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Metamath Proof Explorer

Theorem 3cycld

Description: Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 10-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Hypotheses 3wlkd.p 
|- P = <" A B C D ">
3wlkd.f 
|- F = <" J K L ">
3wlkd.s 
|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
3wlkd.n 
|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
3wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
3wlkd.v 
|- V = ( Vtx ` G )
3wlkd.i 
|- I = ( iEdg ` G )
3trld.n 
|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
3cycld.e 
|- ( ph -> A = D )
Assertion 3cycld 
|- ( ph -> F ( Cycles ` G ) P )

Proof

Step Hyp Ref Expression
1 3wlkd.p 
 |-  P = <" A B C D ">
2 3wlkd.f 
 |-  F = <" J K L ">
3 3wlkd.s 
 |-  ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4 3wlkd.n 
 |-  ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5 3wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6 3wlkd.v 
 |-  V = ( Vtx ` G )
7 3wlkd.i 
 |-  I = ( iEdg ` G )
8 3trld.n 
 |-  ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
9 3cycld.e 
 |-  ( ph -> A = D )
10 1 2 3 4 5 6 7 8 3pthd 
 |-  ( ph -> F ( Paths ` G ) P )
11 1 fveq1i 
 |-  ( P ` 0 ) = ( <" A B C D "> ` 0 )
12 s4fv0 
 |-  ( A e. V -> ( <" A B C D "> ` 0 ) = A )
13 11 12 eqtrid 
 |-  ( A e. V -> ( P ` 0 ) = A )
14 13 ad3antrrr 
 |-  ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> ( P ` 0 ) = A )
15 simpr 
 |-  ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> A = D )
16 2 fveq2i 
 |-  ( # ` F ) = ( # ` <" J K L "> )
17 s3len 
 |-  ( # ` <" J K L "> ) = 3
18 16 17 eqtri 
 |-  ( # ` F ) = 3
19 1 18 fveq12i 
 |-  ( P ` ( # ` F ) ) = ( <" A B C D "> ` 3 )
20 s4fv3 
 |-  ( D e. V -> ( <" A B C D "> ` 3 ) = D )
21 19 20 eqtr2id 
 |-  ( D e. V -> D = ( P ` ( # ` F ) ) )
22 21 adantl 
 |-  ( ( C e. V /\ D e. V ) -> D = ( P ` ( # ` F ) ) )
23 22 ad2antlr 
 |-  ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> D = ( P ` ( # ` F ) ) )
24 14 15 23 3eqtrd 
 |-  ( ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) /\ A = D ) -> ( P ` 0 ) = ( P ` ( # ` F ) ) )
25 3 9 24 syl2anc 
 |-  ( ph -> ( P ` 0 ) = ( P ` ( # ` F ) ) )
26 iscycl 
 |-  ( F ( Cycles ` G ) P <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
27 10 25 26 sylanbrc 
 |-  ( ph -> F ( Cycles ` G ) P )