Step |
Hyp |
Ref |
Expression |
1 |
|
drngnzr |
|- ( R e. DivRing -> R e. NzRing ) |
2 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
3 |
|
eqid |
|- ( Unit ` R ) = ( Unit ` R ) |
4 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
5 |
2 3 4
|
isdrng |
|- ( R e. DivRing <-> ( R e. Ring /\ ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) |
6 |
5
|
simprbi |
|- ( R e. DivRing -> ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
7 |
|
drngring |
|- ( R e. DivRing -> R e. Ring ) |
8 |
|
eqid |
|- ( RLReg ` R ) = ( RLReg ` R ) |
9 |
8 3
|
unitrrg |
|- ( R e. Ring -> ( Unit ` R ) C_ ( RLReg ` R ) ) |
10 |
7 9
|
syl |
|- ( R e. DivRing -> ( Unit ` R ) C_ ( RLReg ` R ) ) |
11 |
6 10
|
eqsstrrd |
|- ( R e. DivRing -> ( ( Base ` R ) \ { ( 0g ` R ) } ) C_ ( RLReg ` R ) ) |
12 |
2 8 4
|
isdomn2 |
|- ( R e. Domn <-> ( R e. NzRing /\ ( ( Base ` R ) \ { ( 0g ` R ) } ) C_ ( RLReg ` R ) ) ) |
13 |
1 11 12
|
sylanbrc |
|- ( R e. DivRing -> R e. Domn ) |