Metamath Proof Explorer


Theorem mndlrid

Description: A monoid's identity element is a two-sided 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 mndlrid
|- ( ( G e. Mnd /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( 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 mndid
 |-  ( G e. Mnd -> E. y e. B A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) )
5 1 3 2 4 mgmlrid
 |-  ( ( G e. Mnd /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )