Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapellkr.h |
|- H = ( LHyp ` K ) |
2 |
|
hdmapellkr.o |
|- O = ( ( ocH ` K ) ` W ) |
3 |
|
hdmapellkr.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
hdmapellkr.v |
|- V = ( Base ` U ) |
5 |
|
hdmapellkr.r |
|- R = ( Scalar ` U ) |
6 |
|
hdmapellkr.z |
|- .0. = ( 0g ` R ) |
7 |
|
hdmapellkr.s |
|- S = ( ( HDMap ` K ) ` W ) |
8 |
|
hdmapellkr.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
9 |
|
hdmapellkr.x |
|- ( ph -> X e. V ) |
10 |
|
hdmapellkr.y |
|- ( ph -> Y e. V ) |
11 |
|
eqid |
|- ( LFnl ` U ) = ( LFnl ` U ) |
12 |
|
eqid |
|- ( LKer ` U ) = ( LKer ` U ) |
13 |
1 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
14 |
|
eqid |
|- ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W ) |
15 |
|
eqid |
|- ( Base ` ( ( LCDual ` K ) ` W ) ) = ( Base ` ( ( LCDual ` K ) ` W ) ) |
16 |
1 3 4 14 15 7 8 9
|
hdmapcl |
|- ( ph -> ( S ` X ) e. ( Base ` ( ( LCDual ` K ) ` W ) ) ) |
17 |
1 14 15 3 11 8 16
|
lcdvbaselfl |
|- ( ph -> ( S ` X ) e. ( LFnl ` U ) ) |
18 |
4 5 6 11 12 13 17 10
|
ellkr2 |
|- ( ph -> ( Y e. ( ( LKer ` U ) ` ( S ` X ) ) <-> ( ( S ` X ) ` Y ) = .0. ) ) |
19 |
1 2 3 4 11 12 7 8 9
|
hdmaplkr |
|- ( ph -> ( ( LKer ` U ) ` ( S ` X ) ) = ( O ` { X } ) ) |
20 |
19
|
eleq2d |
|- ( ph -> ( Y e. ( ( LKer ` U ) ` ( S ` X ) ) <-> Y e. ( O ` { X } ) ) ) |
21 |
18 20
|
bitr3d |
|- ( ph -> ( ( ( S ` X ) ` Y ) = .0. <-> Y e. ( O ` { X } ) ) ) |