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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  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)
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