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Theorem uspgrn2crct 29837
Description: In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
uspgrn2crct ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Proof of Theorem uspgrn2crct
Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crctprop 29824 . . 3 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2 istrl 29728 . . . . . . 7 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
3 uspgrupgr 29209 . . . . . . . . 9 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4 eqid 2734 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2734 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5upgriswlk 29673 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
7 preq2 4738 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8 prcom 4736 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
97, 8eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
109eqcoms 2742 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1110eqeq2d 2745 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
1211anbi2d 630 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃‘0) = (𝑃‘2) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
1312ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
14 eqtr3 2760 . . . . . . . . . . . . . . . . . . . . . 22 ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)))
154, 5uspgrf 29185 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1615adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝐺 ∈ USPGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1716adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1817adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
19 df-f1 6567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun 𝐹))
2019simplbi2 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) → (Fun 𝐹𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
21 wrdf 14553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺))
2220, 21syl11 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝐹 → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
2322adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝐺 ∈ USPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
2423adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
2524imp 406 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺))
26 2nn 12336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℕ
27 lbfzo0 13735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
2826, 27mpbir 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 12539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℕ0
30 1lt2 12434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 13736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1340 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
3534eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐹) = 2 → (1 ∈ (0..^(♯‘𝐹)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝐹) = 2 → ((0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
3833, 37mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
3938ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
40 f1cofveqaeq 7277 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)) ∧ (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹)))) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
4118, 25, 39, 40syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42 0ne1 12334 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ≠ 1
43 eqneqall 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = 1 → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘2)))
4441, 42, 43syl6mpi 67 . . . . . . . . . . . . . . . . . . . . . . 23 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
4544adantll 714 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
4614, 45syl5 34 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝑃‘0) ≠ (𝑃‘2)))
4713, 46sylbid 240 . . . . . . . . . . . . . . . . . . . 20 ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))
4847expimpd 453 . . . . . . . . . . . . . . . . . . 19 (((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph))) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
4948ex 412 . . . . . . . . . . . . . . . . . 18 ((𝑃‘0) = (𝑃‘2) → (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
50 2a1 28 . . . . . . . . . . . . . . . . . 18 ((𝑃‘0) ≠ (𝑃‘2) → (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
5149, 50pm2.61ine 3022 . . . . . . . . . . . . . . . . 17 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
52 fzo0to2pr 13785 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
5453raleqdv 3323 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
55 2wlklem 29699 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
5654, 55bitrdi 287 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
5756anbi2d 630 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐹) = 2 → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
58 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2))
5958neeq2d 2998 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
6057, 59imbi12d 344 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) = 2 → (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) ↔ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
6160adantr 480 . . . . . . . . . . . . . . . . 17 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) ↔ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
6251, 61mpbird 257 . . . . . . . . . . . . . . . 16 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
6362ex 412 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → ((Fun 𝐹𝐺 ∈ USPGraph) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
6463com13 88 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((Fun 𝐹𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
6564expd 415 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
66653adant2 1130 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
676, 66biimtrdi 253 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))))
6867impd 410 . . . . . . . . . 10 (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
6968com23 86 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
703, 69mpcom 38 . . . . . . . 8 (𝐺 ∈ USPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
7170com12 32 . . . . . . 7 ((𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹) → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
722, 71sylbi 217 . . . . . 6 (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
7372imp 406 . . . . 5 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
7473necon2d 2960 . . . 4 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 2))
7574impancom 451 . . 3 ((𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐺 ∈ USPGraph → (♯‘𝐹) ≠ 2))
761, 75syl 17 . 2 (𝐹(Circuits‘𝐺)𝑃 → (𝐺 ∈ USPGraph → (♯‘𝐹) ≠ 2))
7776impcom 407 1 ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  cdif 3959  c0 4338  𝒫 cpw 4604  {csn 4630  {cpr 4632   class class class wbr 5147  ccnv 5687  dom cdm 5688  Fun wfun 6556  wf 6558  1-1wf1 6559  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292  cle 11293  cn 12263  2c2 12318  0cn0 12523  ...cfz 13543  ..^cfzo 13690  chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  UPGraphcupgr 29111  USPGraphcuspgr 29179  Walkscwlks 29628  Trailsctrls 29722  Circuitsccrcts 29816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-uspgr 29181  df-wlks 29631  df-trls 29724  df-crcts 29818
This theorem is referenced by:  usgrn2cycl  29838
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