Step |
Hyp |
Ref |
Expression |
1 |
|
mendsca.a |
|- A = ( MEndo ` M ) |
2 |
|
mendsca.s |
|- S = ( Scalar ` M ) |
3 |
|
fvex |
|- ( Scalar ` M ) e. _V |
4 |
|
eqid |
|- ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) = ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) |
5 |
4
|
algsca |
|- ( ( Scalar ` M ) e. _V -> ( Scalar ` M ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) ) ) |
6 |
3 5
|
mp1i |
|- ( M e. _V -> ( Scalar ` M ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) ) ) |
7 |
|
eqid |
|- ( M LMHom M ) = ( M LMHom M ) |
8 |
|
eqid |
|- ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) = ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) |
9 |
|
eqid |
|- ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) = ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) |
10 |
|
eqid |
|- ( Scalar ` M ) = ( Scalar ` M ) |
11 |
|
eqid |
|- ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) = ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) |
12 |
7 8 9 10 11
|
mendval |
|- ( M e. _V -> ( MEndo ` M ) = ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) ) |
13 |
12
|
fveq2d |
|- ( M e. _V -> ( Scalar ` ( MEndo ` M ) ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x oF ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` ( Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) >. } ) ) ) |
14 |
6 13
|
eqtr4d |
|- ( M e. _V -> ( Scalar ` M ) = ( Scalar ` ( MEndo ` M ) ) ) |
15 |
|
df-sca |
|- Scalar = Slot 5 |
16 |
15
|
str0 |
|- (/) = ( Scalar ` (/) ) |
17 |
|
fvprc |
|- ( -. M e. _V -> ( Scalar ` M ) = (/) ) |
18 |
|
fvprc |
|- ( -. M e. _V -> ( MEndo ` M ) = (/) ) |
19 |
18
|
fveq2d |
|- ( -. M e. _V -> ( Scalar ` ( MEndo ` M ) ) = ( Scalar ` (/) ) ) |
20 |
16 17 19
|
3eqtr4a |
|- ( -. M e. _V -> ( Scalar ` M ) = ( Scalar ` ( MEndo ` M ) ) ) |
21 |
14 20
|
pm2.61i |
|- ( Scalar ` M ) = ( Scalar ` ( MEndo ` M ) ) |
22 |
1
|
fveq2i |
|- ( Scalar ` A ) = ( Scalar ` ( MEndo ` M ) ) |
23 |
21 2 22
|
3eqtr4i |
|- S = ( Scalar ` A ) |