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 ) ) |
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 ) ) |