Metamath Proof Explorer


Theorem srgridm

Description: The unit element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011) (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 srgridm
|- ( ( R e. SRing /\ X e. B ) -> ( X .x. .1. ) = X )

Proof

Step Hyp Ref Expression
1 srgidm.b
 |-  B = ( Base ` R )
2 srgidm.t
 |-  .x. = ( .r ` R )
3 srgidm.u
 |-  .1. = ( 1r ` R )
4 1 2 3 srgidmlem
 |-  ( ( R e. SRing /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )
5 4 simprd
 |-  ( ( R e. SRing /\ X e. B ) -> ( X .x. .1. ) = X )