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Theorem lsmdisj3a 14998
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 )
lsmdisj3b.z  |-  Z  =  (Cntz `  G )
lsmdisj3a.2  |-  ( ph  ->  S  C_  ( Z `  T ) )
Assertion
Ref Expression
lsmdisj3a  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) ) )

Proof of Theorem lsmdisj3a
StepHypRef Expression
1 lsmcntz.s . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
2 lsmcntz.t . . . . . 6  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
3 lsmdisj3a.2 . . . . . 6  |-  ( ph  ->  S  C_  ( Z `  T ) )
4 lsmcntz.p . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
5 lsmdisj3b.z . . . . . . 7  |-  Z  =  (Cntz `  G )
64, 5lsmcom2 14966 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  S  C_  ( Z `
 T ) )  ->  ( S  .(+)  T )  =  ( T 
.(+)  S ) )
71, 2, 3, 6syl3anc 1182 . . . . 5  |-  ( ph  ->  ( S  .(+)  T )  =  ( T  .(+)  S ) )
87ineq1d 3369 . . . 4  |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  ( ( T 
.(+)  S )  i^i  U
) )
98eqeq1d 2291 . . 3  |-  ( ph  ->  ( ( ( S 
.(+)  T )  i^i  U
)  =  {  .0.  }  <-> 
( ( T  .(+)  S )  i^i  U )  =  {  .0.  }
) )
10 incom 3361 . . . . 5  |-  ( S  i^i  T )  =  ( T  i^i  S
)
1110a1i 10 . . . 4  |-  ( ph  ->  ( S  i^i  T
)  =  ( T  i^i  S ) )
1211eqeq1d 2291 . . 3  |-  ( ph  ->  ( ( S  i^i  T )  =  {  .0.  }  <-> 
( T  i^i  S
)  =  {  .0.  } ) )
139, 12anbi12d 691 . 2  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( ( T 
.(+)  S )  i^i  U
)  =  {  .0.  }  /\  ( T  i^i  S )  =  {  .0.  } ) ) )
14 lsmcntz.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
15 lsmdisj.o . . 3  |-  .0.  =  ( 0g `  G )
164, 2, 1, 14, 15lsmdisj2a 14996 . 2  |-  ( ph  ->  ( ( ( ( T  .(+)  S )  i^i  U )  =  {  .0.  }  /\  ( T  i^i  S )  =  {  .0.  } )  <-> 
( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) ) )
1713, 16bitrd 244 1  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-lsm 14947
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