Metamath Proof Explorer

Theorem opprdrng

Description: The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014)

Ref Expression
Hypothesis opprdrng.1 
|- O = ( oppR ` R )
Assertion opprdrng 
|- ( R e. DivRing <-> O e. DivRing )

Proof

Step Hyp Ref Expression
1 opprdrng.1 
 |-  O = ( oppR ` R )
2 1 opprringb 
 |-  ( R e. Ring <-> O e. Ring )
3 2 anbi1i 
 |-  ( ( R e. Ring /\ ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) <-> ( O e. Ring /\ ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) )
4 eqid 
 |-  ( Base ` R ) = ( Base ` R )
5 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
6 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
7 4 5 6 isdrng 
 |-  ( R e. DivRing <-> ( R e. Ring /\ ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) )
8 1 4 opprbas 
 |-  ( Base ` R ) = ( Base ` O )
9 5 1 opprunit 
 |-  ( Unit ` R ) = ( Unit ` O )
10 1 6 oppr0 
 |-  ( 0g ` R ) = ( 0g ` O )
11 8 9 10 isdrng 
 |-  ( O e. DivRing <-> ( O e. Ring /\ ( Unit ` R ) = ( ( Base ` R ) \ { ( 0g ` R ) } ) ) )
12 3 7 11 3bitr4i 
 |-  ( R e. DivRing <-> O e. DivRing )