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Metamath Proof Explorer

Theorem tendocoval

Description: Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

Ref Expression
Hypotheses tendof.h 
|- H = ( LHyp ` K )
tendof.t 
|- T = ( ( LTrn ` K ) ` W )
tendof.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendocoval 
|- ( ( ( 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 tendof.h 
 |-  H = ( LHyp ` K )
2 tendof.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 tendof.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 simp1 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( K e. X /\ W e. H ) )
5 simp2r 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> V e. E )
6 1 2 3 tendof 
 |-  ( ( ( K e. X /\ W e. H ) /\ V e. E ) -> V : T --> T )
7 4 5 6 syl2anc 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> V : T --> T )
8 simp3 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> F e. T )
9 fvco3 
 |-  ( ( V : T --> T /\ F e. T ) -> ( ( U o. V ) ` F ) = ( U ` ( V ` F ) ) )
10 7 8 9 syl2anc 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ F e. T ) -> ( ( U o. V ) ` F ) = ( U ` ( V ` F ) ) )