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Metamath Proof Explorer

Theorem issrgid

Description: Properties showing that an element I is the unity element of a semiring. (Contributed by NM, 7-Aug-2013) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step Hyp Ref Expression
1 srgidm.b 
 |-  B = ( Base ` R )
2 srgidm.t 
 |-  .x. = ( .r ` R )
3 srgidm.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 srgideu 
 |-  ( R e. SRing -> 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. SRing -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
11 5 6 7 10 ismgmid 
 |-  ( R e. SRing -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )