Step |
Hyp |
Ref |
Expression |
1 |
|
dvhvbase.h |
|- H = ( LHyp ` K ) |
2 |
|
dvhvbase.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
dvhvbase.e |
|- E = ( ( TEndo ` K ) ` W ) |
4 |
|
dvhvbase.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
dvhvbase.v |
|- V = ( Base ` U ) |
6 |
|
opelxpi |
|- ( ( F e. T /\ S e. E ) -> <. F , S >. e. ( T X. E ) ) |
7 |
6
|
adantl |
|- ( ( ( K e. X /\ W e. H ) /\ ( F e. T /\ S e. E ) ) -> <. F , S >. e. ( T X. E ) ) |
8 |
1 2 3 4 5
|
dvhvbase |
|- ( ( K e. X /\ W e. H ) -> V = ( T X. E ) ) |
9 |
8
|
adantr |
|- ( ( ( K e. X /\ W e. H ) /\ ( F e. T /\ S e. E ) ) -> V = ( T X. E ) ) |
10 |
7 9
|
eleqtrrd |
|- ( ( ( K e. X /\ W e. H ) /\ ( F e. T /\ S e. E ) ) -> <. F , S >. e. V ) |