Step |
Hyp |
Ref |
Expression |
1 |
|
istrg.1 |
|- M = ( mulGrp ` R ) |
2 |
|
elin |
|- ( R e. ( TopGrp i^i Ring ) <-> ( R e. TopGrp /\ R e. Ring ) ) |
3 |
2
|
anbi1i |
|- ( ( R e. ( TopGrp i^i Ring ) /\ M e. TopMnd ) <-> ( ( R e. TopGrp /\ R e. Ring ) /\ M e. TopMnd ) ) |
4 |
|
fveq2 |
|- ( r = R -> ( mulGrp ` r ) = ( mulGrp ` R ) ) |
5 |
4 1
|
eqtr4di |
|- ( r = R -> ( mulGrp ` r ) = M ) |
6 |
5
|
eleq1d |
|- ( r = R -> ( ( mulGrp ` r ) e. TopMnd <-> M e. TopMnd ) ) |
7 |
|
df-trg |
|- TopRing = { r e. ( TopGrp i^i Ring ) | ( mulGrp ` r ) e. TopMnd } |
8 |
6 7
|
elrab2 |
|- ( R e. TopRing <-> ( R e. ( TopGrp i^i Ring ) /\ M e. TopMnd ) ) |
9 |
|
df-3an |
|- ( ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) <-> ( ( R e. TopGrp /\ R e. Ring ) /\ M e. TopMnd ) ) |
10 |
3 8 9
|
3bitr4i |
|- ( R e. TopRing <-> ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) ) |