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
|
dvhvsca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. ) |
7 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
simprl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> R e. E ) |
9 |
|
xp1st |
|- ( F e. ( T X. E ) -> ( 1st ` F ) e. T ) |
10 |
9
|
ad2antll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( 1st ` F ) e. T ) |
11 |
1 2 3
|
tendocl |
|- ( ( ( K e. HL /\ W e. H ) /\ R e. E /\ ( 1st ` F ) e. T ) -> ( R ` ( 1st ` F ) ) e. T ) |
12 |
7 8 10 11
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R ` ( 1st ` F ) ) e. T ) |
13 |
|
xp2nd |
|- ( F e. ( T X. E ) -> ( 2nd ` F ) e. E ) |
14 |
13
|
ad2antll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( 2nd ` F ) e. E ) |
15 |
1 3
|
tendococl |
|- ( ( ( K e. HL /\ W e. H ) /\ R e. E /\ ( 2nd ` F ) e. E ) -> ( R o. ( 2nd ` F ) ) e. E ) |
16 |
7 8 14 15
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R o. ( 2nd ` F ) ) e. E ) |
17 |
|
opelxpi |
|- ( ( ( R ` ( 1st ` F ) ) e. T /\ ( R o. ( 2nd ` F ) ) e. E ) -> <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. e. ( T X. E ) ) |
18 |
12 16 17
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. e. ( T X. E ) ) |
19 |
6 18
|
eqeltrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) e. ( T X. E ) ) |