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Theorem lsmcntzr 15239
Description: The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
lsmcntz.p  |-  .(+)  =  (
LSSum `  G )
lsmcntz.s  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
lsmcntz.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmcntz.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
lsmcntz.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
lsmcntzr  |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+) 
U ) )  <->  ( S  C_  ( Z `  T
)  /\  S  C_  ( Z `  U )
) ) )

Proof of Theorem lsmcntzr
StepHypRef Expression
1 lsmcntz.p . . 3  |-  .(+)  =  (
LSSum `  G )
2 lsmcntz.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
3 lsmcntz.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 lsmcntz.s . . 3  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
5 lsmcntz.z . . 3  |-  Z  =  (Cntz `  G )
61, 2, 3, 4, 5lsmcntz 15238 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  ( T  C_  ( Z `  S
)  /\  U  C_  ( Z `  S )
) ) )
7 subgrcl 14876 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
8 grpmnd 14744 . . . . 5  |-  ( G  e.  Grp  ->  G  e.  Mnd )
94, 7, 83syl 19 . . . 4  |-  ( ph  ->  G  e.  Mnd )
10 eqid 2387 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14872 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
122, 11syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
1310subgss 14872 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
143, 13syl 16 . . . 4  |-  ( ph  ->  U  C_  ( Base `  G ) )
1510, 1lsmssv 15204 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  C_  ( Base `  G ) )
169, 12, 14, 15syl3anc 1184 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
C_  ( Base `  G
) )
1710subgss 14872 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
184, 17syl 16 . . 3  |-  ( ph  ->  S  C_  ( Base `  G ) )
1910, 5cntzrec 15059 . . 3  |-  ( ( ( T  .(+)  U ) 
C_  ( Base `  G
)  /\  S  C_  ( Base `  G ) )  ->  ( ( T 
.(+)  U )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( T  .(+)  U ) ) ) )
2016, 18, 19syl2anc 643 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  S  C_  ( Z `  ( T  .(+) 
U ) ) ) )
2110, 5cntzrec 15059 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  S )  <->  S  C_  ( Z `  T )
) )
2212, 18, 21syl2anc 643 . . 3  |-  ( ph  ->  ( T  C_  ( Z `  S )  <->  S 
C_  ( Z `  T ) ) )
2310, 5cntzrec 15059 . . . 4  |-  ( ( U  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( U  C_  ( Z `  S )  <->  S  C_  ( Z `  U )
) )
2414, 18, 23syl2anc 643 . . 3  |-  ( ph  ->  ( U  C_  ( Z `  S )  <->  S 
C_  ( Z `  U ) ) )
2522, 24anbi12d 692 . 2  |-  ( ph  ->  ( ( T  C_  ( Z `  S )  /\  U  C_  ( Z `  S )
)  <->  ( S  C_  ( Z `  T )  /\  S  C_  ( Z `  U )
) ) )
266, 20, 253bitr3d 275 1  |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+) 
U ) )  <->  ( S  C_  ( Z `  T
)  /\  S  C_  ( Z `  U )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   Mndcmnd 14611   Grpcgrp 14612  SubGrpcsubg 14865  Cntzccntz 15041   LSSumclsm 15195
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-subg 14868  df-cntz 15043  df-lsm 15197
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