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Theorem cntzrecd 14987
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 2283 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14622 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
64subgss 14622 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
7 cntzrecd.z . . . . 5  |-  Z  =  (Cntz `  G )
84, 7cntzrec 14809 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  U )  <->  U  C_  ( Z `  T )
) )
95, 6, 8syl2an 463 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  C_  ( Z `  U
)  <->  U  C_  ( Z `
 T ) ) )
102, 3, 9syl2anc 642 . 2  |-  ( ph  ->  ( T  C_  ( Z `  U )  <->  U 
C_  ( Z `  T ) ) )
111, 10mpbid 201 1  |-  ( ph  ->  U  C_  ( Z `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   Basecbs 13148  SubGrpcsubg 14615  Cntzccntz 14791
This theorem is referenced by:  subgdisj2  15001  pj2f  15007  pj1id  15008  dprdcntz2  15273  dmdprdsplit2lem  15280  dmdprdsplit2  15281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-subg 14618  df-cntz 14793
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