4cycl4v4e - Metamath Proof Explorer

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Theorem 4cycl4v4e 26194

Description: If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)

**Assertion**
Ref	Expression
4cycl4v4e	⊢ ((Fun 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))

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

Proof of Theorem 4cycl4v4e Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycliswlk 26160	. . . . 5 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → 𝐹(𝑉 Walks 𝐸)𝑃)
2		wlkbprop 26051	. . . . 5 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3	1, 2	syl 17	. . . 4 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
4		iscycl 26153	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
5		ispth 26098	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
6		istrl 26067	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
7		fzo0to42pr 12422	. . . . . . . . . . . . . . . . . . 19 ⊢ (0..^4) = ({0, 1} ∪ {2, 3})
8	7	raleqi 3119	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
9		ralunb 3756	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
10		2wlklem 26094	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
11		2z 11286	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
12		3z 11287	. . . . . . . . . . . . . . . . . . . 20 ⊢ 3 ∈ ℤ
13		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝐹‘𝑘) = (𝐹‘2))
14	13	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘2)))
15		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
16		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
17		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 + 1) = 3
18	16, 17	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
19	18	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
20	15, 19	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
21	14, 20	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 2 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
22		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝐹‘𝑘) = (𝐹‘3))
23	22	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 3 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘3)))
24		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝑃‘𝑘) = (𝑃‘3))
25		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 3 → (𝑘 + 1) = (3 + 1))
26		3p1e4 11030	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (3 + 1) = 4
27	25, 26	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 3 → (𝑘 + 1) = 4)
28	27	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝑃‘(𝑘 + 1)) = (𝑃‘4))
29	24, 28	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 3 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘3), (𝑃‘4)})
30	23, 29	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 3 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
31	21, 30	ralprg 4181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
32	11, 12, 31	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
33	10, 32	anbi12i 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
34	8, 9, 33	3bitri 285	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
35		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘4) = (𝑃‘0) → {(𝑃‘3), (𝑃‘4)} = {(𝑃‘3), (𝑃‘0)})
36	35	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘4) → {(𝑃‘3), (𝑃‘4)} = {(𝑃‘3), (𝑃‘0)})
37	36	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘4) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))
38	37	anbi2d 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘4) → (((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}) ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})))
39	38	anbi2d 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) = (𝑃‘4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))))
40		4pos 10993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 < 4
41		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐹) = 4 → (0 < (#‘𝐹) ↔ 0 < 4))
42	40, 41	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐹) = 4 → 0 < (#‘𝐹))
43		0nn0 11184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℕ₀
44	42, 43	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 4 → (0 ∈ ℕ₀ ∧ 0 < (#‘𝐹)))
45		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (0 ∈ ℕ₀ ∧ 0 < (#‘𝐹))) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
46	44, 45	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
47		1lt4 11076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 1 < 4
48		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐹) = 4 → (1 < (#‘𝐹) ↔ 1 < 4))
49	47, 48	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐹) = 4 → 1 < (#‘𝐹))
50		1nn0 11185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 1 ∈ ℕ₀
51	49, 50	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 4 → (1 ∈ ℕ₀ ∧ 1 < (#‘𝐹)))
52		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (1 ∈ ℕ₀ ∧ 1 < (#‘𝐹))) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
53	51, 52	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
54	46, 53	anim12d 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸)))
55		2lt4 11075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 < 4
56		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐹) = 4 → (2 < (#‘𝐹) ↔ 2 < 4))
57	55, 56	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐹) = 4 → 2 < (#‘𝐹))
58		2nn0 11186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 2 ∈ ℕ₀
59	57, 58	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 4 → (2 ∈ ℕ₀ ∧ 2 < (#‘𝐹)))
60		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (2 ∈ ℕ₀ ∧ 2 < (#‘𝐹))) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} → {(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸))
61	59, 60	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} → {(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸))
62		3lt4 11074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 3 < 4
63		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐹) = 4 → (3 < (#‘𝐹) ↔ 3 < 4))
64	62, 63	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐹) = 4 → 3 < (#‘𝐹))
65		3nn0 11187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 3 ∈ ℕ₀
66	64, 65	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 4 → (3 ∈ ℕ₀ ∧ 3 < (#‘𝐹)))
67		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (3 ∈ ℕ₀ ∧ 3 < (#‘𝐹))) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)} → {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))
68	66, 67	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)} → {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))
69	61, 68	anim12d 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → (((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}) → ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)))
70	54, 69	anim12d 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))))
71	70	adantlrr 753	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))))
72	71	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) ∧ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)))
73		4z 11288	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℤ
74		3re 10971	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 3 ∈ ℝ
75		4re 10974	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 4 ∈ ℝ
76	74, 75, 62	ltleii 10039	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 3 ≤ 4
77		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (4 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 4 ∈ ℤ ∧ 3 ≤ 4))
78	12, 73, 76, 77	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ (ℤ_≥‘3)
79		4fvwrd4 12328	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((4 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...4)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
80	78, 79	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
81		preq12 4214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
82	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
83	82	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
84		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘1) = 𝑏)
85		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘2) = 𝑐)
86	84, 85	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
87	86	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ↔ {𝑏, 𝑐} ∈ ran 𝐸))
88	83, 87	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ↔ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
89		preq12 4214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → {(𝑃‘2), (𝑃‘3)} = {𝑐, 𝑑})
90	89	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘2), (𝑃‘3)} = {𝑐, 𝑑})
91	90	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ↔ {𝑐, 𝑑} ∈ ran 𝐸))
92		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘3) = 𝑑)
93		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘0) = 𝑎)
94	92, 93	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘3), (𝑃‘0)} = {𝑑, 𝑎})
95	94	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸 ↔ {𝑑, 𝑎} ∈ ran 𝐸))
96	91, 95	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸) ↔ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))
97	88, 96	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ↔ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
98	97	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) → ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
99	98	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) → (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
100	99	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
101	100	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
102	101	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
103	72, 80, 102	syl2im 39	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) ∧ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))) → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
104	103	exp41 636	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Fun 𝐸 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((#‘𝐹) = 4 → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
105	104	com14 94	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((#‘𝐹) = 4 → (Fun 𝐸 → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
106	105	com35 96	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
107	39, 106	syl6bi 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘0) = (𝑃‘4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
108	107	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → ((𝑃‘0) = (𝑃‘4) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
109	108	com24 93	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → (𝑃:(0...4)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
110	34, 109	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝑃:(0...4)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
111	110	com13 86	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
112	111	3imp 1249	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
113	112	com14 94	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
114		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 4 → (𝑃‘(#‘𝐹)) = (𝑃‘4))
115	114	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘4)))
116		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 4 → (0...(#‘𝐹)) = (0...4))
117	116	feq2d 5944	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 4 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...4)⟶𝑉))
118		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 4 → (0..^(#‘𝐹)) = (0..^4))
119	118	raleqdv 3121	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 4 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
120	117, 119	3anbi23d 1394	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 4 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
121	120	imbi1d 330	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 4 → ((((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))) ↔ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
122	121	imbi2d 329	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))) ↔ (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
123	113, 115, 122	3imtr4d 282	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
124	123	com14 94	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
125	124	2a1d 26	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
126	6, 125	syl6bi 242	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))))
127	126	3impd 1273	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
128	5, 127	sylbid 229	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
129	128	impd 446	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
130	4, 129	sylbid 229	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
131	130	3adant1 1072	. . . 4 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
132	3, 131	mpcom 37	. . 3 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
133	132	com12 32	. 2 ⊢ (Fun 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
134	133	3imp 1249	1 ⊢ ((Fun 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {cpr 4127 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 2c2 10947 3c3 10948 4c4 10949 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Walks cwalk 26026 Trails ctrail 26027 Paths cpath 26028 Cycles ccycl 26035

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-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-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-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-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-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-wlk 26036 df-trail 26037 df-pth 26038 df-cycl 26041

This theorem is referenced by: (None)

W3C validator