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Metamath Proof Explorer

Theorem erngplus2-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 erngplus2-rN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( ( U .+ V ) ` F ) = ( ( 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 erngplus-rN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .+ V ) = ( f e. T |-> ( ( U ` f ) o. ( V ` f ) ) ) )
7 6 3adantr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( U .+ V ) = ( f e. T |-> ( ( U ` f ) o. ( V ` f ) ) ) )
8 fveq2 
 |-  ( f = F -> ( U ` f ) = ( U ` F ) )
9 fveq2 
 |-  ( f = F -> ( V ` f ) = ( V ` F ) )
10 8 9 coeq12d 
 |-  ( f = F -> ( ( U ` f ) o. ( V ` f ) ) = ( ( U ` F ) o. ( V ` F ) ) )
11 10 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) /\ f = F ) -> ( ( U ` f ) o. ( V ` f ) ) = ( ( U ` F ) o. ( V ` F ) ) )
12 simpr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> F e. T )
13 fvex 
 |-  ( U ` F ) e. _V
14 fvex 
 |-  ( V ` F ) e. _V
15 13 14 coex 
 |-  ( ( U ` F ) o. ( V ` F ) ) e. _V
16 15 a1i 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( ( U ` F ) o. ( V ` F ) ) e. _V )
17 7 11 12 16 fvmptd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ F e. T ) ) -> ( ( U .+ V ) ` F ) = ( ( U ` F ) o. ( V ` F ) ) )