Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uspgrn2crct Structured version   Visualization version   GIF version

Theorem uspgrn2crct 41011
 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.)
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 40998 . . 3 (𝐹(CircuitS‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2 trlis1wlk 40905 . . . . . . 7 (𝐹(TrailS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
3 wlkv 40815 . . . . . . . 8 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4 isTrl 40904 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun 𝐹)))
5 uspgrupgr 40406 . . . . . . . . . . . . . . 15 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
6 eqid 2610 . . . . . . . . . . . . . . . . . 18 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2610 . . . . . . . . . . . . . . . . . 18 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7upgriswlk 40849 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
9 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
10 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
119, 10syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1211eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
1312eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
1413anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((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)})))
1514ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((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)})))
16 eqtr3 2631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)))
176, 7uspgrf 40384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1817adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((Fun 𝐹𝐺 ∈ USPGraph ) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1918adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
2019adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
21 wrdf 13165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺))
22 df-f1 5809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun 𝐹))
2322simplbi2 653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) → (Fun 𝐹𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
2421, 23syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝐹 ∈ Word dom (iEdg‘𝐺) → (Fun 𝐹𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
2524com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Fun 𝐹 → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
2625adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((Fun 𝐹𝐺 ∈ USPGraph ) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
2726adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
2827imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺))
29 2nn 11062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ ℕ
30 lbfzo0 12375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
3129, 30mpbir 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ (0..^2)
32 1nn0 11185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1 ∈ ℕ0
33 1lt2 11071 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1 < 2
34 elfzo0 12376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
3532, 29, 33, 34mpbir3an 1237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ (0..^2)
3631, 35pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
37 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
3837eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ↔ 0 ∈ (0..^2)))
3937eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘𝐹) = 2 → (1 ∈ (0..^(#‘𝐹)) ↔ 1 ∈ (0..^2)))
4038, 39anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝐹) = 2 → ((0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
4136, 40mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
4241ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
43 f1cofveqaeq 40323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((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))
4420, 28, 42, 43syl21anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
45 0ne1 10965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 ≠ 1
46 eqneqall 2793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 = 1 → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘2)))
4744, 45, 46syl6mpi 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
4847adantll 746 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((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)))
4916, 48syl5 33 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((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)))
5015, 49sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((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)))
5150expimpd 627 . . . . . . . . . . . . . . . . . . . . . . . 24 (((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)))
5251ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 ((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))))
53 2a1 28 . . . . . . . . . . . . . . . . . . . . . . 23 ((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))))
5452, 53pm2.61ine 2865 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘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)))
55 fzo0to2pr 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0..^2) = {0, 1}
5637, 55syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
5756raleqdv 3121 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
58 2wlklem 26094 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀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)}))
5957, 58syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘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)})))
6059anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘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)}))))
61 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
6261neeq2d 2842 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
6360, 62imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘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))))
6463adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘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))))
6554, 64mpbird 246 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝐹) = 2 ∧ (Fun 𝐹𝐺 ∈ USPGraph )) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
6665ex 449 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) = 2 → ((Fun 𝐹𝐺 ∈ USPGraph ) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
6766com13 86 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((Fun 𝐹𝐺 ∈ USPGraph ) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
6867expd 451 . . . . . . . . . . . . . . . . . 18 ((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))))))
69683adant2 1073 . . . . . . . . . . . . . . . . 17 ((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))))))
708, 69syl6bi 242 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
71703expib 1260 . . . . . . . . . . . . . . 15 (𝐺 ∈ UPGraph → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
725, 71syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ USPGraph → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
7372com25 97 . . . . . . . . . . . . 13 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
7473pm2.43i 50 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
7574com14 94 . . . . . . . . . . 11 ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
76753adant1 1072 . . . . . . . . . 10 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun 𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
7776impd 446 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹(1Walks‘𝐺)𝑃 ∧ Fun 𝐹) → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
784, 77sylbid 229 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
793, 78syl 17 . . . . . . 7 (𝐹(1Walks‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
802, 79mpcom 37 . . . . . 6 (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
8180imp 444 . . . . 5 ((𝐹(TrailS‘𝐺)𝑃𝐺 ∈ USPGraph ) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
8281necon2d 2805 . . . 4 ((𝐹(TrailS‘𝐺)𝑃𝐺 ∈ USPGraph ) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 2))
8382impancom 455 . . 3 ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐺 ∈ USPGraph → (#‘𝐹) ≠ 2))
841, 83syl 17 . 2 (𝐹(CircuitS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → (#‘𝐹) ≠ 2))
8584impcom 445 1 ((𝐺 ∈ USPGraph ∧ 𝐹(CircuitS‘𝐺)𝑃) → (#‘𝐹) ≠ 2)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747   USPGraph cuspgr 40378  1Walksc1wlks 40796  TrailSctrls 40899  CircuitSccrcts 40990 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-ifp 1007  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-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801  df-trls 40901  df-crcts 40992 This theorem is referenced by:  usgrn2cycl  41012
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