Metamath Proof Explorer

Theorem 3pthond

Description: A 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 ) )
Assertion 3pthond 
|- ( ph -> F ( A ( PathsOn ` 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 1 2 3 4 5 6 7 8 3trlond 
 |-  ( ph -> F ( A ( TrailsOn ` G ) D ) P )
10 1 2 3 4 5 6 7 8 3pthd 
 |-  ( ph -> F ( Paths ` G ) P )
11 3 simplld 
 |-  ( ph -> A e. V )
12 3 simprrd 
 |-  ( ph -> D e. V )
13 s3cli 
 |-  <" J K L "> e. Word _V
14 2 13 eqeltri 
 |-  F e. Word _V
15 s4cli 
 |-  <" A B C D "> e. Word _V
16 1 15 eqeltri 
 |-  P e. Word _V
17 14 16 pm3.2i 
 |-  ( F e. Word _V /\ P e. Word _V )
18 17 a1i 
 |-  ( ph -> ( F e. Word _V /\ P e. Word _V ) )
19 6 ispthson 
 |-  ( ( ( A e. V /\ D e. V ) /\ ( F e. Word _V /\ P e. Word _V ) ) -> ( F ( A ( PathsOn ` G ) D ) P <-> ( F ( A ( TrailsOn ` G ) D ) P /\ F ( Paths ` G ) P ) ) )
20 11 12 18 19 syl21anc 
 |-  ( ph -> ( F ( A ( PathsOn ` G ) D ) P <-> ( F ( A ( TrailsOn ` G ) D ) P /\ F ( Paths ` G ) P ) ) )
21 9 10 20 mpbir2and 
 |-  ( ph -> F ( A ( PathsOn ` G ) D ) P )