| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fveq2 |
|- ( a = W -> ( subringAlg ` a ) = ( subringAlg ` W ) ) |
| 2 |
|
fveq2 |
|- ( a = W -> ( Base ` a ) = ( Base ` W ) ) |
| 3 |
1 2
|
fveq12d |
|- ( a = W -> ( ( subringAlg ` a ) ` ( Base ` a ) ) = ( ( subringAlg ` W ) ` ( Base ` W ) ) ) |
| 4 |
|
df-rgmod |
|- ringLMod = ( a e. _V |-> ( ( subringAlg ` a ) ` ( Base ` a ) ) ) |
| 5 |
|
fvex |
|- ( ( subringAlg ` W ) ` ( Base ` W ) ) e. _V |
| 6 |
3 4 5
|
fvmpt |
|- ( W e. _V -> ( ringLMod ` W ) = ( ( subringAlg ` W ) ` ( Base ` W ) ) ) |
| 7 |
|
0fv |
|- ( (/) ` ( Base ` W ) ) = (/) |
| 8 |
7
|
eqcomi |
|- (/) = ( (/) ` ( Base ` W ) ) |
| 9 |
|
fvprc |
|- ( -. W e. _V -> ( ringLMod ` W ) = (/) ) |
| 10 |
|
fvprc |
|- ( -. W e. _V -> ( subringAlg ` W ) = (/) ) |
| 11 |
10
|
fveq1d |
|- ( -. W e. _V -> ( ( subringAlg ` W ) ` ( Base ` W ) ) = ( (/) ` ( Base ` W ) ) ) |
| 12 |
8 9 11
|
3eqtr4a |
|- ( -. W e. _V -> ( ringLMod ` W ) = ( ( subringAlg ` W ) ` ( Base ` W ) ) ) |
| 13 |
6 12
|
pm2.61i |
|- ( ringLMod ` W ) = ( ( subringAlg ` W ) ` ( Base ` W ) ) |