Step |
Hyp |
Ref |
Expression |
1 |
|
cntzrecd.z |
|- Z = ( Cntz ` G ) |
2 |
|
cntzrecd.t |
|- ( ph -> T e. ( SubGrp ` G ) ) |
3 |
|
cntzrecd.u |
|- ( ph -> U e. ( SubGrp ` G ) ) |
4 |
|
cntzrecd.s |
|- ( ph -> T C_ ( Z ` U ) ) |
5 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
6 |
5
|
subgss |
|- ( T e. ( SubGrp ` G ) -> T C_ ( Base ` G ) ) |
7 |
5
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
8 |
5 1
|
cntzrec |
|- ( ( T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( T C_ ( Z ` U ) <-> U C_ ( Z ` T ) ) ) |
9 |
6 7 8
|
syl2an |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( T C_ ( Z ` U ) <-> U C_ ( Z ` T ) ) ) |
10 |
2 3 9
|
syl2anc |
|- ( ph -> ( T C_ ( Z ` U ) <-> U C_ ( Z ` T ) ) ) |
11 |
4 10
|
mpbid |
|- ( ph -> U C_ ( Z ` T ) ) |