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

Metamath Proof Explorer

Theorem isringid

Description: Properties showing that an element I is the unity element of a ring. (Contributed by NM, 7-Aug-2013)

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

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 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
6 4 3 ringidval 
 |-  .1. = ( 0g ` ( mulGrp ` R ) )
7 4 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
8 1 2 ringideu 
 |-  ( R e. Ring -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
9 reurex 
 |-  ( E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
10 8 9 syl 
 |-  ( R e. Ring -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
11 5 6 7 10 ismgmid 
 |-  ( R e. Ring -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )