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Theorem lsmcntzr 15005
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 15004 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  ( T  C_  ( Z `  S
)  /\  U  C_  ( Z `  S )
) ) )
7 subgrcl 14642 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
8 grpmnd 14510 . . . . 5  |-  ( G  e.  Grp  ->  G  e.  Mnd )
94, 7, 83syl 18 . . . 4  |-  ( ph  ->  G  e.  Mnd )
10 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14638 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
122, 11syl 15 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
1310subgss 14638 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
143, 13syl 15 . . . 4  |-  ( ph  ->  U  C_  ( Base `  G ) )
1510, 1lsmssv 14970 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  C_  ( Base `  G ) )
169, 12, 14, 15syl3anc 1182 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
C_  ( Base `  G
) )
1710subgss 14638 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
184, 17syl 15 . . 3  |-  ( ph  ->  S  C_  ( Base `  G ) )
1910, 5cntzrec 14825 . . 3  |-  ( ( ( T  .(+)  U ) 
C_  ( Base `  G
)  /\  S  C_  ( Base `  G ) )  ->  ( ( T 
.(+)  U )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( T  .(+)  U ) ) ) )
2016, 18, 19syl2anc 642 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  S  C_  ( Z `  ( T  .(+) 
U ) ) ) )
2110, 5cntzrec 14825 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  S )  <->  S  C_  ( Z `  T )
) )
2212, 18, 21syl2anc 642 . . 3  |-  ( ph  ->  ( T  C_  ( Z `  S )  <->  S 
C_  ( Z `  T ) ) )
2310, 5cntzrec 14825 . . . 4  |-  ( ( U  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( U  C_  ( Z `  S )  <->  S  C_  ( Z `  U )
) )
2414, 18, 23syl2anc 642 . . 3  |-  ( ph  ->  ( U  C_  ( Z `  S )  <->  S 
C_  ( Z `  U ) ) )
2522, 24anbi12d 691 . 2  |-  ( ph  ->  ( ( T  C_  ( Z `  S )  /\  U  C_  ( Z `  S )
)  <->  ( S  C_  ( Z `  T )  /\  S  C_  ( Z `  U )
) ) )
266, 20, 253bitr3d 274 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Mndcmnd 14377   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-cntz 14809  df-lsm 14963
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