Metamath Proof Explorer


Theorem opprnzr

Description: The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015)

Ref Expression
Hypothesis opprnzr.1
|- O = ( oppR ` R )
Assertion opprnzr
|- ( R e. NzRing -> O e. NzRing )

Proof

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 )