Metamath Proof Explorer


Theorem elcntz

Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015)

Ref Expression
Hypotheses cntzfval.b B=BaseM
cntzfval.p +˙=+M
cntzfval.z Z=CntzM
Assertion elcntz SBAZSABySA+˙y=y+˙A

Proof

Step Hyp Ref Expression
1 cntzfval.b B=BaseM
2 cntzfval.p +˙=+M
3 cntzfval.z Z=CntzM
4 1 2 3 cntzval SBZS=xB|ySx+˙y=y+˙x
5 4 eleq2d SBAZSAxB|ySx+˙y=y+˙x
6 oveq1 x=Ax+˙y=A+˙y
7 oveq2 x=Ay+˙x=y+˙A
8 6 7 eqeq12d x=Ax+˙y=y+˙xA+˙y=y+˙A
9 8 ralbidv x=AySx+˙y=y+˙xySA+˙y=y+˙A
10 9 elrab AxB|ySx+˙y=y+˙xABySA+˙y=y+˙A
11 5 10 bitrdi SBAZSABySA+˙y=y+˙A