Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
2 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
3 |
|
eqid |
|- ( LSSum ` R ) = ( LSSum ` R ) |
4 |
1 2 3
|
lsmfval |
|- ( R e. V -> ( LSSum ` R ) = ( t e. ~P ( Base ` R ) , u e. ~P ( Base ` R ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` R ) y ) ) ) ) |
5 |
|
fvex |
|- ( ringLMod ` R ) e. _V |
6 |
|
rlmbas |
|- ( Base ` R ) = ( Base ` ( ringLMod ` R ) ) |
7 |
|
rlmplusg |
|- ( +g ` R ) = ( +g ` ( ringLMod ` R ) ) |
8 |
|
eqid |
|- ( LSSum ` ( ringLMod ` R ) ) = ( LSSum ` ( ringLMod ` R ) ) |
9 |
6 7 8
|
lsmfval |
|- ( ( ringLMod ` R ) e. _V -> ( LSSum ` ( ringLMod ` R ) ) = ( t e. ~P ( Base ` R ) , u e. ~P ( Base ` R ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` R ) y ) ) ) ) |
10 |
5 9
|
mp1i |
|- ( R e. V -> ( LSSum ` ( ringLMod ` R ) ) = ( t e. ~P ( Base ` R ) , u e. ~P ( Base ` R ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` R ) y ) ) ) ) |
11 |
4 10
|
eqtr4d |
|- ( R e. V -> ( LSSum ` R ) = ( LSSum ` ( ringLMod ` R ) ) ) |