Step |
Hyp |
Ref |
Expression |
1 |
|
fnmgp |
|- mulGrp Fn _V |
2 |
|
ssv |
|- Ring C_ _V |
3 |
|
fnssres |
|- ( ( mulGrp Fn _V /\ Ring C_ _V ) -> ( mulGrp |` Ring ) Fn Ring ) |
4 |
1 2 3
|
mp2an |
|- ( mulGrp |` Ring ) Fn Ring |
5 |
|
fvres |
|- ( a e. Ring -> ( ( mulGrp |` Ring ) ` a ) = ( mulGrp ` a ) ) |
6 |
|
eqid |
|- ( mulGrp ` a ) = ( mulGrp ` a ) |
7 |
6
|
ringmgp |
|- ( a e. Ring -> ( mulGrp ` a ) e. Mnd ) |
8 |
5 7
|
eqeltrd |
|- ( a e. Ring -> ( ( mulGrp |` Ring ) ` a ) e. Mnd ) |
9 |
8
|
rgen |
|- A. a e. Ring ( ( mulGrp |` Ring ) ` a ) e. Mnd |
10 |
|
ffnfv |
|- ( ( mulGrp |` Ring ) : Ring --> Mnd <-> ( ( mulGrp |` Ring ) Fn Ring /\ A. a e. Ring ( ( mulGrp |` Ring ) ` a ) e. Mnd ) ) |
11 |
4 9 10
|
mpbir2an |
|- ( mulGrp |` Ring ) : Ring --> Mnd |