Step |
Hyp |
Ref |
Expression |
1 |
|
idrnghm.b |
|- B = ( Base ` R ) |
2 |
|
id |
|- ( R e. Rng -> R e. Rng ) |
3 |
2 2
|
jca |
|- ( R e. Rng -> ( R e. Rng /\ R e. Rng ) ) |
4 |
|
rngabl |
|- ( R e. Rng -> R e. Abel ) |
5 |
|
ablgrp |
|- ( R e. Abel -> R e. Grp ) |
6 |
1
|
idghm |
|- ( R e. Grp -> ( _I |` B ) e. ( R GrpHom R ) ) |
7 |
4 5 6
|
3syl |
|- ( R e. Rng -> ( _I |` B ) e. ( R GrpHom R ) ) |
8 |
|
eqid |
|- ( mulGrp ` R ) = ( mulGrp ` R ) |
9 |
8
|
rngmgp |
|- ( R e. Rng -> ( mulGrp ` R ) e. Smgrp ) |
10 |
|
sgrpmgm |
|- ( ( mulGrp ` R ) e. Smgrp -> ( mulGrp ` R ) e. Mgm ) |
11 |
8 1
|
mgpbas |
|- B = ( Base ` ( mulGrp ` R ) ) |
12 |
11
|
idmgmhm |
|- ( ( mulGrp ` R ) e. Mgm -> ( _I |` B ) e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` R ) ) ) |
13 |
9 10 12
|
3syl |
|- ( R e. Rng -> ( _I |` B ) e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` R ) ) ) |
14 |
7 13
|
jca |
|- ( R e. Rng -> ( ( _I |` B ) e. ( R GrpHom R ) /\ ( _I |` B ) e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` R ) ) ) ) |
15 |
8 8
|
isrnghmmul |
|- ( ( _I |` B ) e. ( R RngHomo R ) <-> ( ( R e. Rng /\ R e. Rng ) /\ ( ( _I |` B ) e. ( R GrpHom R ) /\ ( _I |` B ) e. ( ( mulGrp ` R ) MgmHom ( mulGrp ` R ) ) ) ) ) |
16 |
3 14 15
|
sylanbrc |
|- ( R e. Rng -> ( _I |` B ) e. ( R RngHomo R ) ) |