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Metamath Proof Explorer

Theorem tendospdi1

Description: Forward distributive 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 tendospdi1 
|- ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> ( U ` ( F o. G ) ) = ( ( U ` F ) o. ( U ` G ) ) )

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 simpll 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> K e. V )
5 simplr 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> W e. H )
6 simpr1 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> U e. E )
7 simpr2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> F e. T )
8 simpr3 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> G e. T )
9 1 2 3 tendovalco 
 |-  ( ( ( K e. V /\ W e. H /\ U e. E ) /\ ( F e. T /\ G e. T ) ) -> ( U ` ( F o. G ) ) = ( ( U ` F ) o. ( U ` G ) ) )
10 4 5 6 7 8 9 syl32anc 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( U e. E /\ F e. T /\ G e. T ) ) -> ( U ` ( F o. G ) ) = ( ( U ` F ) o. ( U ` G ) ) )