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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 ) )