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Theorem uspgrn2crct 28753
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 28740 . . 3 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2 istrl 28644 . . . . . . 7 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
3 uspgrupgr 28127 . . . . . . . . 9 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4 eqid 2736 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2736 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5upgriswlk 28589 . . . . . . . . . . . 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 4695 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8 prcom 4693 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
97, 8eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
109eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1110eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
1211anbi2d 629 . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . . . . . . . . . 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 28105 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1615adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝐺 ∈ USPGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1716adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1817adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 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 6501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun 𝐹))
2019simplbi2 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) → (Fun 𝐹𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
21 wrdf 14407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝐺 ∈ USPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
2423adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
2524imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺))
26 2nn 12226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℕ
27 lbfzo0 13612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
2826, 27mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 12429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℕ0
30 1lt2 12324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 13613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1341 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
3534eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝐹) = 2 → (1 ∈ (0..^(♯‘𝐹)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝐹) = 2 → ((0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
3833, 37mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
3938ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
40 f1cofveqaeq 7205 . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((♯‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42 0ne1 12224 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ≠ 1
43 eqneqall 2954 . . . . . . . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . 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 3028 . . . . . . . . . . . . . . . . 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 13657 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
5453raleqdv 3313 . . . . . . . . . . . . . . . . . . . . 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 28615 . . . . . . . . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2))
5958neeq2d 3004 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 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 1131 . . . . . . . . . . . 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, 66syl6bi 252 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))))
6867impd 411 . . . . . . . . . 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 216 . . . . . 6 (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
7372imp 407 . . . . 5 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
7473necon2d 2966 . . . 4 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 2))
7574impancom 452 . . 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 408 1 ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  cdif 3907  c0 4282  𝒫 cpw 4560  {csn 4586  {cpr 4588   class class class wbr 5105  ccnv 5632  dom cdm 5633  Fun wfun 6490  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cn 12153  2c2 12208  0cn0 12413  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402  Vtxcvtx 27947  iEdgciedg 27948  UPGraphcupgr 28031  USPGraphcuspgr 28099  Walkscwlks 28544  Trailsctrls 28638  Circuitsccrcts 28732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-edg 27999  df-uhgr 28009  df-upgr 28033  df-uspgr 28101  df-wlks 28547  df-trls 28640  df-crcts 28734
This theorem is referenced by:  usgrn2cycl  28754
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