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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 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({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.
StepHypRef Expression
1 cycliswlk 26160 . . . . 5 (𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)
2 wlkbprop 26051 . . . . 5 (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
31, 2syl 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‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
7 fzo0to42pr 12422 . . . . . . . . . . . . . . . . . . 19 (0..^4) = ({0, 1} ∪ {2, 3})
87raleqi 3119 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
9 ralunb 3756 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
10 2wlklem 26094 . . . . . . . . . . . . . . . . . . 19 (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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 → (𝐹𝑘) = (𝐹‘2))
1413fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 2 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘2)))
15 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 2 → (𝑃𝑘) = (𝑃‘2))
16 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
17 2p1e3 11028 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 + 1) = 3
1816, 17syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 2 → (𝑘 + 1) = 3)
1918fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
2015, 19preq12d 4220 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 2 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
2114, 20eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 2 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
22 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 3 → (𝐹𝑘) = (𝐹‘3))
2322fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 3 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘3)))
24 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 3 → (𝑃𝑘) = (𝑃‘3))
25 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 3 → (𝑘 + 1) = (3 + 1))
26 3p1e4 11030 . . . . . . . . . . . . . . . . . . . . . . . . 25 (3 + 1) = 4
2725, 26syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 3 → (𝑘 + 1) = 4)
2827fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 3 → (𝑃‘(𝑘 + 1)) = (𝑃‘4))
2924, 28preq12d 4220 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 3 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘3), (𝑃‘4)})
3023, 29eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 3 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
3121, 30ralprg 4181 . . . . . . . . . . . . . . . . . . . 20 ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
3211, 12, 31mp2an 704 . . . . . . . . . . . . . . . . . . 19 (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
3310, 32anbi12i 729 . . . . . . . . . . . . . . . . . 18 ((∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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)})))
348, 9, 333bitri 285 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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)})
3635eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘0) = (𝑃‘4) → {(𝑃‘3), (𝑃‘4)} = {(𝑃‘3), (𝑃‘0)})
3736eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃‘0) = (𝑃‘4) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))
3837anbi2d 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)})))
3938anbi2d 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))
4240, 41mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 4 → 0 < (#‘𝐹))
43 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℕ0
4442, 43jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 4 → (0 ∈ ℕ0 ∧ 0 < (#‘𝐹)))
45 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (0 ∈ ℕ0 ∧ 0 < (#‘𝐹))) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
4644, 45sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ 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))
4947, 48mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 4 → 1 < (#‘𝐹))
50 1nn0 11185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 ∈ ℕ0
5149, 50jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 4 → (1 ∈ ℕ0 ∧ 1 < (#‘𝐹)))
52 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (1 ∈ ℕ0 ∧ 1 < (#‘𝐹))) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
5351, 52sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
5446, 53anim12d 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Fun 𝐸𝐹 ∈ 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))
5755, 56mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 4 → 2 < (#‘𝐹))
58 2nn0 11186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 ∈ ℕ0
5957, 58jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 4 → (2 ∈ ℕ0 ∧ 2 < (#‘𝐹)))
60 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (2 ∈ ℕ0 ∧ 2 < (#‘𝐹))) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} → {(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸))
6159, 60sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ 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))
6462, 63mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 4 → 3 < (#‘𝐹))
65 3nn0 11187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 ∈ ℕ0
6664, 65jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 4 → (3 ∈ ℕ0 ∧ 3 < (#‘𝐹)))
67 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (3 ∈ ℕ0 ∧ 3 < (#‘𝐹))) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)} → {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))
6866, 67sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)} → {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸))
6961, 68anim12d 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Fun 𝐸𝐹 ∈ 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)))
7054, 69anim12d 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Fun 𝐸𝐹 ∈ 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))))
7170adantlrr 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))))
7271imp 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
7674, 75, 62ltleii 10039 . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 ≤ 4
77 eluz2 11569 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (4 ∈ (ℤ‘3) ↔ (3 ∈ ℤ ∧ 4 ∈ ℤ ∧ 3 ≤ 4))
7812, 73, 76, 77mpbir3an 1237 . . . . . . . . . . . . . . . . . . . . . . . . 25 4 ∈ (ℤ‘3)
79 4fvwrd4 12328 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((4 ∈ (ℤ‘3) ∧ 𝑃:(0...4)⟶𝑉) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
8078, 79mpan 702 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃:(0...4)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((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})
8281adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
8382eleq1d 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)
8684, 85preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
8786eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ↔ {𝑏, 𝑐} ∈ ran 𝐸))
8883, 87anbi12d 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})
9089adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘2), (𝑃‘3)} = {𝑐, 𝑑})
9190eleq1d 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)
9492, 93preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘3), (𝑃‘0)} = {𝑑, 𝑎})
9594eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸 ↔ {𝑑, 𝑎} ∈ ran 𝐸))
9691, 95anbi12d 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)))
9788, 96anbi12d 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))))
9897biimpcd 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))))
9998reximdv 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)) → (∃𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
10099reximdv 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)) → (∃𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
101100reximdv 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)) → (∃𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
102101reximdv 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)) → (∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
10372, 80, 102syl2im 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
104103exp41 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
105104com14 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
106105com35 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
10739, 106syl6bi 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
108107com12 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
109108com24 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
11034, 109sylbi 206 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝑃:(0...4)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
111110com13 86 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → (𝑃:(0...4)⟶𝑉 → (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
1121113imp 1249 . . . . . . . . . . . . . 14 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
113112com14 94 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘4) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({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))
115114eqeq2d 2620 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘4)))
116 oveq2 6557 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 4 → (0...(#‘𝐹)) = (0...4))
117116feq2d 5944 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 4 → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...4)⟶𝑉))
118 oveq2 6557 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 4 → (0..^(#‘𝐹)) = (0..^4))
119118raleqdv 3121 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 4 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
120117, 1193anbi23d 1394 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 4 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
121120imbi1d 330 . . . . . . . . . . . . . 14 ((#‘𝐹) = 4 → ((((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({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‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
122121imbi2d 329 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → ((Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({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‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
123113, 115, 1223imtr4d 282 . . . . . . . . . . . 12 ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
124123com14 94 . . . . . . . . . . 11 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
1251242a1d 26 . . . . . . . . . 10 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun (𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))))
1266, 125syl6bi 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))))
1271263impd 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 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
1285, 127sylbid 229 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))))
129128impd 446 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
1304, 129sylbid 229 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
1311303adant1 1072 . . . 4 (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))))
1323, 131mpcom 37 . . 3 (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
133132com12 32 . 2 (Fun 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))))
1341333imp 1249 1 ((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({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  0cn0 11169  cz 11254  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)
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