Metamath Proof Explorer


Theorem mndrid

Description: The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011)

Ref Expression
Hypotheses mndlrid.b
|- B = ( Base ` G )
mndlrid.p
|- .+ = ( +g ` G )
mndlrid.o
|- .0. = ( 0g ` G )
Assertion mndrid
|- ( ( G e. Mnd /\ X e. B ) -> ( X .+ .0. ) = X )

Proof

Step Hyp Ref Expression
1 mndlrid.b
 |-  B = ( Base ` G )
2 mndlrid.p
 |-  .+ = ( +g ` G )
3 mndlrid.o
 |-  .0. = ( 0g ` G )
4 1 2 3 mndlrid
 |-  ( ( G e. Mnd /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )
5 4 simprd
 |-  ( ( G e. Mnd /\ X e. B ) -> ( X .+ .0. ) = X )