Metamath Proof Explorer


Theorem srgideu

Description: The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses srgcl.b
|- B = ( Base ` R )
srgcl.t
|- .x. = ( .r ` R )
Assertion srgideu
|- ( R e. SRing -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Proof

Step Hyp Ref Expression
1 srgcl.b
 |-  B = ( Base ` R )
2 srgcl.t
 |-  .x. = ( .r ` R )
3 eqid
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 srgmgp
 |-  ( R e. SRing -> ( mulGrp ` R ) e. Mnd )
5 3 1 mgpbas
 |-  B = ( Base ` ( mulGrp ` R ) )
6 3 2 mgpplusg
 |-  .x. = ( +g ` ( mulGrp ` R ) )
7 5 6 mndideu
 |-  ( ( mulGrp ` R ) e. Mnd -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )
8 4 7 syl
 |-  ( R e. SRing -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )