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Theorem lsmdisj3 15008
Description: Association of the disjointness constraint in 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 ) )
lsmdisj.o  |-  .0.  =  ( 0g `  G )
lsmdisj.i  |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }
)
lsmdisj2.i  |-  ( ph  ->  ( S  i^i  T
)  =  {  .0.  } )
lsmdisj3.z  |-  Z  =  (Cntz `  G )
lsmdisj3.s  |-  ( ph  ->  S  C_  ( Z `  T ) )
Assertion
Ref Expression
lsmdisj3  |-  ( ph  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }
)

Proof of Theorem lsmdisj3
StepHypRef Expression
1 lsmcntz.p . 2  |-  .(+)  =  (
LSSum `  G )
2 lsmcntz.t . 2  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
3 lsmcntz.s . 2  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
4 lsmcntz.u . 2  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5 lsmdisj.o . 2  |-  .0.  =  ( 0g `  G )
6 lsmdisj3.s . . . . 5  |-  ( ph  ->  S  C_  ( Z `  T ) )
7 lsmdisj3.z . . . . . 6  |-  Z  =  (Cntz `  G )
81, 7lsmcom2 14982 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  S  C_  ( Z `
 T ) )  ->  ( S  .(+)  T )  =  ( T 
.(+)  S ) )
93, 2, 6, 8syl3anc 1182 . . . 4  |-  ( ph  ->  ( S  .(+)  T )  =  ( T  .(+)  S ) )
109ineq1d 3382 . . 3  |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  ( ( T 
.(+)  S )  i^i  U
) )
11 lsmdisj.i . . 3  |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }
)
1210, 11eqtr3d 2330 . 2  |-  ( ph  ->  ( ( T  .(+)  S )  i^i  U )  =  {  .0.  }
)
13 incom 3374 . . 3  |-  ( T  i^i  S )  =  ( S  i^i  T
)
14 lsmdisj2.i . . 3  |-  ( ph  ->  ( S  i^i  T
)  =  {  .0.  } )
1513, 14syl5eq 2340 . 2  |-  ( ph  ->  ( T  i^i  S
)  =  {  .0.  } )
161, 2, 3, 4, 5, 12, 15lsmdisj2 15007 1  |-  ( ph  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961
This theorem is referenced by:  dmdprdsplit2lem  15296
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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