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Theorem lsmdisj3a 15321
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 15289 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  S  C_  ( Z `
 T ) )  ->  ( S  .(+)  T )  =  ( T 
.(+)  S ) )
71, 2, 3, 6syl3anc 1184 . . . . 5  |-  ( ph  ->  ( S  .(+)  T )  =  ( T  .(+)  S ) )
87ineq1d 3541 . . . 4  |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  ( ( T 
.(+)  S )  i^i  U
) )
98eqeq1d 2444 . . 3  |-  ( ph  ->  ( ( ( S 
.(+)  T )  i^i  U
)  =  {  .0.  }  <-> 
( ( T  .(+)  S )  i^i  U )  =  {  .0.  }
) )
10 incom 3533 . . . . 5  |-  ( S  i^i  T )  =  ( T  i^i  S
)
1110a1i 11 . . . 4  |-  ( ph  ->  ( S  i^i  T
)  =  ( T  i^i  S ) )
1211eqeq1d 2444 . . 3  |-  ( ph  ->  ( ( S  i^i  T )  =  {  .0.  }  <-> 
( T  i^i  S
)  =  {  .0.  } ) )
139, 12anbi12d 692 . 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 15319 . 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 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   {csn 3814   ` cfv 5454  (class class class)co 6081   0gc0g 13723  SubGrpcsubg 14938  Cntzccntz 15114   LSSumclsm 15268
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-subg 14941  df-cntz 15116  df-lsm 15270
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