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Metamath Proof Explorer

Theorem erngmul-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.m-r 
|- .x. = ( .r ` D )
Assertion erngmul-rN 
|- ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .x. V ) = ( V o. U ) )

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.m-r 
 |-  .x. = ( .r ` D )
6 1 2 3 4 5 erngfmul-rN 
 |-  ( ( K e. X /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( t o. s ) ) )
7 6 adantr 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> .x. = ( s e. E , t e. E |-> ( t o. s ) ) )
8 7 oveqd 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .x. V ) = ( U ( s e. E , t e. E |-> ( t o. s ) ) V ) )
9 coexg 
 |-  ( ( V e. E /\ U e. E ) -> ( V o. U ) e. _V )
10 9 ancoms 
 |-  ( ( U e. E /\ V e. E ) -> ( V o. U ) e. _V )
11 coeq2 
 |-  ( s = U -> ( t o. s ) = ( t o. U ) )
12 coeq1 
 |-  ( t = V -> ( t o. U ) = ( V o. U ) )
13 eqid 
 |-  ( s e. E , t e. E |-> ( t o. s ) ) = ( s e. E , t e. E |-> ( t o. s ) )
14 11 12 13 ovmpog 
 |-  ( ( U e. E /\ V e. E /\ ( V o. U ) e. _V ) -> ( U ( s e. E , t e. E |-> ( t o. s ) ) V ) = ( V o. U ) )
15 10 14 mpd3an3 
 |-  ( ( U e. E /\ V e. E ) -> ( U ( s e. E , t e. E |-> ( t o. s ) ) V ) = ( V o. U ) )
16 15 adantl 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U ( s e. E , t e. E |-> ( t o. s ) ) V ) = ( V o. U ) )
17 8 16 eqtrd 
 |-  ( ( ( K e. X /\ W e. H ) /\ ( U e. E /\ V e. E ) ) -> ( U .x. V ) = ( V o. U ) )