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Theorem cntzrecd 17018
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 2402 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
54subgss 16524 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
64subgss 16524 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
7 cntzrecd.z . . . . 5  |-  Z  =  (Cntz `  G )
84, 7cntzrec 16693 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  U )  <->  U  C_  ( Z `  T )
) )
95, 6, 8syl2an 475 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  C_  ( Z `  U
)  <->  U  C_  ( Z `
 T ) ) )
102, 3, 9syl2anc 659 . 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 1405    e. wcel 1842    C_ wss 3413   ` cfv 5568   Basecbs 14839  SubGrpcsubg 16517  Cntzccntz 16675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-subg 16520  df-cntz 16677
This theorem is referenced by:  subgdisj2  17032  pj2f  17038  pj1id  17039  dprdcntz2  17404  dmdprdsplit2lem  17412  dmdprdsplit2  17413
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