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Metamath Proof Explorer

Theorem erngplus-rN

Description: Ring addition operation. (Contributed by NM, 10-Jun-2013) (New usage is discouraged.)

Ref Expression
Hypotheses erngset.h-r 
|- H = ( LHyp ` K )
erngset.t-r 
|- T = ( ( LTrn ` K ) ` W )
erngset.e-r 
|- E = ( ( TEndo ` K ) ` W )
erngset.d-r 
|- D = ( ( EDRingR ` K ) ` W )
erng.p-r 
|- .+ = ( +g ` D )
Assertion erngplus-rN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .+ V ) = ( f e. T |-> ( ( U ` f ) o. ( V ` f ) ) ) )

Proof

Step Hyp Ref Expression
1 erngset.h-r 
 |-  H = ( LHyp ` K )
2 erngset.t-r 
 |-  T = ( ( LTrn ` K ) ` W )
3 erngset.e-r 
 |-  E = ( ( TEndo ` K ) ` W )
4 erngset.d-r 
 |-  D = ( ( EDRingR ` K ) ` W )
5 erng.p-r 
 |-  .+ = ( +g ` D )
6 1 2 3 4 5 erngfplus-rN 
 |-  ( ( K e. HL /\ W e. H ) -> .+ = ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) )
7 6 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( U .+ V ) = ( U ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) V ) )
8 eqid 
 |-  ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) = ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) )
9 8 2 tendopl 
 |-  ( ( U e. E /\ V e. E ) -> ( U ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) V ) = ( f e. T |-> ( ( U ` f ) o. ( V ` f ) ) ) )
10 7 9 sylan9eq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .+ V ) = ( f e. T |-> ( ( U ` f ) o. ( V ` f ) ) ) )