cycl3grtri - Mathbox for Alexander van der Vekens

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Theorem cycl3grtri 49250

Description: The vertices of a cycle of size 3 are a triangle in a graph. (Contributed by AV, 5-Oct-2025.)

**Hypotheses**
Ref	Expression
cycl3grtri.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
cycl3grtri.c	⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
cycl3grtri.n	⊢ (𝜑 → (♯‘𝐹) = 3)

**Assertion**
Ref	Expression
cycl3grtri	⊢ (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺))

Proof of Theorem cycl3grtri Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycl3grtri.n	. 2 ⊢ (𝜑 → (♯‘𝐹) = 3)
2		cycl3grtri.c	. 2 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
3		cyclprop 30410	. . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
4		tpeq1 4703	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑃‘0) → {𝑥, 𝑦, 𝑧} = {(𝑃‘0), 𝑦, 𝑧})
5	4	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑃‘0) → (ran 𝑃 = {𝑥, 𝑦, 𝑧} ↔ ran 𝑃 = {(𝑃‘0), 𝑦, 𝑧}))
6		preq1 4694	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃‘0) → {𝑥, 𝑦} = {(𝑃‘0), 𝑦})
7	6	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑃‘0) → ({𝑥, 𝑦} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), 𝑦} ∈ (Edg‘𝐺)))
8		preq1 4694	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃‘0) → {𝑥, 𝑧} = {(𝑃‘0), 𝑧})
9	8	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑃‘0) → ({𝑥, 𝑧} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺)))
10	7, 9	3anbi12d 1466	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑃‘0) → (({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺)) ↔ ({(𝑃‘0), 𝑦} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺))))
11	5, 10	3anbi13d 1467	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑃‘0) → ((ran 𝑃 = {𝑥, 𝑦, 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺))) ↔ (ran 𝑃 = {(𝑃‘0), 𝑦, 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), 𝑦} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺)))))
12		tpeq2 4704	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑃‘1) → {(𝑃‘0), 𝑦, 𝑧} = {(𝑃‘0), (𝑃‘1), 𝑧})
13	12	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑃‘1) → (ran 𝑃 = {(𝑃‘0), 𝑦, 𝑧} ↔ ran 𝑃 = {(𝑃‘0), (𝑃‘1), 𝑧}))
14		preq2 4695	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑃‘1) → {(𝑃‘0), 𝑦} = {(𝑃‘0), (𝑃‘1)})
15	14	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑃‘1) → ({(𝑃‘0), 𝑦} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
16		preq1 4694	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑃‘1) → {𝑦, 𝑧} = {(𝑃‘1), 𝑧})
17	16	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑃‘1) → ({𝑦, 𝑧} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), 𝑧} ∈ (Edg‘𝐺)))
18	15, 17	3anbi13d 1467	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑃‘1) → (({(𝑃‘0), 𝑦} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺)) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), 𝑧} ∈ (Edg‘𝐺))))
19	13, 18	3anbi13d 1467	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑃‘1) → ((ran 𝑃 = {(𝑃‘0), 𝑦, 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), 𝑦} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺))) ↔ (ran 𝑃 = {(𝑃‘0), (𝑃‘1), 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), 𝑧} ∈ (Edg‘𝐺)))))
20		tpeq3 4705	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑃‘2) → {(𝑃‘0), (𝑃‘1), 𝑧} = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
21	20	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑃‘2) → (ran 𝑃 = {(𝑃‘0), (𝑃‘1), 𝑧} ↔ ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)}))
22		preq2 4695	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑃‘2) → {(𝑃‘0), 𝑧} = {(𝑃‘0), (𝑃‘2)})
23	22	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑃‘2) → ({(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
24		preq2 4695	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑃‘2) → {(𝑃‘1), 𝑧} = {(𝑃‘1), (𝑃‘2)})
25	24	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑃‘2) → ({(𝑃‘1), 𝑧} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
26	23, 25	3anbi23d 1468	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑃‘2) → (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), 𝑧} ∈ (Edg‘𝐺)) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
27	21, 26	3anbi13d 1467	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃‘2) → ((ran 𝑃 = {(𝑃‘0), (𝑃‘1), 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), 𝑧} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), 𝑧} ∈ (Edg‘𝐺))) ↔ (ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))))
28		pthiswlk 30340	. . . . . . . . . . . . . 14 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
29		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
30	29	wlkp 30227	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
31		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
32		3nn0 12651	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 ∈ ℕ₀
33		0elfz 13785	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
34	32, 33	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ (0...3)
35		oveq2 7451	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 3 → (0...(♯‘𝐹)) = (0...3))
36	34, 35	eleqtrrid 2871	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 3 → 0 ∈ (0...(♯‘𝐹)))
37	36	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 0 ∈ (0...(♯‘𝐹)))
38	31, 37	ffvelcdmd 7109	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘0) ∈ (Vtx‘𝐺))
39	38	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘0) ∈ (Vtx‘𝐺)))
40	28, 30, 39	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝐹(Paths‘𝐺)𝑃 → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘0) ∈ (Vtx‘𝐺)))
41	40	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘0) ∈ (Vtx‘𝐺)))
42	41	imp 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘0) ∈ (Vtx‘𝐺))
43		1nn0 12649	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ₀
44		1le3 12584	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ≤ 3
45		elfz2nn0 13779	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
46	43, 32, 44, 45	mpbir3an 1360	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ (0...3)
47	46, 35	eleqtrrid 2871	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 3 → 1 ∈ (0...(♯‘𝐹)))
48	47	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 1 ∈ (0...(♯‘𝐹)))
49	31, 48	ffvelcdmd 7109	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘1) ∈ (Vtx‘𝐺))
50	49	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘1) ∈ (Vtx‘𝐺)))
51	28, 30, 50	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝐹(Paths‘𝐺)𝑃 → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘1) ∈ (Vtx‘𝐺)))
52	51	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘1) ∈ (Vtx‘𝐺)))
53	52	imp 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘1) ∈ (Vtx‘𝐺))
54		2nn0 12650	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
55		2re 12444	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
56		3re 12450	. . . . . . . . . . . . . . . . . . . 20 ⊢ 3 ∈ ℝ
57		2lt3 12543	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 < 3
58	55, 56, 57	ltleii 11460	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ 3
59		elfz2nn0 13779	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
60	54, 32, 58, 59	mpbir3an 1360	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ (0...3)
61	60, 35	eleqtrrid 2871	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 3 → 2 ∈ (0...(♯‘𝐹)))
62	61	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 2 ∈ (0...(♯‘𝐹)))
63	31, 62	ffvelcdmd 7109	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘2) ∈ (Vtx‘𝐺))
64	63	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘2) ∈ (Vtx‘𝐺)))
65	28, 30, 64	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝐹(Paths‘𝐺)𝑃 → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘2) ∈ (Vtx‘𝐺)))
66	65	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (𝑃‘2) ∈ (Vtx‘𝐺)))
67	66	imp 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃‘2) ∈ (Vtx‘𝐺))
68		fdm 6741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
69		elnn0uz 13033	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (ℤ_≥‘0))
70	32, 69	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 3 ∈ (ℤ_≥‘0)
71		fzisfzounsn 13942	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (3 ∈ (ℤ_≥‘0) → (0...3) = ((0..^3) ∪ {3}))
72	70, 71	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...3) = ((0..^3) ∪ {3})
73		fzo0to3tp 13914	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0..^3) = {0, 1, 2}
74	73	uneq1i 4166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0..^3) ∪ {3}) = ({0, 1, 2} ∪ {3})
75	72, 74	eqtri 2786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...3) = ({0, 1, 2} ∪ {3})
76	35, 75	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → (0...(♯‘𝐹)) = ({0, 1, 2} ∪ {3}))
77	76	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (0...(♯‘𝐹)) = ({0, 1, 2} ∪ {3}))
78	68, 77	sylan9eq 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → dom 𝑃 = ({0, 1, 2} ∪ {3}))
79	78	imaeq2d 6064	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃 “ dom 𝑃) = (𝑃 “ ({0, 1, 2} ∪ {3})))
80		imadmrn 6079	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 “ dom 𝑃) = ran 𝑃
81		imaundi 6153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 “ ({0, 1, 2} ∪ {3})) = ((𝑃 “ {0, 1, 2}) ∪ (𝑃 “ {3}))
82	79, 80, 81	3eqtr3g 2821	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ran 𝑃 = ((𝑃 “ {0, 1, 2}) ∪ (𝑃 “ {3})))
83		ffn 6731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
84	83	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 𝑃 Fn (0...(♯‘𝐹)))
85	84, 37, 48, 62	fnimatpd 6992	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃 “ {0, 1, 2}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
86		nn0fz0 13786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
87	32, 86	mpbi 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ (0...3)
88	87, 35	eleqtrrid 2871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → 3 ∈ (0...(♯‘𝐹)))
89	88	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → 3 ∈ (0...(♯‘𝐹)))
90		fnsnfv 6987	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ 3 ∈ (0...(♯‘𝐹))) → {(𝑃‘3)} = (𝑃 “ {3}))
91	83, 89, 90	syl2an 608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → {(𝑃‘3)} = (𝑃 “ {3}))
92	91	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (𝑃 “ {3}) = {(𝑃‘3)})
93	85, 92	uneq12d 4171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ((𝑃 “ {0, 1, 2}) ∪ (𝑃 “ {3})) = ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}))
94		fveq2 6908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
95	94	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
96		sneq 4594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘3) = (𝑃‘0) → {(𝑃‘3)} = {(𝑃‘0)})
97	96	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘3)} = {(𝑃‘0)})
98	97	uneq2d 4170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘3) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}) = ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘0)}))
99		snsstp1 4775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘0)} ⊆ {(𝑃‘0), (𝑃‘1), (𝑃‘2)}
100	99	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘0)} ⊆ {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
101		ssequn2 4190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({(𝑃‘0)} ⊆ {(𝑃‘0), (𝑃‘1), (𝑃‘2)} ↔ ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘0)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
102	100, 101	sylib 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘3) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘0)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
103	98, 102	eqtrd 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) = (𝑃‘3) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
104	95, 103	biimtrdi 256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)}))
105	104	impcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
106	105	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∪ {(𝑃‘3)}) = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
107	82, 93, 106	3eqtrd 2802	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
108	107	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)}))
109	28, 30, 108	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝐹(Paths‘𝐺)𝑃 → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)}))
110	109	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)}))
111	110	imp 412	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)})
112		breq2 5110	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 3 → (1 ≤ (♯‘𝐹) ↔ 1 ≤ 3))
113	44, 112	mpbiri 261	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 3 → 1 ≤ (♯‘𝐹))
114	113	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 1 ≤ (♯‘𝐹))
115	2	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → 𝐹(Cycles‘𝐺)𝑃)
116		cyclnumvtx 30418	. . . . . . . . . . . . . 14 ⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran 𝑃) = (♯‘𝐹))
117	114, 115, 116	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (♯‘ran 𝑃) = (♯‘𝐹))
118	1	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (♯‘𝐹) = 3)
119	117, 118	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (♯‘ran 𝑃) = 3)
120		cycl3grtri.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
121		cycl3grtrilem 49249	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
122	120, 121	sylanl1 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
123	111, 119, 122	3jca 1146	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (ran 𝑃 = {(𝑃‘0), (𝑃‘1), (𝑃‘2)} ∧ (♯‘ran 𝑃) = 3 ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
124	11, 19, 27, 42, 53, 67, 123	3rspcedvdw 3597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ∃𝑥 ∈ (Vtx‘𝐺)∃𝑦 ∈ (Vtx‘𝐺)∃𝑧 ∈ (Vtx‘𝐺)(ran 𝑃 = {𝑥, 𝑦, 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺))))
125		eqid 2763	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
126	29, 125	isgrtri 49246	. . . . . . . . . 10 ⊢ (ran 𝑃 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥 ∈ (Vtx‘𝐺)∃𝑦 ∈ (Vtx‘𝐺)∃𝑧 ∈ (Vtx‘𝐺)(ran 𝑃 = {𝑥, 𝑦, 𝑧} ∧ (♯‘ran 𝑃) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐺))))
127	124, 126	sylibr 237	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ran 𝑃 ∈ (GrTriangles‘𝐺))
128	127	exp32 426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ((♯‘𝐹) = 3 → ran 𝑃 ∈ (GrTriangles‘𝐺))))
129	128	com23 87	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹(Paths‘𝐺)𝑃) → ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ran 𝑃 ∈ (GrTriangles‘𝐺))))
130	129	expcom 419	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝜑 → ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ran 𝑃 ∈ (GrTriangles‘𝐺)))))
131	130	com24 96	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ((♯‘𝐹) = 3 → (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺)))))
132	131	imp 412	. . . 4 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 3 → (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺))))
133	3, 132	syl 18	. . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → ((♯‘𝐹) = 3 → (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺))))
134	133	com13 89	. 2 ⊢ (𝜑 → ((♯‘𝐹) = 3 → (𝐹(Cycles‘𝐺)𝑃 → ran 𝑃 ∈ (GrTriangles‘𝐺))))
135	1, 2, 134	mp2d 50	1 ⊢ (𝜑 → ran 𝑃 ∈ (GrTriangles‘𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 ⊆ wss 3900 ∪ cun 4152 {csn 4588 {cpr 4590 {ctp 4592 class class class wbr 5106 dom cdm 5655 ran crn 5656 “ cima 5658 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 0cc0 11227 1c1 11228 ≤ cle 11371 2c2 12424 3c3 12425 ℕ₀cn0 12633 ℤ_≥cuz 12992 ...cfz 13666 ..^cfzo 13815 ♯chash 14501 Vtxcvtx 29604 Edgcedg 29655 UPGraphcupgr 29688 Walkscwlks 30207 Pathscpths 30325 Cyclesccycls 30402 GrTrianglescgrtri 49240

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-3o 8506 df-oadd 8508 df-er 8745 df-map 8877 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-xnn0 12707 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502 df-word 14686 df-edg 29656 df-uhgr 29666 df-upgr 29690 df-wlks 30210 df-trls 30305 df-pths 30329 df-cycls 30404 df-grtri 49241

This theorem is used by: (None)

W3C validator