Description: Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | cntri.b | |- B = ( Base ` M ) |
|
cntri.p | |- .+ = ( +g ` M ) |
||
cntri.z | |- Z = ( Cntr ` M ) |
||
Assertion | cntri | |- ( ( X e. Z /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntri.b | |- B = ( Base ` M ) |
|
2 | cntri.p | |- .+ = ( +g ` M ) |
|
3 | cntri.z | |- Z = ( Cntr ` M ) |
|
4 | eqid | |- ( Cntz ` M ) = ( Cntz ` M ) |
|
5 | 1 4 | cntrval | |- ( ( Cntz ` M ) ` B ) = ( Cntr ` M ) |
6 | 3 5 | eqtr4i | |- Z = ( ( Cntz ` M ) ` B ) |
7 | 6 | eleq2i | |- ( X e. Z <-> X e. ( ( Cntz ` M ) ` B ) ) |
8 | 2 4 | cntzi | |- ( ( X e. ( ( Cntz ` M ) ` B ) /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) ) |
9 | 7 8 | sylanb | |- ( ( X e. Z /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) ) |