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Theorem uspgrn2crct 26569
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 26556 . . 3 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2 istrl 26462 . . . . . . 7 (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))
3 uspgrupgr 25964 . . . . . . . . 9 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
4 eqid 2621 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2621 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5upgriswlk 26406 . . . . . . . . . . . 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 4239 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8 prcom 4237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
97, 8syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
109eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1110eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
1211anbi2d 739 . . . . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . . . . . . . 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 25942 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺))
26 2nn 11129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℕ
27 lbfzo0 12448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
2826, 27mpbir 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 11252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℕ0
30 1lt2 11138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 12449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1242 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 6612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
3534eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 2 → (1 ∈ (0..^(#‘𝐹)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
40 f1cofveqaeq 6469 . . . . . . . . . . . . . . . . . . . . . . . . 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 1322 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42 0ne1 11032 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ≠ 1
43 eqneqall 2801 . . . . . . . . . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . 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 2873 . . . . . . . . . . . . . . . . 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 12494 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
5453raleqdv 3133 . . . . . . . . . . . . . . . . . . . . 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 26432 . . . . . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
5958neeq2d 2850 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
6057, 59imbi12d 334 . . . . . . . . . . . . . . . . . 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 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 450 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 1078 . . . . . . . . . . . 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 447 . . . . . . . . . 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 445 . . . . 5 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph ) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
7473necon2d 2813 . . . 4 ((𝐹(Trails‘𝐺)𝑃𝐺 ∈ USPGraph ) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 2))
7574impancom 456 . . 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 446 1 ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (#‘𝐹) ≠ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  cdif 3552  c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150   class class class wbr 4613  ccnv 5073  dom cdm 5074  Fun wfun 5841  wf 5843  1-1wf1 5844  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  cle 10019  cn 10964  2c2 11014  0cn0 11236  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230  Vtxcvtx 25774  iEdgciedg 25775   UPGraph cupgr 25871   USPGraph cuspgr 25936  Walkscwlks 26362  Trailsctrls 26456  Circuitsccrcts 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-edg 25840  df-uhgr 25849  df-upgr 25873  df-uspgr 25938  df-wlks 26365  df-trls 26458  df-crcts 26550
This theorem is referenced by:  usgrn2cycl  26570
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