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Metamath Proof Explorer

Theorem irrednegb

Description: An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014)

Ref Expression
Hypotheses irredn0.i 
|- I = ( Irred ` R )
irredneg.n 
|- N = ( invg ` R )
irrednegb.b 
|- B = ( Base ` R )
Assertion irrednegb 
|- ( ( R e. Ring /\ X e. B ) -> ( X e. I <-> ( N ` X ) e. I ) )

Proof

Step Hyp Ref Expression
1 irredn0.i 
 |-  I = ( Irred ` R )
2 irredneg.n 
 |-  N = ( invg ` R )
3 irrednegb.b 
 |-  B = ( Base ` R )
4 1 2 irredneg 
 |-  ( ( R e. Ring /\ X e. I ) -> ( N ` X ) e. I )
5 4 adantlr 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ X e. I ) -> ( N ` X ) e. I )
6 ringgrp 
 |-  ( R e. Ring -> R e. Grp )
7 3 2 grpinvinv 
 |-  ( ( R e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
8 6 7 sylan 
 |-  ( ( R e. Ring /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
9 8 adantr 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ ( N ` X ) e. I ) -> ( N ` ( N ` X ) ) = X )
10 1 2 irredneg 
 |-  ( ( R e. Ring /\ ( N ` X ) e. I ) -> ( N ` ( N ` X ) ) e. I )
11 10 adantlr 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ ( N ` X ) e. I ) -> ( N ` ( N ` X ) ) e. I )
12 9 11 eqeltrrd 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ ( N ` X ) e. I ) -> X e. I )
13 5 12 impbida 
 |-  ( ( R e. Ring /\ X e. B ) -> ( X e. I <-> ( N ` X ) e. I ) )