Metamath Proof Explorer

Theorem umgrwlknloop

Description: In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017) (Revised by AV, 3-Jan-2021)

Ref Expression
Assertion umgrwlknloop 
|- ( ( G e. UMGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Proof

Step Hyp Ref Expression
1 umgrupgr 
 |-  ( G e. UMGraph -> G e. UPGraph )
2 eqid 
 |-  ( Edg ` G ) = ( Edg ` G )
3 2 upgrwlkvtxedg 
 |-  ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) )
4 1 3 sylan 
 |-  ( ( G e. UMGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) )
5 2 umgredgne 
 |-  ( ( G e. UMGraph /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
6 5 ex 
 |-  ( G e. UMGraph -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
7 6 adantr 
 |-  ( ( G e. UMGraph /\ F ( Walks ` G ) P ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
8 7 ralimdv 
 |-  ( ( G e. UMGraph /\ F ( Walks ` G ) P ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. ( Edg ` G ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
9 4 8 mpd 
 |-  ( ( G e. UMGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )