Step |
Hyp |
Ref |
Expression |
1 |
|
prjspnerlem.e |
|- .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) } |
2 |
|
prjspnerlem.w |
|- W = ( K freeLMod ( 0 ... N ) ) |
3 |
|
prjspnerlem.b |
|- B = ( ( Base ` W ) \ { ( 0g ` W ) } ) |
4 |
|
prjspnerlem.s |
|- S = ( Base ` K ) |
5 |
|
prjspnerlem.x |
|- .x. = ( .s ` W ) |
6 |
|
ovex |
|- ( 0 ... N ) e. _V |
7 |
2
|
frlmsca |
|- ( ( K e. DivRing /\ ( 0 ... N ) e. _V ) -> K = ( Scalar ` W ) ) |
8 |
6 7
|
mpan2 |
|- ( K e. DivRing -> K = ( Scalar ` W ) ) |
9 |
8
|
fveq2d |
|- ( K e. DivRing -> ( Base ` K ) = ( Base ` ( Scalar ` W ) ) ) |
10 |
4 9
|
syl5eq |
|- ( K e. DivRing -> S = ( Base ` ( Scalar ` W ) ) ) |
11 |
10
|
rexeqdv |
|- ( K e. DivRing -> ( E. l e. S x = ( l .x. y ) <-> E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) ) |
12 |
11
|
anbi2d |
|- ( K e. DivRing -> ( ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) <-> ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) ) ) |
13 |
12
|
opabbidv |
|- ( K e. DivRing -> { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. S x = ( l .x. y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) } ) |
14 |
1 13
|
syl5eq |
|- ( K e. DivRing -> .~ = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ E. l e. ( Base ` ( Scalar ` W ) ) x = ( l .x. y ) ) } ) |