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Theorem ablcntzd 15392
Description: All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
ablcntzd.z  |-  Z  =  (Cntz `  G )
ablcntzd.a  |-  ( ph  ->  G  e.  Abel )
ablcntzd.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
ablcntzd.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
Assertion
Ref Expression
ablcntzd  |-  ( ph  ->  T  C_  ( Z `  U ) )

Proof of Theorem ablcntzd
StepHypRef Expression
1 ablcntzd.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 eqid 2380 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
32subgss 14865 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
41, 3syl 16 . 2  |-  ( ph  ->  T  C_  ( Base `  G ) )
5 ablcntzd.a . . . 4  |-  ( ph  ->  G  e.  Abel )
6 ablcmn 15338 . . . 4  |-  ( G  e.  Abel  ->  G  e. CMnd
)
75, 6syl 16 . . 3  |-  ( ph  ->  G  e. CMnd )
8 ablcntzd.u . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
92subgss 14865 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
108, 9syl 16 . . 3  |-  ( ph  ->  U  C_  ( Base `  G ) )
11 ablcntzd.z . . . 4  |-  Z  =  (Cntz `  G )
122, 11cntzcmn 15379 . . 3  |-  ( ( G  e. CMnd  /\  U  C_  ( Base `  G
) )  ->  ( Z `  U )  =  ( Base `  G
) )
137, 10, 12syl2anc 643 . 2  |-  ( ph  ->  ( Z `  U
)  =  ( Base `  G ) )
144, 13sseqtr4d 3321 1  |-  ( ph  ->  T  C_  ( Z `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3256   ` cfv 5387   Basecbs 13389  SubGrpcsubg 14858  Cntzccntz 15034  CMndccmn 15332   Abelcabel 15333
This theorem is referenced by:  lsmsubg2  15394  ablfacrp2  15545  ablfac1b  15548  pgpfaclem1  15559  pgpfaclem2  15560  pj1lmhm  16092  pj1lmhm2  16093  lvecindp  16130  lvecindp2  16131  pjdm2  16854  pjf2  16857  pjfo  16858  lshpsmreu  29275  lshpkrlem5  29280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-subg 14861  df-cntz 15036  df-cmn 15334  df-abl 15335
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