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Theorem lsmdisj3b 15098
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 )
lsmdisj3b.2  |-  ( ph  ->  T  C_  ( Z `  U ) )
Assertion
Ref Expression
lsmdisj3b  |-  ( 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 lsmdisj3b
StepHypRef Expression
1 lsmcntz.p . . 3  |-  .(+)  =  (
LSSum `  G )
2 lsmcntz.s . . 3  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
3 lsmcntz.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 lsmcntz.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
5 lsmdisj.o . . 3  |-  .0.  =  ( 0g `  G )
61, 2, 3, 4, 5lsmdisj2b 15096 . 2  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( S  i^i  ( U  .(+)  T ) )  =  {  .0.  }  /\  ( U  i^i  T )  =  {  .0.  } ) ) )
7 lsmdisj3b.2 . . . . . 6  |-  ( ph  ->  T  C_  ( Z `  U ) )
8 lsmdisj3b.z . . . . . . 7  |-  Z  =  (Cntz `  G )
91, 8lsmcom2 15065 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
104, 3, 7, 9syl3anc 1182 . . . . 5  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
1110ineq2d 3446 . . . 4  |-  ( ph  ->  ( S  i^i  ( T  .(+)  U ) )  =  ( S  i^i  ( U  .(+)  T ) ) )
1211eqeq1d 2366 . . 3  |-  ( ph  ->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  <-> 
( S  i^i  ( U  .(+)  T ) )  =  {  .0.  }
) )
13 incom 3437 . . . . 5  |-  ( T  i^i  U )  =  ( U  i^i  T
)
1413a1i 10 . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  ( U  i^i  T ) )
1514eqeq1d 2366 . . 3  |-  ( ph  ->  ( ( T  i^i  U )  =  {  .0.  }  <-> 
( U  i^i  T
)  =  {  .0.  } ) )
1612, 15anbi12d 691 . 2  |-  ( ph  ->  ( ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )  <-> 
( ( S  i^i  ( U  .(+)  T ) )  =  {  .0.  }  /\  ( U  i^i  T )  =  {  .0.  } ) ) )
176, 16bitr4d 247 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 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   {csn 3716   ` cfv 5337  (class class class)co 5945   0gc0g 13499  SubGrpcsubg 14714  Cntzccntz 14890   LSSumclsm 15044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-subg 14717  df-cntz 14892  df-oppg 14918  df-lsm 15046
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