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Theorem 4cycl4dv 26195
 Description: In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
Assertion
Ref Expression
4cycl4dv ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)))))

Proof of Theorem 4cycl4dv
StepHypRef Expression
1 usgrafun 25878 . . . . 5 (𝑉 USGrph 𝐸 → Fun 𝐸)
2 4pos 10993 . . . . . . . . . . . . . . 15 0 < 4
3 breq2 4587 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 4 → (0 < (#‘𝐹) ↔ 0 < 4))
42, 3mpbiri 247 . . . . . . . . . . . . . 14 ((#‘𝐹) = 4 → 0 < (#‘𝐹))
5 0nn0 11184 . . . . . . . . . . . . . 14 0 ∈ ℕ0
64, 5jctil 558 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → (0 ∈ ℕ0 ∧ 0 < (#‘𝐹)))
7 nvnencycllem 26171 . . . . . . . . . . . . 13 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (0 ∈ ℕ0 ∧ 0 < (#‘𝐹))) → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))
86, 7sylan2 490 . . . . . . . . . . . 12 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))
9 1lt4 11076 . . . . . . . . . . . . . . 15 1 < 4
10 breq2 4587 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 4 → (1 < (#‘𝐹) ↔ 1 < 4))
119, 10mpbiri 247 . . . . . . . . . . . . . 14 ((#‘𝐹) = 4 → 1 < (#‘𝐹))
12 1nn0 11185 . . . . . . . . . . . . . 14 1 ∈ ℕ0
1311, 12jctil 558 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → (1 ∈ ℕ0 ∧ 1 < (#‘𝐹)))
14 nvnencycllem 26171 . . . . . . . . . . . . 13 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (1 ∈ ℕ0 ∧ 1 < (#‘𝐹))) → ((𝐸‘(𝐹‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} ∈ ran 𝐸))
1513, 14sylan2 490 . . . . . . . . . . . 12 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} ∈ ran 𝐸))
168, 15anim12d 584 . . . . . . . . . . 11 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
17 2lt4 11075 . . . . . . . . . . . . . . 15 2 < 4
18 breq2 4587 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 4 → (2 < (#‘𝐹) ↔ 2 < 4))
1917, 18mpbiri 247 . . . . . . . . . . . . . 14 ((#‘𝐹) = 4 → 2 < (#‘𝐹))
20 2nn0 11186 . . . . . . . . . . . . . 14 2 ∈ ℕ0
2119, 20jctil 558 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → (2 ∈ ℕ0 ∧ 2 < (#‘𝐹)))
22 nvnencycllem 26171 . . . . . . . . . . . . 13 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (2 ∈ ℕ0 ∧ 2 < (#‘𝐹))) → ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} → {𝐶, 𝐷} ∈ ran 𝐸))
2321, 22sylan2 490 . . . . . . . . . . . 12 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} → {𝐶, 𝐷} ∈ ran 𝐸))
24 3lt4 11074 . . . . . . . . . . . . . . 15 3 < 4
25 breq2 4587 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 4 → (3 < (#‘𝐹) ↔ 3 < 4))
2624, 25mpbiri 247 . . . . . . . . . . . . . 14 ((#‘𝐹) = 4 → 3 < (#‘𝐹))
27 3nn0 11187 . . . . . . . . . . . . . 14 3 ∈ ℕ0
2826, 27jctil 558 . . . . . . . . . . . . 13 ((#‘𝐹) = 4 → (3 ∈ ℕ0 ∧ 3 < (#‘𝐹)))
29 nvnencycllem 26171 . . . . . . . . . . . . 13 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (3 ∈ ℕ0 ∧ 3 < (#‘𝐹))) → ((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} → {𝐷, 𝐴} ∈ ran 𝐸))
3028, 29sylan2 490 . . . . . . . . . . . 12 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} → {𝐷, 𝐴} ∈ ran 𝐸))
3123, 30anim12d 584 . . . . . . . . . . 11 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → (((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}) → ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))
3216, 31anim12d 584 . . . . . . . . . 10 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 4) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸))))
3332ex 449 . . . . . . . . 9 ((Fun 𝐸𝐹 ∈ Word dom 𝐸) → ((#‘𝐹) = 4 → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))))
3433expcom 450 . . . . . . . 8 (𝐹 ∈ Word dom 𝐸 → (Fun 𝐸 → ((#‘𝐹) = 4 → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸))))))
3534com23 84 . . . . . . 7 (𝐹 ∈ Word dom 𝐸 → ((#‘𝐹) = 4 → (Fun 𝐸 → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸))))))
3635imp 444 . . . . . 6 ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 4) → (Fun 𝐸 → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))))
37363adant2 1073 . . . . 5 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4) → (Fun 𝐸 → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))))
381, 37mpan9 485 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸))))
3938imp 444 . . 3 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))
40 simpl 472 . . . 4 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)))
41 usgraedgrn 25910 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴𝐵)
4241ex 449 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸𝐴𝐵))
4342ad2antrr 758 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → ({𝐴, 𝐵} ∈ ran 𝐸𝐴𝐵))
4443com12 32 . . . . . . . 8 ({𝐴, 𝐵} ∈ ran 𝐸 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐵))
4544ad2antrr 758 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐵))
4645imp 444 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐴𝐵)
47 preq1 4212 . . . . . . . . . . . . . 14 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
4847eqeq2d 2620 . . . . . . . . . . . . 13 (𝐴 = 𝐶 → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ↔ (𝐸‘(𝐹‘0)) = {𝐶, 𝐵}))
49 prcom 4211 . . . . . . . . . . . . . . 15 {𝐵, 𝐶} = {𝐶, 𝐵}
5049eqeq2i 2622 . . . . . . . . . . . . . 14 ((𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ↔ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵})
5150a1i 11 . . . . . . . . . . . . 13 (𝐴 = 𝐶 → ((𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ↔ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵}))
5248, 51anbi12d 743 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ↔ ((𝐸‘(𝐹‘0)) = {𝐶, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵})))
5352adantr 480 . . . . . . . . . . 11 ((𝐴 = 𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ↔ ((𝐸‘(𝐹‘0)) = {𝐶, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵})))
54 eqtr3 2631 . . . . . . . . . . . . 13 (((𝐸‘(𝐹‘0)) = {𝐶, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵}) → (𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)))
55 usgraf1 25889 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
56 wrdf 13165 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ Word dom 𝐸𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
57 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐹) = 4 → (0..^(#‘𝐹)) = (0..^4))
5857feq2d 5944 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸𝐹:(0..^4)⟶dom 𝐸))
59 4nn 11064 . . . . . . . . . . . . . . . . . . . . 21 4 ∈ ℕ
60 lbfzo0 12375 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ (0..^4) ↔ 4 ∈ ℕ)
6159, 60mpbir 220 . . . . . . . . . . . . . . . . . . . 20 0 ∈ (0..^4)
62 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0..^4)⟶dom 𝐸 ∧ 0 ∈ (0..^4)) → (𝐹‘0) ∈ dom 𝐸)
6361, 62mpan2 703 . . . . . . . . . . . . . . . . . . 19 (𝐹:(0..^4)⟶dom 𝐸 → (𝐹‘0) ∈ dom 𝐸)
64 elfzo0 12376 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (0..^4) ↔ (1 ∈ ℕ0 ∧ 4 ∈ ℕ ∧ 1 < 4))
6512, 59, 9, 64mpbir3an 1237 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (0..^4)
66 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0..^4)⟶dom 𝐸 ∧ 1 ∈ (0..^4)) → (𝐹‘1) ∈ dom 𝐸)
6765, 66mpan2 703 . . . . . . . . . . . . . . . . . . 19 (𝐹:(0..^4)⟶dom 𝐸 → (𝐹‘1) ∈ dom 𝐸)
6863, 67jca 553 . . . . . . . . . . . . . . . . . 18 (𝐹:(0..^4)⟶dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
6958, 68syl6bi 242 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸)))
7056, 69mpan9 485 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 4) → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
71703adant2 1073 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4) → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
72 f1veqaeq 6418 . . . . . . . . . . . . . . 15 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸)) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
7355, 71, 72syl2an 493 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
74 df-f1 5809 . . . . . . . . . . . . . . . . . . 19 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
75 f1eq2 6010 . . . . . . . . . . . . . . . . . . . . . 22 ((0..^(#‘𝐹)) = (0..^4) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝐹:(0..^4)–1-1→dom 𝐸))
7657, 75syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝐹:(0..^4)–1-1→dom 𝐸))
77 f1veqaeq 6418 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹:(0..^4)–1-1→dom 𝐸 ∧ (0 ∈ (0..^4) ∧ 1 ∈ (0..^4))) → ((𝐹‘0) = (𝐹‘1) → 0 = 1))
7861, 65, 77mpanr12 717 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:(0..^4)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) → 0 = 1))
79 ax-1ne0 9884 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ≠ 0
8079nesymi 2839 . . . . . . . . . . . . . . . . . . . . . . 23 ¬ 0 = 1
8180pm2.21i 115 . . . . . . . . . . . . . . . . . . . . . 22 (0 = 1 → 𝐴𝐶)
8278, 81syl6 34 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:(0..^4)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶))
8376, 82syl6bi 242 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶)))
8483com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ((#‘𝐹) = 4 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶)))
8574, 84sylbir 224 . . . . . . . . . . . . . . . . . 18 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹) → ((#‘𝐹) = 4 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶)))
8685ex 449 . . . . . . . . . . . . . . . . 17 (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → (Fun 𝐹 → ((#‘𝐹) = 4 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶))))
8756, 86syl 17 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word dom 𝐸 → (Fun 𝐹 → ((#‘𝐹) = 4 → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶))))
88873imp 1249 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4) → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶))
8988adantl 481 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐹‘0) = (𝐹‘1) → 𝐴𝐶))
9073, 89syld 46 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → 𝐴𝐶))
9154, 90syl5 33 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → (((𝐸‘(𝐹‘0)) = {𝐶, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵}) → 𝐴𝐶))
9291adantl 481 . . . . . . . . . . 11 ((𝐴 = 𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘0)) = {𝐶, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐶, 𝐵}) → 𝐴𝐶))
9353, 92sylbid 229 . . . . . . . . . 10 ((𝐴 = 𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) → 𝐴𝐶))
9493adantrd 483 . . . . . . . . 9 ((𝐴 = 𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → 𝐴𝐶))
9594expimpd 627 . . . . . . . 8 (𝐴 = 𝐶 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐶))
96 ax-1 6 . . . . . . . 8 (𝐴𝐶 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐶))
9795, 96pm2.61ine 2865 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐶)
9897adantl 481 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐴𝐶)
99 usgraedgrn 25910 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) → 𝐷𝐴)
10099necomd 2837 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) → 𝐴𝐷)
101100ex 449 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → ({𝐷, 𝐴} ∈ ran 𝐸𝐴𝐷))
102101ad2antrr 758 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → ({𝐷, 𝐴} ∈ ran 𝐸𝐴𝐷))
103102com12 32 . . . . . . . . 9 ({𝐷, 𝐴} ∈ ran 𝐸 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐷))
104103adantl 481 . . . . . . . 8 (({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐷))
105104adantl 481 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐴𝐷))
106105imp 444 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐴𝐷)
10746, 98, 1063jca 1235 . . . . 5 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → (𝐴𝐵𝐴𝐶𝐴𝐷))
108 usgraedgrn 25910 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐵𝐶)
109108ex 449 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸𝐵𝐶))
110109ad2antrr 758 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → ({𝐵, 𝐶} ∈ ran 𝐸𝐵𝐶))
111110com12 32 . . . . . . . . 9 ({𝐵, 𝐶} ∈ ran 𝐸 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐶))
112111adantl 481 . . . . . . . 8 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐶))
113112adantr 480 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐶))
114113imp 444 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐵𝐶)
115 prcom 4211 . . . . . . . . . . . . . . . . . . . . 21 {𝐷, 𝐴} = {𝐴, 𝐷}
116115eqeq2i 2622 . . . . . . . . . . . . . . . . . . . 20 ((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ↔ (𝐸‘(𝐹‘3)) = {𝐴, 𝐷})
117116a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐵 = 𝐷 → ((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ↔ (𝐸‘(𝐹‘3)) = {𝐴, 𝐷}))
118 preq2 4213 . . . . . . . . . . . . . . . . . . . 20 (𝐵 = 𝐷 → {𝐴, 𝐵} = {𝐴, 𝐷})
119118eqeq2d 2620 . . . . . . . . . . . . . . . . . . 19 (𝐵 = 𝐷 → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ↔ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷}))
120117, 119anbi12d 743 . . . . . . . . . . . . . . . . . 18 (𝐵 = 𝐷 → (((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐵}) ↔ ((𝐸‘(𝐹‘3)) = {𝐴, 𝐷} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷})))
121120adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐵}) ↔ ((𝐸‘(𝐹‘3)) = {𝐴, 𝐷} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷})))
122 eqtr3 2631 . . . . . . . . . . . . . . . . . . 19 (((𝐸‘(𝐹‘3)) = {𝐴, 𝐷} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷}) → (𝐸‘(𝐹‘3)) = (𝐸‘(𝐹‘0)))
123 elfzo0 12376 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (3 ∈ (0..^4) ↔ (3 ∈ ℕ0 ∧ 4 ∈ ℕ ∧ 3 < 4))
12427, 59, 24, 123mpbir3an 1237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 ∈ (0..^4)
125 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹:(0..^4)⟶dom 𝐸 ∧ 3 ∈ (0..^4)) → (𝐹‘3) ∈ dom 𝐸)
126124, 125mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:(0..^4)⟶dom 𝐸 → (𝐹‘3) ∈ dom 𝐸)
127126, 63jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:(0..^4)⟶dom 𝐸 → ((𝐹‘3) ∈ dom 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸))
12858, 127syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → ((𝐹‘3) ∈ dom 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸)))
12956, 128mpan9 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 4) → ((𝐹‘3) ∈ dom 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸))
1301293adant2 1073 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4) → ((𝐹‘3) ∈ dom 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸))
131 f1veqaeq 6418 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ((𝐹‘3) ∈ dom 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸)) → ((𝐸‘(𝐹‘3)) = (𝐸‘(𝐹‘0)) → (𝐹‘3) = (𝐹‘0)))
13255, 130, 131syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐸‘(𝐹‘3)) = (𝐸‘(𝐹‘0)) → (𝐹‘3) = (𝐹‘0)))
133 f1veqaeq 6418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:(0..^4)–1-1→dom 𝐸 ∧ (3 ∈ (0..^4) ∧ 0 ∈ (0..^4))) → ((𝐹‘3) = (𝐹‘0) → 3 = 0))
134124, 61, 133mpanr12 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐹:(0..^4)–1-1→dom 𝐸 → ((𝐹‘3) = (𝐹‘0) → 3 = 0))
135 3ne0 10992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 ≠ 0
136135neii 2784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ¬ 3 = 0
137136pm2.21i 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (3 = 0 → 𝐵𝐷)
138134, 137syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:(0..^4)–1-1→dom 𝐸 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷))
13976, 138syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝐹) = 4 → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷)))
140139com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → ((#‘𝐹) = 4 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷)))
14174, 140sylbir 224 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹) → ((#‘𝐹) = 4 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷)))
142141ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → (Fun 𝐹 → ((#‘𝐹) = 4 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷))))
14356, 142syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ Word dom 𝐸 → (Fun 𝐹 → ((#‘𝐹) = 4 → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷))))
1441433imp 1249 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4) → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷))
145144adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐹‘3) = (𝐹‘0) → 𝐵𝐷))
146132, 145syld 46 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((𝐸‘(𝐹‘3)) = (𝐸‘(𝐹‘0)) → 𝐵𝐷))
147122, 146syl5 33 . . . . . . . . . . . . . . . . . 18 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → (((𝐸‘(𝐹‘3)) = {𝐴, 𝐷} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷}) → 𝐵𝐷))
148147adantl 481 . . . . . . . . . . . . . . . . 17 ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘3)) = {𝐴, 𝐷} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐷}) → 𝐵𝐷))
149121, 148sylbid 229 . . . . . . . . . . . . . . . 16 ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → (((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐵}) → 𝐵𝐷))
150149com12 32 . . . . . . . . . . . . . . 15 (((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} ∧ (𝐸‘(𝐹‘0)) = {𝐴, 𝐵}) → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷))
151150ex 449 . . . . . . . . . . . . . 14 ((𝐸‘(𝐹‘3)) = {𝐷, 𝐴} → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷)))
152151adantl 481 . . . . . . . . . . . . 13 (((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}) → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷)))
153152com12 32 . . . . . . . . . . . 12 ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} → (((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}) → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷)))
154153adantr 480 . . . . . . . . . . 11 (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) → (((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}) → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷)))
155154imp 444 . . . . . . . . . 10 ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → 𝐵𝐷))
156155com12 32 . . . . . . . . 9 ((𝐵 = 𝐷 ∧ (𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4))) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → 𝐵𝐷))
157156expimpd 627 . . . . . . . 8 (𝐵 = 𝐷 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐷))
158 ax-1 6 . . . . . . . 8 (𝐵𝐷 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐷))
159157, 158pm2.61ine 2865 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐵𝐷)
160159adantl 481 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐵𝐷)
161 usgraedgrn 25910 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ {𝐶, 𝐷} ∈ ran 𝐸) → 𝐶𝐷)
162161ex 449 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → ({𝐶, 𝐷} ∈ ran 𝐸𝐶𝐷))
163162ad2antrr 758 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → ({𝐶, 𝐷} ∈ ran 𝐸𝐶𝐷))
164163com12 32 . . . . . . . . 9 ({𝐶, 𝐷} ∈ ran 𝐸 → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐶𝐷))
165164adantr 480 . . . . . . . 8 (({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐶𝐷))
166165adantl 481 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → 𝐶𝐷))
167166imp 444 . . . . . 6 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → 𝐶𝐷)
168114, 160, 1673jca 1235 . . . . 5 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → (𝐵𝐶𝐵𝐷𝐶𝐷))
169107, 168jca 553 . . . 4 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)))
17040, 169jca 553 . . 3 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})))) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷))))
17139, 170mpancom 700 . 2 (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) ∧ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴}))) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷))))
172171ex 449 1 ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   < clt 9953  ℕcn 10897  2c2 10947  3c3 10948  4c4 10949  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859 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-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-oadd 7451  df-er 7629  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-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862 This theorem is referenced by:  4cycl4dv4e  26196
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