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Theorem ablcntzd 15460
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 2435 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
32subgss 14933 . . 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 15406 . . . 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 14933 . . . 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 15447 . . 3  |-  ( ( G  e. CMnd  /\  U  C_  ( Base `  G
) )  ->  ( Z `  U )  =  ( Base `  G
) )
137, 10, 12syl2anc 643 . 2  |-  ( ph  ->  ( Z `  U
)  =  ( Base `  G ) )
144, 13sseqtr4d 3377 1  |-  ( ph  ->  T  C_  ( Z `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5445   Basecbs 13457  SubGrpcsubg 14926  Cntzccntz 15102  CMndccmn 15400   Abelcabel 15401
This theorem is referenced by:  lsmsubg2  15462  ablfacrp2  15613  ablfac1b  15616  pgpfaclem1  15627  pgpfaclem2  15628  pj1lmhm  16160  pj1lmhm2  16161  lvecindp  16198  lvecindp2  16199  pjdm2  16926  pjf2  16929  pjfo  16930  lshpsmreu  29746  lshpkrlem5  29751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-subg 14929  df-cntz 15104  df-cmn 15402  df-abl 15403
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