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Metamath Proof Explorer

Theorem opprirred

Description: Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014)

Ref Expression
Hypotheses opprirred.1 
|- S = ( oppR ` R )
opprirred.2 
|- I = ( Irred ` R )
Assertion opprirred 
|- I = ( Irred ` S )

Proof

Step Hyp Ref Expression
1 opprirred.1 
 |-  S = ( oppR ` R )
2 opprirred.2 
 |-  I = ( Irred ` R )
3 ralcom 
 |-  ( A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x <-> A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x )
4 eqid 
 |-  ( Base ` R ) = ( Base ` R )
5 eqid 
 |-  ( .r ` R ) = ( .r ` R )
6 eqid 
 |-  ( .r ` S ) = ( .r ` S )
7 4 5 1 6 opprmul 
 |-  ( y ( .r ` S ) z ) = ( z ( .r ` R ) y )
8 7 neeq1i 
 |-  ( ( y ( .r ` S ) z ) =/= x <-> ( z ( .r ` R ) y ) =/= x )
9 8 2ralbii 
 |-  ( A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( y ( .r ` S ) z ) =/= x <-> A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x )
10 3 9 bitr4i 
 |-  ( A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x <-> A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( y ( .r ` S ) z ) =/= x )
11 10 anbi2i 
 |-  ( ( x e. ( ( Base ` R ) \ ( Unit ` R ) ) /\ A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x ) <-> ( x e. ( ( Base ` R ) \ ( Unit ` R ) ) /\ A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( y ( .r ` S ) z ) =/= x ) )
12 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
13 eqid 
 |-  ( ( Base ` R ) \ ( Unit ` R ) ) = ( ( Base ` R ) \ ( Unit ` R ) )
14 4 12 2 13 5 isirred 
 |-  ( x e. I <-> ( x e. ( ( Base ` R ) \ ( Unit ` R ) ) /\ A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) ( z ( .r ` R ) y ) =/= x ) )
15 1 4 opprbas 
 |-  ( Base ` R ) = ( Base ` S )
16 12 1 opprunit 
 |-  ( Unit ` R ) = ( Unit ` S )
17 eqid 
 |-  ( Irred ` S ) = ( Irred ` S )
18 15 16 17 13 6 isirred 
 |-  ( x e. ( Irred ` S ) <-> ( x e. ( ( Base ` R ) \ ( Unit ` R ) ) /\ A. y e. ( ( Base ` R ) \ ( Unit ` R ) ) A. z e. ( ( Base ` R ) \ ( Unit ` R ) ) ( y ( .r ` S ) z ) =/= x ) )
19 11 14 18 3bitr4i 
 |-  ( x e. I <-> x e. ( Irred ` S ) )
20 19 eqriv 
 |-  I = ( Irred ` S )