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Theorem cntzrecd 15003
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 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14638 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
64subgss 14638 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
7 cntzrecd.z . . . . 5  |-  Z  =  (Cntz `  G )
84, 7cntzrec 14825 . . . 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 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Basecbs 13164  SubGrpcsubg 14631  Cntzccntz 14807
This theorem is referenced by:  subgdisj2  15017  pj2f  15023  pj1id  15024  dprdcntz2  15289  dmdprdsplit2lem  15296  dmdprdsplit2  15297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-subg 14634  df-cntz 14809
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