This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tendospass

Description: Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013)

Ref Expression
Hypotheses tendosp.h 
|- H = ( LHyp ` K )
tendosp.t 
|- T = ( ( LTrn ` K ) ` W )
tendosp.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendospass 
|- ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( ( U o. V ) ` F ) = ( U ` ( V ` F ) ) )

Proof

Step Hyp Ref Expression
1 tendosp.h 
 |-  H = ( LHyp ` K )
2 tendosp.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 tendosp.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 1 2 3 tendof 
 |-  ( ( ( K e. X /\ W e. H ) /\ V e. E ) -> V : T --> T )
5 4 3ad2antr2 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> V : T --> T )
6 simpr3 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> F e. T )
7 fvco3 
 |-  ( ( V : T --> T /\ F e. T ) -> ( ( U o. V ) ` F ) = ( U ` ( V ` F ) ) )
8 5 6 7 syl2anc 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( ( U o. V ) ` F ) = ( U ` ( V ` F ) ) )