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Metamath Proof Explorer

Theorem tendopl2

Description: Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013)

Ref Expression
Hypotheses tendoplcbv.p 
|- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
tendopl2.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion tendopl2 
|- ( ( U e. E /\ V e. E /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )

Proof

Step Hyp Ref Expression
1 tendoplcbv.p 
 |-  P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
2 tendopl2.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 1 2 tendopl 
 |-  ( ( U e. E /\ V e. E ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )
4 3 3adant3 
 |-  ( ( U e. E /\ V e. E /\ F e. T ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )
5 fveq2 
 |-  ( g = F -> ( U ` g ) = ( U ` F ) )
6 fveq2 
 |-  ( g = F -> ( V ` g ) = ( V ` F ) )
7 5 6 coeq12d 
 |-  ( g = F -> ( ( U ` g ) o. ( V ` g ) ) = ( ( U ` F ) o. ( V ` F ) ) )
8 7 adantl 
 |-  ( ( ( U e. E /\ V e. E /\ F e. T ) /\ g = F ) -> ( ( U ` g ) o. ( V ` g ) ) = ( ( U ` F ) o. ( V ` F ) ) )
9 simp3 
 |-  ( ( U e. E /\ V e. E /\ F e. T ) -> F e. T )
10 fvex 
 |-  ( U ` F ) e. _V
11 fvex 
 |-  ( V ` F ) e. _V
12 10 11 coex 
 |-  ( ( U ` F ) o. ( V ` F ) ) e. _V
13 12 a1i 
 |-  ( ( U e. E /\ V e. E /\ F e. T ) -> ( ( U ` F ) o. ( V ` F ) ) e. _V )
14 4 8 9 13 fvmptd 
 |-  ( ( U e. E /\ V e. E /\ F e. T ) -> ( ( U P V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )