Metamath Proof Explorer

Theorem 2wlkond

Description: A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 30-Jan-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 2wlkd.p 
|- P = <" A B C ">
2wlkd.f 
|- F = <" J K ">
2wlkd.s 
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
2wlkd.n 
|- ( ph -> ( A =/= B /\ B =/= C ) )
2wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
2wlkd.v 
|- V = ( Vtx ` G )
2wlkd.i 
|- I = ( iEdg ` G )
Assertion 2wlkond 
|- ( ph -> F ( A ( WalksOn ` G ) C ) P )

Proof

Step Hyp Ref Expression
1 2wlkd.p 
 |-  P = <" A B C ">
2 2wlkd.f 
 |-  F = <" J K ">
3 2wlkd.s 
 |-  ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4 2wlkd.n 
 |-  ( ph -> ( A =/= B /\ B =/= C ) )
5 2wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6 2wlkd.v 
 |-  V = ( Vtx ` G )
7 2wlkd.i 
 |-  I = ( iEdg ` G )
8 1 2 3 4 5 6 7 2wlkd 
 |-  ( ph -> F ( Walks ` G ) P )
9 3 simp1d 
 |-  ( ph -> A e. V )
10 1 fveq1i 
 |-  ( P ` 0 ) = ( <" A B C "> ` 0 )
11 s3fv0 
 |-  ( A e. V -> ( <" A B C "> ` 0 ) = A )
12 10 11 syl5eq 
 |-  ( A e. V -> ( P ` 0 ) = A )
13 9 12 syl 
 |-  ( ph -> ( P ` 0 ) = A )
14 2 fveq2i 
 |-  ( # ` F ) = ( # ` <" J K "> )
15 s2len 
 |-  ( # ` <" J K "> ) = 2
16 14 15 eqtri 
 |-  ( # ` F ) = 2
17 1 16 fveq12i 
 |-  ( P ` ( # ` F ) ) = ( <" A B C "> ` 2 )
18 3 simp3d 
 |-  ( ph -> C e. V )
19 s3fv2 
 |-  ( C e. V -> ( <" A B C "> ` 2 ) = C )
20 18 19 syl 
 |-  ( ph -> ( <" A B C "> ` 2 ) = C )
21 17 20 syl5eq 
 |-  ( ph -> ( P ` ( # ` F ) ) = C )
22 3simpb 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
23 3 22 syl 
 |-  ( ph -> ( A e. V /\ C e. V ) )
24 s2cli 
 |-  <" J K "> e. Word _V
25 2 24 eqeltri 
 |-  F e. Word _V
26 s3cli 
 |-  <" A B C "> e. Word _V
27 1 26 eqeltri 
 |-  P e. Word _V
28 25 27 pm3.2i 
 |-  ( F e. Word _V /\ P e. Word _V )
29 6 iswlkon 
 |-  ( ( ( A e. V /\ C e. V ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( WalksOn ` G ) C ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
30 23 28 29 sylancl 
 |-  ( ph -> ( F ( A ( WalksOn ` G ) C ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
31 8 13 21 30 mpbir3and 
 |-  ( ph -> F ( A ( WalksOn ` G ) C ) P )