Metamath Proof Explorer

Theorem 2trlond

Description: A trail 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 )
2trld.n 
|- ( ph -> J =/= K )
Assertion 2trlond 
|- ( ph -> F ( A ( TrailsOn ` 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 2trld.n 
 |-  ( ph -> J =/= K )
9 1 2 3 4 5 6 7 2wlkond 
 |-  ( ph -> F ( A ( WalksOn ` G ) C ) P )
10 1 2 3 4 5 6 7 8 2trld 
 |-  ( ph -> F ( Trails ` G ) P )
11 3 simp1d 
 |-  ( ph -> A e. V )
12 3 simp3d 
 |-  ( ph -> C e. V )
13 s2cli 
 |-  <" J K "> e. Word _V
14 2 13 eqeltri 
 |-  F e. Word _V
15 14 a1i 
 |-  ( ph -> F e. Word _V )
16 s3cli 
 |-  <" A B C "> e. Word _V
17 1 16 eqeltri 
 |-  P e. Word _V
18 17 a1i 
 |-  ( ph -> P e. Word _V )
19 6 istrlson 
 |-  ( ( ( A e. V /\ C e. V ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( TrailsOn ` G ) C ) P <-> ( F ( A ( WalksOn ` G ) C ) P /\ F ( Trails ` G ) P ) ) )
20 11 12 15 18 19 syl22anc 
 |-  ( ph -> ( F ( A ( TrailsOn ` G ) C ) P <-> ( F ( A ( WalksOn ` G ) C ) P /\ F ( Trails ` G ) P ) ) )
21 9 10 20 mpbir2and 
 |-  ( ph -> F ( A ( TrailsOn ` G ) C ) P )