Step |
Hyp |
Ref |
Expression |
1 |
|
dvhfvsca.h |
|- H = ( LHyp ` K ) |
2 |
|
dvhfvsca.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
dvhfvsca.e |
|- E = ( ( TEndo ` K ) ` W ) |
4 |
|
dvhfvsca.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
dvhfvsca.s |
|- .x. = ( .s ` U ) |
6 |
1 2 3 4 5
|
dvhfvsca |
|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) ) |
7 |
6
|
oveqd |
|- ( ( K e. V /\ W e. H ) -> ( R .x. F ) = ( R ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) F ) ) |
8 |
|
eqid |
|- ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) |
9 |
8
|
dvhvscaval |
|- ( ( R e. E /\ F e. ( T X. E ) ) -> ( R ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. ) |
10 |
7 9
|
sylan9eq |
|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. ) |