . . . . . . . 13 ⊢ ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾)
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
72		df-neg 11443	. . . . . . . . . . . 12 ⊢ -𝐾 = (0 − 𝐾)
73	71, 72	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
74	73	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
75	63, 69, 74	3netr4d 3033	. . . . . . . . 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 7443	. . . . . . . . 9 ⊢ (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) ∈ V
78	38, 77	opthne 5464	. . . . . . . 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 7425	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = (0 + 𝐾))
82	81, 67	eqtrd 2796	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) + 𝐾) = 𝐾)
83	82	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁) = (𝐾 mod 𝑁))
84	83	opeq2d 4844	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) + 𝐾) mod 𝑁)⟩ = ⟨1, (𝐾 mod 𝑁)⟩)
85	80	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = (0 − 𝐾))
86	85, 72	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → ((2nd ‘⟨1, 0⟩) − 𝐾) = -𝐾)
87	86	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℕ → (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁) = (-𝐾 mod 𝑁))
88	87	opeq2d 4844	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → ⟨1, (((2nd ‘⟨1, 0⟩) − 𝐾) mod 𝑁)⟩ = ⟨1, (-𝐾 mod 𝑁)⟩)
89	84, 88	preq12d 4706	. . . . . . . 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 2761	. . . . . . . . 9 ⊢ {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}
91	3, 20, 90	gpg3kgrtriexlem6 48820	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩} ∈ (Edg‘𝐺))
92	89, 91	eqeltrd 2861	. . . . . . 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 3594	. . . 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	22, 27, 96	rspcedvd 3582	. 2 ⊢ (𝐾 ∈ ℕ → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
98		gpgusgra 48789	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
99	20, 98	eqeltrid 2865	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
100		eqid 2761	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
101		eqid 2761	. . . 4 ⊢ (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
102	34, 100, 101	usgrgrtrirex 48682	. . 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	33, 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 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3087  {cpr 4590  {ctp 4592  ⟨cop 4594   × cxp 5659  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984  0cc0 11099  1c1 11100   + caddc 11102   · cmul 11104   − cmin 11440  -cneg 11441   / cdiv 11870  ℕcn 12232  2c2 12294  3c3 12295  ℤ_≥cuz 12861  ..^cfzo 13682  ⌈cceil 13824   mod cmo 13902  Vtxcvtx 29312  Edgcedg 29363  USGraphcusgr 29465   NeighbVtx cnbgr 29648  GrTrianglescgrtri 48669   gPetersenGr cgpg 48772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  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 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-3o 8454  df-oadd 8456  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-xnn0 12577  df-z 12591  df-dec 12711  df-uz 12862  df-rp 13016  df-ico 13377  df-fz 13535  df-fzo 13683  df-fl 13825  df-ceil 13826  df-mod 13903  df-hash 14367  df-dvds 16310  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-edgf 29305  df-vtx 29314  df-iedg 29315  df-edg 29364  df-uhgr 29374  df-upgr 29398  df-umgr 29399  df-uspgr 29466  df-usgr 29467  df-nbgr 29649  df-grtri 48670  df-gpg 48773

This theorem is referenced by: (None)