MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uspgrn2crct Structured version   Visualization version   GIF version

Theorem uspgrn2crct 26756
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 26743 . . 3 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2 istrl 26649 . . . . . . 7 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
3 uspgrupgr 26116 . . . . . . . . 9 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4 eqid 2651 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2651 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5upgriswlk 26593 . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8 prcom 4299 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
97, 8syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
109eqcoms 2659 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1110eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
1211anbi2d 740 . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . . . . . . . . . . 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 26094 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun 𝐹))
2019simplbi2 654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) → (Fun 𝐹𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
21 wrdf 13342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺))
26 2nn 11223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℕ
27 lbfzo0 12547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
2826, 27mpbir 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 11346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℕ0
30 1lt2 11232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1263 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
3534eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 2 → (1 ∈ (0..^(#‘𝐹)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝐹) = 2 → ((0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
3833, 37mpbiri 248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
3938ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
40 f1cofveqaeq 6555 . . . . . . . . . . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42 0ne1 11126 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ≠ 1
43 eqneqall 2834 . . . . . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . 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 2906 . . . . . . . . . . . . . . . . 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 12593 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
5453raleqdv 3174 . . . . . . . . . . . . . . . . . . . . 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 26619 . . . . . . . . . . . . . . . . . . . . 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, 55syl6bb 276 . . . . . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
5958neeq2d 2883 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
6057, 59imbi12d 333 . . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 1100 . . . . . . . . . . . 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 243 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
6867impd 446 . . . . . . . . . 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 207 . . . . . 6 (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
7372imp 444 . . . . 5 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
7473necon2d 2846 . . . 4 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 2))
7574impancom 455 . . 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 445 1 ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (#‘𝐹) ≠ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212   class class class wbr 4685  ccnv 5142  dom cdm 5143  Fun wfun 5920  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cn 11058  2c2 11108  0cn0 11330  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  UPGraphcupgr 26020  USPGraphcuspgr 26088  Walkscwlks 26548  Trailsctrls 26643  Circuitsccrcts 26735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551  df-trls 26645  df-crcts 26737
This theorem is referenced by:  usgrn2cycl  26757
  Copyright terms: Public domain W3C validator