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Metamath Proof Explorer

Theorem dvhmulr

Description: Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvhfmul.h 
|- H = ( LHyp ` K )
dvhfmul.t 
|- T = ( ( LTrn ` K ) ` W )
dvhfmul.e 
|- E = ( ( TEndo ` K ) ` W )
dvhfmul.u 
|- U = ( ( DVecH ` K ) ` W )
dvhfmul.f 
|- F = ( Scalar ` U )
dvhfmul.m 
|- .x. = ( .r ` F )
Assertion dvhmulr 
|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .x. S ) = ( R o. S ) )

Proof

Step Hyp Ref Expression
1 dvhfmul.h 
 |-  H = ( LHyp ` K )
2 dvhfmul.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvhfmul.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvhfmul.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvhfmul.f 
 |-  F = ( Scalar ` U )
6 dvhfmul.m 
 |-  .x. = ( .r ` F )
7 1 2 3 4 5 6 dvhfmulr 
 |-  ( ( K e. V /\ W e. H ) -> .x. = ( r e. E , s e. E |-> ( r o. s ) ) )
8 7 oveqd 
 |-  ( ( K e. V /\ W e. H ) -> ( R .x. S ) = ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) )
9 coexg 
 |-  ( ( R e. E /\ S e. E ) -> ( R o. S ) e. _V )
10 coeq1 
 |-  ( r = R -> ( r o. s ) = ( R o. s ) )
11 coeq2 
 |-  ( s = S -> ( R o. s ) = ( R o. S ) )
12 eqid 
 |-  ( r e. E , s e. E |-> ( r o. s ) ) = ( r e. E , s e. E |-> ( r o. s ) )
13 10 11 12 ovmpog 
 |-  ( ( R e. E /\ S e. E /\ ( R o. S ) e. _V ) -> ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) = ( R o. S ) )
14 9 13 mpd3an3 
 |-  ( ( R e. E /\ S e. E ) -> ( R ( r e. E , s e. E |-> ( r o. s ) ) S ) = ( R o. S ) )
15 8 14 sylan9eq 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .x. S ) = ( R o. S ) )