Step |
Hyp |
Ref |
Expression |
1 |
|
prjsprel.1 |
|- .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. K x = ( l .x. y ) ) } |
2 |
|
prjspertr.b |
|- B = ( ( Base ` V ) \ { ( 0g ` V ) } ) |
3 |
|
prjspertr.s |
|- S = ( Scalar ` V ) |
4 |
|
prjspertr.x |
|- .x. = ( .s ` V ) |
5 |
|
prjspertr.k |
|- K = ( Base ` S ) |
6 |
|
prjspreln0.z |
|- .0. = ( 0g ` S ) |
7 |
|
eqid |
|- ( Base ` V ) = ( Base ` V ) |
8 |
|
lveclmod |
|- ( V e. LVec -> V e. LMod ) |
9 |
8
|
3ad2ant1 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> V e. LMod ) |
10 |
|
eldifi |
|- ( N e. ( K \ { .0. } ) -> N e. K ) |
11 |
10
|
3ad2ant3 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> N e. K ) |
12 |
|
difss |
|- ( ( Base ` V ) \ { ( 0g ` V ) } ) C_ ( Base ` V ) |
13 |
2 12
|
eqsstri |
|- B C_ ( Base ` V ) |
14 |
13
|
sseli |
|- ( X e. B -> X e. ( Base ` V ) ) |
15 |
14
|
3ad2ant2 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> X e. ( Base ` V ) ) |
16 |
7 3 4 5 9 11 15
|
lmodvscld |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) e. ( Base ` V ) ) |
17 |
|
eldifsni |
|- ( N e. ( K \ { .0. } ) -> N =/= .0. ) |
18 |
17
|
3ad2ant3 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> N =/= .0. ) |
19 |
|
eldifsni |
|- ( X e. ( ( Base ` V ) \ { ( 0g ` V ) } ) -> X =/= ( 0g ` V ) ) |
20 |
19 2
|
eleq2s |
|- ( X e. B -> X =/= ( 0g ` V ) ) |
21 |
20
|
3ad2ant2 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> X =/= ( 0g ` V ) ) |
22 |
|
eqid |
|- ( 0g ` V ) = ( 0g ` V ) |
23 |
|
simp1 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> V e. LVec ) |
24 |
7 4 3 5 6 22 23 11 15
|
lvecvsn0 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( ( N .x. X ) =/= ( 0g ` V ) <-> ( N =/= .0. /\ X =/= ( 0g ` V ) ) ) ) |
25 |
18 21 24
|
mpbir2and |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) =/= ( 0g ` V ) ) |
26 |
|
nelsn |
|- ( ( N .x. X ) =/= ( 0g ` V ) -> -. ( N .x. X ) e. { ( 0g ` V ) } ) |
27 |
25 26
|
syl |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> -. ( N .x. X ) e. { ( 0g ` V ) } ) |
28 |
16 27
|
eldifd |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) e. ( ( Base ` V ) \ { ( 0g ` V ) } ) ) |
29 |
28 2
|
eleqtrrdi |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) e. B ) |
30 |
|
simp2 |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> X e. B ) |
31 |
|
oveq1 |
|- ( N = m -> ( N .x. X ) = ( m .x. X ) ) |
32 |
31
|
eqcoms |
|- ( m = N -> ( N .x. X ) = ( m .x. X ) ) |
33 |
|
tbtru |
|- ( ( N .x. X ) = ( m .x. X ) <-> ( ( N .x. X ) = ( m .x. X ) <-> T. ) ) |
34 |
32 33
|
sylib |
|- ( m = N -> ( ( N .x. X ) = ( m .x. X ) <-> T. ) ) |
35 |
34
|
adantl |
|- ( ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) /\ m = N ) -> ( ( N .x. X ) = ( m .x. X ) <-> T. ) ) |
36 |
|
trud |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> T. ) |
37 |
11 35 36
|
rspcedvd |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> E. m e. K ( N .x. X ) = ( m .x. X ) ) |
38 |
1
|
prjsprel |
|- ( ( N .x. X ) .~ X <-> ( ( ( N .x. X ) e. B /\ X e. B ) /\ E. m e. K ( N .x. X ) = ( m .x. X ) ) ) |
39 |
29 30 37 38
|
syl21anbrc |
|- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) .~ X ) |