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Metamath Proof Explorer

Theorem 3wlkd

Description: Construction of a walk from two given edges in a graph. (Contributed by AV, 7-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 )
Assertion 3wlkd 
|- ( ph -> F ( Walks ` 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 s4cli 
 |-  <" A B C D "> e. Word _V
9 1 8 eqeltri 
 |-  P e. Word _V
10 9 a1i 
 |-  ( ph -> P e. Word _V )
11 s3cli 
 |-  <" J K L "> e. Word _V
12 2 11 eqeltri 
 |-  F e. Word _V
13 12 a1i 
 |-  ( ph -> F e. Word _V )
14 1 2 3wlkdlem1 
 |-  ( # ` P ) = ( ( # ` F ) + 1 )
15 14 a1i 
 |-  ( ph -> ( # ` P ) = ( ( # ` F ) + 1 ) )
16 1 2 3 4 5 3wlkdlem10 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
17 1 2 3 4 3wlkdlem5 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
18 6 1vgrex 
 |-  ( A e. V -> G e. _V )
19 18 ad2antrr 
 |-  ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> G e. _V )
20 3 19 syl 
 |-  ( ph -> G e. _V )
21 1 2 3 3wlkdlem4 
 |-  ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )
22 10 13 15 16 17 20 6 7 21 wlkd 
 |-  ( ph -> F ( Walks ` G ) P )