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Theorem cntzrecd 16485
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 2460 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
54subgss 15990 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
64subgss 15990 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
7 cntzrecd.z . . . . 5  |-  Z  =  (Cntz `  G )
84, 7cntzrec 16159 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  U )  <->  U  C_  ( Z `  T )
) )
95, 6, 8syl2an 477 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  C_  ( Z `  U
)  <->  U  C_  ( Z `
 T ) ) )
102, 3, 9syl2anc 661 . 2  |-  ( ph  ->  ( T  C_  ( Z `  U )  <->  U 
C_  ( Z `  T ) ) )
111, 10mpbid 210 1  |-  ( ph  ->  U  C_  ( Z `  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762    C_ wss 3469   ` cfv 5579   Basecbs 14479  SubGrpcsubg 15983  Cntzccntz 16141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-subg 15986  df-cntz 16143
This theorem is referenced by:  subgdisj2  16499  pj2f  16505  pj1id  16506  dprdcntz2  16869  dmdprdsplit2lem  16877  dmdprdsplit2  16878
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