| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ldual0vs.f |
|- F = ( LFnl ` W ) |
| 2 |
|
ldual0vs.r |
|- R = ( Scalar ` W ) |
| 3 |
|
ldual0vs.z |
|- .0. = ( 0g ` R ) |
| 4 |
|
ldual0vs.d |
|- D = ( LDual ` W ) |
| 5 |
|
ldual0vs.t |
|- .x. = ( .s ` D ) |
| 6 |
|
ldual0vs.o |
|- O = ( 0g ` D ) |
| 7 |
|
ldual0vs.w |
|- ( ph -> W e. LMod ) |
| 8 |
|
ldual0vs.g |
|- ( ph -> G e. F ) |
| 9 |
|
eqid |
|- ( Scalar ` D ) = ( Scalar ` D ) |
| 10 |
|
eqid |
|- ( 0g ` ( Scalar ` D ) ) = ( 0g ` ( Scalar ` D ) ) |
| 11 |
2 3 4 9 10 7
|
ldual0 |
|- ( ph -> ( 0g ` ( Scalar ` D ) ) = .0. ) |
| 12 |
11
|
oveq1d |
|- ( ph -> ( ( 0g ` ( Scalar ` D ) ) .x. G ) = ( .0. .x. G ) ) |
| 13 |
4 7
|
lduallmod |
|- ( ph -> D e. LMod ) |
| 14 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
| 15 |
1 4 14 7 8
|
ldualelvbase |
|- ( ph -> G e. ( Base ` D ) ) |
| 16 |
14 9 5 10 6
|
lmod0vs |
|- ( ( D e. LMod /\ G e. ( Base ` D ) ) -> ( ( 0g ` ( Scalar ` D ) ) .x. G ) = O ) |
| 17 |
13 15 16
|
syl2anc |
|- ( ph -> ( ( 0g ` ( Scalar ` D ) ) .x. G ) = O ) |
| 18 |
12 17
|
eqtr3d |
|- ( ph -> ( .0. .x. G ) = O ) |