Metamath Proof Explorer

Theorem tendopl

Description: Value 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 tendopl 
|- ( ( U e. E /\ V e. E ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )

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 fveq1 
 |-  ( u = U -> ( u ` g ) = ( U ` g ) )
4 3 coeq1d 
 |-  ( u = U -> ( ( u ` g ) o. ( v ` g ) ) = ( ( U ` g ) o. ( v ` g ) ) )
5 4 mpteq2dv 
 |-  ( u = U -> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) = ( g e. T |-> ( ( U ` g ) o. ( v ` g ) ) ) )
6 fveq1 
 |-  ( v = V -> ( v ` g ) = ( V ` g ) )
7 6 coeq2d 
 |-  ( v = V -> ( ( U ` g ) o. ( v ` g ) ) = ( ( U ` g ) o. ( V ` g ) ) )
8 7 mpteq2dv 
 |-  ( v = V -> ( g e. T |-> ( ( U ` g ) o. ( v ` g ) ) ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )
9 1 tendoplcbv 
 |-  P = ( u e. E , v e. E |-> ( g e. T |-> ( ( u ` g ) o. ( v ` g ) ) ) )
10 2 fvexi 
 |-  T e. _V
11 10 mptex 
 |-  ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) e. _V
12 5 8 9 11 ovmpo 
 |-  ( ( U e. E /\ V e. E ) -> ( U P V ) = ( g e. T |-> ( ( U ` g ) o. ( V ` g ) ) ) )