upgr3v3e3cycl - Mathbox for Alexander van der Vekens

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Theorem upgr3v3e3cycl 41347

Description: If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)

**Hypotheses**
Ref	Expression
upgr3v3e3cycl.e	⊢ 𝐸 = (Edg‘𝐺)
upgr3v3e3cycl.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
upgr3v3e3cycl	⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))

Distinct variable groups: 𝐸,𝑎,𝑏,𝑐 𝑃,𝑎,𝑏,𝑐 𝑉,𝑎,𝑏,𝑐 Allowed substitution hints: 𝐹(𝑎,𝑏,𝑐) 𝐺(𝑎,𝑏,𝑐)

Proof of Theorem upgr3v3e3cycl Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cyclprop 40999	. . 3 ⊢ (𝐹(CycleS‘𝐺)𝑃 → (𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2		pthis1wlk 40933	. . . . 5 ⊢ (𝐹(PathS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
3		upgr3v3e3cycl.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
4	3	upgr1wlkvtxedg 40853	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸)
5		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 3 → (𝑃‘(#‘𝐹)) = (𝑃‘3))
6	5	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
7	6	anbi2d 736	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 3 → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ↔ (𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3))))
8		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = (0..^3))
9		fzo0to3tp 12421	. . . . . . . . . . . . . . . 16 ⊢ (0..^3) = {0, 1, 2}
10	8, 9	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = {0, 1, 2})
11	10	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 3 → (∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ∀𝑘 ∈ {0, 1, 2} {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸))
12		c0ex 9913	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
13		1ex 9914	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
14		2ex 10969	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ V
15		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
16		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
17		0p1e1 11009	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 + 1) = 1
18	16, 17	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
19	18	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
20	15, 19	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
21	20	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ 𝐸))
22		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
23		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
24		1p1e2 11011	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 + 1) = 2
25	23, 24	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2)
26	25	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
27	22, 26	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
28	27	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸))
29		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
30		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
31		2p1e3 11028	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
32	30, 31	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
33	32	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
34	29, 33	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
35	34	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 2 → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))
36	12, 13, 14, 21, 28, 35	raltp 4187	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ {0, 1, 2} {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))
37	11, 36	syl6bb 275	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 3 → (∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)))
38	7, 37	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 3 → (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) ↔ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))))
39		upgr3v3e3cycl.v	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 = (Vtx‘𝐺)
40	39	1wlkp 40821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
41		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐹) = 3 → (0...(#‘𝐹)) = (0...3))
42	41	feq2d 5944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐹) = 3 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...3)⟶𝑉))
43		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...3)⟶𝑉 → 𝑃:(0...3)⟶𝑉)
44		3nn0 11187	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 3 ∈ ℕ₀
45		0elfz 12305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
46	44, 45	mp1i 13	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...3)⟶𝑉 → 0 ∈ (0...3))
47	43, 46	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘0) ∈ 𝑉)
48		1nn0 11185	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
49		1lt3 11073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 < 3
50		fvffz0 12326	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 1 < 3) ∧ 𝑃:(0...3)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
51	50	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 1 < 3) → (𝑃:(0...3)⟶𝑉 → (𝑃‘1) ∈ 𝑉))
52	44, 48, 49, 51	mp3an 1416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘1) ∈ 𝑉)
53		2nn0 11186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ₀
54		2lt3 11072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 < 3
55		fvffz0 12326	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((3 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 < 3) ∧ 𝑃:(0...3)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
56	55	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 < 3) → (𝑃:(0...3)⟶𝑉 → (𝑃‘2) ∈ 𝑉))
57	44, 53, 54, 56	mp3an 1416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...3)⟶𝑉 → (𝑃‘2) ∈ 𝑉)
58	47, 52, 57	3jca 1235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃:(0...3)⟶𝑉 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
59	42, 58	syl6bi 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐹) = 3 → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
60	59	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → ((#‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
61	2, 40, 60	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹(PathS‘𝐺)𝑃 → ((#‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
62	61	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((#‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
63	62	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ((#‘𝐹) = 3 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
64	63	impcom 445	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐹) = 3 ∧ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
65		preq2 4213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘3) = (𝑃‘0) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
66	65	eqcoms 2618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
67	66	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
68	67	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ({(𝑃‘2), (𝑃‘3)} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
69	68	3anbi3d 1397	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸)))
70	69	biimpa 500	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
71	70	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐹) = 3 ∧ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
72		simpll 786	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → 𝐹(PathS‘𝐺)𝑃)
73		breq2 4587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐹) = 3 → (1 < (#‘𝐹) ↔ 1 < 3))
74	49, 73	mpbiri 247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐹) = 3 → 1 < (#‘𝐹))
75	74	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → 1 < (#‘𝐹))
76		3nn 11063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ
77		lbfzo0 12375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
78	76, 77	mpbir 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ (0..^3)
79	78, 8	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐹) = 3 → 0 ∈ (0..^(#‘𝐹)))
80	79	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → 0 ∈ (0..^(#‘𝐹)))
81		pthdadjvtx 40936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 0 ∈ (0..^(#‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(0 + 1)))
82		1e0p1 11428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 = (0 + 1)
83	82	fveq2i 6106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
84	83	neeq2i 2847	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘(0 + 1)))
85	81, 84	sylibr 223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 0 ∈ (0..^(#‘𝐹))) → (𝑃‘0) ≠ (𝑃‘1))
86	72, 75, 80, 85	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → (𝑃‘0) ≠ (𝑃‘1))
87		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ (0..^3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ ∧ 1 < 3))
88	48, 76, 49, 87	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ (0..^3)
89	88, 8	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐹) = 3 → 1 ∈ (0..^(#‘𝐹)))
90	89	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → 1 ∈ (0..^(#‘𝐹)))
91		pthdadjvtx 40936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 1 ∈ (0..^(#‘𝐹))) → (𝑃‘1) ≠ (𝑃‘(1 + 1)))
92		df-2 10956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 = (1 + 1)
93	92	fveq2i 6106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘2) = (𝑃‘(1 + 1))
94	93	neeq2i 2847	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘1) ≠ (𝑃‘2) ↔ (𝑃‘1) ≠ (𝑃‘(1 + 1)))
95	91, 94	sylibr 223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 1 ∈ (0..^(#‘𝐹))) → (𝑃‘1) ≠ (𝑃‘2))
96	72, 75, 90, 95	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → (𝑃‘1) ≠ (𝑃‘2))
97		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ (0..^3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ ∧ 2 < 3))
98	53, 76, 54, 97	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (0..^3)
99	98, 8	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐹) = 3 → 2 ∈ (0..^(#‘𝐹)))
100	99	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → 2 ∈ (0..^(#‘𝐹)))
101		pthdadjvtx 40936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 2 ∈ (0..^(#‘𝐹))) → (𝑃‘2) ≠ (𝑃‘(2 + 1)))
102	72, 75, 100, 101	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → (𝑃‘2) ≠ (𝑃‘(2 + 1)))
103		neeq2 2845	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘3)))
104		df-3 10957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 3 = (2 + 1)
105	104	fveq2i 6106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃‘3) = (𝑃‘(2 + 1))
106	105	neeq2i 2847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘2) ≠ (𝑃‘3) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1)))
107	103, 106	syl6bb 275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
108	107	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
109	108	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘(2 + 1))))
110	102, 109	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → (𝑃‘2) ≠ (𝑃‘0))
111	86, 96, 110	3jca 1235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ (#‘𝐹) = 3) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))
112	111	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) → ((#‘𝐹) = 3 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
113	112	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ((#‘𝐹) = 3 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
114	113	impcom 445	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐹) = 3 ∧ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))
115		preq1 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑃‘0) → {𝑎, 𝑏} = {(𝑃‘0), 𝑏})
116	115	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑃‘0), 𝑏} ∈ 𝐸))
117		preq2 4213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑃‘0) → {𝑐, 𝑎} = {𝑐, (𝑃‘0)})
118	117	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → ({𝑐, 𝑎} ∈ 𝐸 ↔ {𝑐, (𝑃‘0)} ∈ 𝐸))
119	116, 118	3anbi13d 1393	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑃‘0) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ↔ ({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸)))
120		neeq1 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → (𝑎 ≠ 𝑏 ↔ (𝑃‘0) ≠ 𝑏))
121		neeq2 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑃‘0) → (𝑐 ≠ 𝑎 ↔ 𝑐 ≠ (𝑃‘0)))
122	120, 121	3anbi13d 1393	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑃‘0) → ((𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎) ↔ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))))
123	119, 122	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑃‘0) → ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)) ↔ (({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)))))
124		preq2 4213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑃‘1) → {(𝑃‘0), 𝑏} = {(𝑃‘0), (𝑃‘1)})
125	124	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ({(𝑃‘0), 𝑏} ∈ 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ 𝐸))
126		preq1 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑃‘1) → {𝑏, 𝑐} = {(𝑃‘1), 𝑐})
127	126	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ({𝑏, 𝑐} ∈ 𝐸 ↔ {(𝑃‘1), 𝑐} ∈ 𝐸))
128	125, 127	3anbi12d 1392	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑃‘1) → (({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸)))
129		neeq2 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → ((𝑃‘0) ≠ 𝑏 ↔ (𝑃‘0) ≠ (𝑃‘1)))
130		neeq1 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑃‘1) → (𝑏 ≠ 𝑐 ↔ (𝑃‘1) ≠ 𝑐))
131	129, 130	3anbi12d 1392	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑃‘1) → (((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))))
132	128, 131	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑃‘1) → ((({(𝑃‘0), 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))) ↔ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)))))
133		preq2 4213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑃‘2) → {(𝑃‘1), 𝑐} = {(𝑃‘1), (𝑃‘2)})
134	133	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ({(𝑃‘1), 𝑐} ∈ 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸))
135		preq1 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑃‘2) → {𝑐, (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
136	135	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ({𝑐, (𝑃‘0)} ∈ 𝐸 ↔ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸))
137	134, 136	3anbi23d 1394	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑃‘2) → (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸)))
138		neeq2 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → ((𝑃‘1) ≠ 𝑐 ↔ (𝑃‘1) ≠ (𝑃‘2)))
139		neeq1 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑃‘2) → (𝑐 ≠ (𝑃‘0) ↔ (𝑃‘2) ≠ (𝑃‘0)))
140	138, 139	3anbi23d 1394	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑃‘2) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0))))
141	137, 140	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑃‘2) → ((({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), 𝑐} ∈ 𝐸 ∧ {𝑐, (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ 𝑐 ∧ 𝑐 ≠ (𝑃‘0))) ↔ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))))
142	123, 132, 141	rspc3ev 3297	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉) ∧ (({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ 𝐸) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘2) ≠ (𝑃‘0)))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))
143	64, 71, 114, 142	syl12anc 1316	. . . . . . . . . . . . 13 ⊢ (((#‘𝐹) = 3 ∧ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))
144	143	ex 449	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 3 → (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘3)) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ 𝐸 ∧ {(𝑃‘2), (𝑃‘3)} ∈ 𝐸)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))
145	38, 144	sylbid 229	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 3 → (((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))
146	145	expd 451	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 3 → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
147	146	com13 86	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸 → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
148	4, 147	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
149	148	expcom 450	. . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
150	149	com23 84	. . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐺 ∈ UPGraph → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
151	150	expd 451	. . . . 5 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐹(PathS‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐺 ∈ UPGraph → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))))
152	2, 151	mpcom 37	. . . 4 ⊢ (𝐹(PathS‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐺 ∈ UPGraph → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))))))
153	152	imp 444	. . 3 ⊢ ((𝐹(PathS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐺 ∈ UPGraph → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
154	1, 153	syl 17	. 2 ⊢ (𝐹(CycleS‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))))
155	154	3imp21 1269	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {cpr 4127 {ctp 4129 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ℕcn 10897 2c2 10947 3c3 10948 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 UPGraph cupgr 25747 Edgcedga 25792 1Walksc1wlks 40796 PathScpths 40919 CycleSccycls 40991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-1wlks 40800 df-wlks 40801 df-trls 40901 df-pths 40923 df-cycls 40993

This theorem is referenced by: umgr3v3e3cycl 41351

W3C validator