This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ringidmlem

Description: Lemma for ringlidm and ringridm . (Contributed by NM, 15-Sep-2011) (Revised by Mario Carneiro, 27-Dec-2014)

Ref Expression
Hypotheses ringidm.b 
|- B = ( Base ` R )
ringidm.t 
|- .x. = ( .r ` R )
ringidm.u 
|- .1. = ( 1r ` R )
Assertion ringidmlem 
|- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )

Proof

Step Hyp Ref Expression
1 ringidm.b 
 |-  B = ( Base ` R )
2 ringidm.t 
 |-  .x. = ( .r ` R )
3 ringidm.u 
 |-  .1. = ( 1r ` R )
4 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
5 4 ringmgp 
 |-  ( R e. Ring -> ( mulGrp ` R ) e. Mnd )
6 4 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
7 4 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
8 4 3 ringidval 
 |-  .1. = ( 0g ` ( mulGrp ` R ) )
9 6 7 8 mndlrid 
 |-  ( ( ( mulGrp ` R ) e. Mnd /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )
10 5 9 sylan 
 |-  ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )