Step |
Hyp |
Ref |
Expression |
1 |
|
mgpval.1 |
|- M = ( mulGrp ` R ) |
2 |
|
mgpval.2 |
|- .x. = ( .r ` R ) |
3 |
|
id |
|- ( r = R -> r = R ) |
4 |
|
fveq2 |
|- ( r = R -> ( .r ` r ) = ( .r ` R ) ) |
5 |
4 2
|
eqtr4di |
|- ( r = R -> ( .r ` r ) = .x. ) |
6 |
5
|
opeq2d |
|- ( r = R -> <. ( +g ` ndx ) , ( .r ` r ) >. = <. ( +g ` ndx ) , .x. >. ) |
7 |
3 6
|
oveq12d |
|- ( r = R -> ( r sSet <. ( +g ` ndx ) , ( .r ` r ) >. ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) ) |
8 |
|
df-mgp |
|- mulGrp = ( r e. _V |-> ( r sSet <. ( +g ` ndx ) , ( .r ` r ) >. ) ) |
9 |
|
ovex |
|- ( R sSet <. ( +g ` ndx ) , .x. >. ) e. _V |
10 |
7 8 9
|
fvmpt |
|- ( R e. _V -> ( mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) ) |
11 |
|
fvprc |
|- ( -. R e. _V -> ( mulGrp ` R ) = (/) ) |
12 |
|
reldmsets |
|- Rel dom sSet |
13 |
12
|
ovprc1 |
|- ( -. R e. _V -> ( R sSet <. ( +g ` ndx ) , .x. >. ) = (/) ) |
14 |
11 13
|
eqtr4d |
|- ( -. R e. _V -> ( mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) ) |
15 |
10 14
|
pm2.61i |
|- ( mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) |
16 |
1 15
|
eqtri |
|- M = ( R sSet <. ( +g ` ndx ) , .x. >. ) |