Step |
Hyp |
Ref |
Expression |
1 |
|
dvafmul.h |
|- H = ( LHyp ` K ) |
2 |
|
dvafmul.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
dvafmul.e |
|- E = ( ( TEndo ` K ) ` W ) |
4 |
|
dvafmul.u |
|- U = ( ( DVecA ` K ) ` W ) |
5 |
|
dvafmul.f |
|- F = ( Scalar ` U ) |
6 |
|
dvafmul.p |
|- .x. = ( .r ` F ) |
7 |
|
eqid |
|- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W ) |
8 |
1 7 4 5
|
dvasca |
|- ( ( K e. V /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) ) |
9 |
8
|
fveq2d |
|- ( ( K e. V /\ W e. H ) -> ( .r ` F ) = ( .r ` ( ( EDRing ` K ) ` W ) ) ) |
10 |
6 9
|
syl5eq |
|- ( ( K e. V /\ W e. H ) -> .x. = ( .r ` ( ( EDRing ` K ) ` W ) ) ) |
11 |
|
eqid |
|- ( .r ` ( ( EDRing ` K ) ` W ) ) = ( .r ` ( ( EDRing ` K ) ` W ) ) |
12 |
1 2 3 7 11
|
erngfmul |
|- ( ( K e. V /\ W e. H ) -> ( .r ` ( ( EDRing ` K ) ` W ) ) = ( s e. E , t e. E |-> ( s o. t ) ) ) |
13 |
10 12
|
eqtrd |
|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( s o. t ) ) ) |