Metamath Proof Explorer

Theorem crctisclwlk

Description: A circuit is a closed walk. (Contributed by AV, 17-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion crctisclwlk 
|- ( F ( Circuits ` G ) P -> F ( ClWalks ` G ) P )

Proof

Step Hyp Ref Expression
1 crctprop 
 |-  ( F ( Circuits ` G ) P -> ( F ( Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
2 trliswlk 
 |-  ( F ( Trails ` G ) P -> F ( Walks ` G ) P )
3 isclwlk 
 |-  ( F ( ClWalks ` G ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
4 3 biimpri 
 |-  ( ( F ( Walks ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> F ( ClWalks ` G ) P )
5 2 4 sylan 
 |-  ( ( F ( Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> F ( ClWalks ` G ) P )
6 1 5 syl 
 |-  ( F ( Circuits ` G ) P -> F ( ClWalks ` G ) P )