Metamath Proof Explorer


Theorem cntzrecd

Description: Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses cntzrecd.z
|- Z = ( Cntz ` G )
cntzrecd.t
|- ( ph -> T e. ( SubGrp ` G ) )
cntzrecd.u
|- ( ph -> U e. ( SubGrp ` G ) )
cntzrecd.s
|- ( ph -> T C_ ( Z ` U ) )
Assertion cntzrecd
|- ( ph -> U C_ ( Z ` T ) )

Proof

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 ) )