Step |
Hyp |
Ref |
Expression |
1 |
|
opprnzr.1 |
|- O = ( oppR ` R ) |
2 |
|
nzrring |
|- ( R e. NzRing -> R e. Ring ) |
3 |
1
|
opprring |
|- ( R e. Ring -> O e. Ring ) |
4 |
2 3
|
syl |
|- ( R e. NzRing -> O e. Ring ) |
5 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
6 |
5
|
isnzr2 |
|- ( R e. NzRing <-> ( R e. Ring /\ 2o ~<_ ( Base ` R ) ) ) |
7 |
6
|
simprbi |
|- ( R e. NzRing -> 2o ~<_ ( Base ` R ) ) |
8 |
1 5
|
opprbas |
|- ( Base ` R ) = ( Base ` O ) |
9 |
8
|
isnzr2 |
|- ( O e. NzRing <-> ( O e. Ring /\ 2o ~<_ ( Base ` R ) ) ) |
10 |
4 7 9
|
sylanbrc |
|- ( R e. NzRing -> O e. NzRing ) |