Metamath Proof Explorer


Theorem elcntzsn

Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016)

Ref Expression
Hypotheses cntzfval.b
|- B = ( Base ` M )
cntzfval.p
|- .+ = ( +g ` M )
cntzfval.z
|- Z = ( Cntz ` M )
Assertion elcntzsn
|- ( Y e. B -> ( X e. ( Z ` { Y } ) <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) ) )

Proof

Step Hyp Ref Expression
1 cntzfval.b
 |-  B = ( Base ` M )
2 cntzfval.p
 |-  .+ = ( +g ` M )
3 cntzfval.z
 |-  Z = ( Cntz ` M )
4 1 2 3 cntzsnval
 |-  ( Y e. B -> ( Z ` { Y } ) = { x e. B | ( x .+ Y ) = ( Y .+ x ) } )
5 4 eleq2d
 |-  ( Y e. B -> ( X e. ( Z ` { Y } ) <-> X e. { x e. B | ( x .+ Y ) = ( Y .+ x ) } ) )
6 oveq1
 |-  ( x = X -> ( x .+ Y ) = ( X .+ Y ) )
7 oveq2
 |-  ( x = X -> ( Y .+ x ) = ( Y .+ X ) )
8 6 7 eqeq12d
 |-  ( x = X -> ( ( x .+ Y ) = ( Y .+ x ) <-> ( X .+ Y ) = ( Y .+ X ) ) )
9 8 elrab
 |-  ( X e. { x e. B | ( x .+ Y ) = ( Y .+ x ) } <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) )
10 5 9 bitrdi
 |-  ( Y e. B -> ( X e. ( Z ` { Y } ) <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) ) )