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Theorem cntzrecd 15312
Description: Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
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
Ref Expression
cntzrecd  |-  ( ph  ->  U  C_  ( Z `  T ) )

Proof of Theorem cntzrecd
StepHypRef Expression
1 cntzrecd.s . 2  |-  ( ph  ->  T  C_  ( Z `  U ) )
2 cntzrecd.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
3 cntzrecd.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 eqid 2438 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14947 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
64subgss 14947 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
7 cntzrecd.z . . . . 5  |-  Z  =  (Cntz `  G )
84, 7cntzrec 15134 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  U )  <->  U  C_  ( Z `  T )
) )
95, 6, 8syl2an 465 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  C_  ( Z `  U
)  <->  U  C_  ( Z `
 T ) ) )
102, 3, 9syl2anc 644 . 2  |-  ( ph  ->  ( T  C_  ( Z `  U )  <->  U 
C_  ( Z `  T ) ) )
111, 10mpbid 203 1  |-  ( ph  ->  U  C_  ( Z `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5456   Basecbs 13471  SubGrpcsubg 14940  Cntzccntz 15116
This theorem is referenced by:  subgdisj2  15326  pj2f  15332  pj1id  15333  dprdcntz2  15598  dmdprdsplit2lem  15605  dmdprdsplit2  15606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-subg 14943  df-cntz 15118
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