Metamath Proof Explorer

Theorem 3spthond

Description: A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021) (Revised by AV, 24-Mar-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 ) )
3spthd.n 
|- ( ph -> A =/= D )
Assertion 3spthond 
|- ( ph -> F ( A ( SPathsOn ` G ) D ) 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 3spthd.n 
 |-  ( ph -> A =/= D )
10 1 2 3 4 5 6 7 8 3trlond 
 |-  ( ph -> F ( A ( TrailsOn ` G ) D ) P )
11 1 2 3 4 5 6 7 8 9 3spthd 
 |-  ( ph -> F ( SPaths ` G ) P )
12 3 simplld 
 |-  ( ph -> A e. V )
13 3 simprrd 
 |-  ( ph -> D e. V )
14 s3cli 
 |-  <" J K L "> e. Word _V
15 2 14 eqeltri 
 |-  F e. Word _V
16 s4cli 
 |-  <" A B C D "> e. Word _V
17 1 16 eqeltri 
 |-  P e. Word _V
18 15 17 pm3.2i 
 |-  ( F e. Word _V /\ P e. Word _V )
19 18 a1i 
 |-  ( ph -> ( F e. Word _V /\ P e. Word _V ) )
20 6 isspthson 
 |-  ( ( ( A e. V /\ D e. V ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( SPathsOn ` G ) D ) P <-> ( F ( A ( TrailsOn ` G ) D ) P /\ F ( SPaths ` G ) P ) ) )
21 12 13 19 20 syl21anc 
 |-  ( ph -> ( F ( A ( SPathsOn ` G ) D ) P <-> ( F ( A ( TrailsOn ` G ) D ) P /\ F ( SPaths ` G ) P ) ) )
22 10 11 21 mpbir2and 
 |-  ( ph -> F ( A ( SPathsOn ` G ) D ) P )