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Theorem lsmdisj2a 16205
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 )
Assertion
Ref Expression
lsmdisj2a  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) ) )

Proof of Theorem lsmdisj2a
StepHypRef Expression
1 lsmcntz.p . . . 4  |-  .(+)  =  (
LSSum `  G )
2 lsmcntz.s . . . . 5  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
32adantr 465 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  S  e.  (SubGrp `  G )
)
4 lsmcntz.t . . . . 5  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
54adantr 465 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  T  e.  (SubGrp `  G )
)
6 lsmcntz.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  U  e.  (SubGrp `  G )
)
8 lsmdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
9 simprl 755 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  (
( S  .(+)  T )  i^i  U )  =  {  .0.  } )
10 simprr 756 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  ( S  i^i  T )  =  {  .0.  } )
111, 3, 5, 7, 8, 9, 10lsmdisj2 16200 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
121, 3, 5, 7, 8, 9lsmdisj 16199 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  (
( S  i^i  U
)  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )
1312simpld 459 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  ( S  i^i  U )  =  {  .0.  } )
1411, 13jca 532 . 2  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  (
( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U
)  =  {  .0.  } ) )
15 incom 3564 . . . 4  |-  ( ( S  .(+)  T )  i^i  U )  =  ( U  i^i  ( S 
.(+)  T ) )
162adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  S  e.  (SubGrp `  G ) )
176adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  U  e.  (SubGrp `  G ) )
184adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  T  e.  (SubGrp `  G ) )
19 incom 3564 . . . . . 6  |-  ( ( S  .(+)  U )  i^i  T )  =  ( T  i^i  ( S 
.(+)  U ) )
20 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
2119, 20syl5eq 2487 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  .(+)  U )  i^i 
T )  =  {  .0.  } )
22 simprr 756 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  U )  =  {  .0.  } )
231, 16, 17, 18, 8, 21, 22lsmdisj2 16200 . . . 4  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( U  i^i  ( S  .(+)  T ) )  =  {  .0.  } )
2415, 23syl5eq 2487 . . 3  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  .(+)  T )  i^i 
U )  =  {  .0.  } )
25 incom 3564 . . . 4  |-  ( S  i^i  T )  =  ( T  i^i  S
)
261, 18, 16, 17, 8, 20lsmdisjr 16202 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( T  i^i  S )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  }
) )
2726simpld 459 . . . 4  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( T  i^i  S )  =  {  .0.  } )
2825, 27syl5eq 2487 . . 3  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  T )  =  {  .0.  } )
2924, 28jca 532 . 2  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )
3014, 29impbida 828 1  |-  ( ph  ->  ( ( ( ( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  } )  <-> 
( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348   {csn 3898   ` cfv 5439  (class class class)co 6112   0gc0g 14399  SubGrpcsubg 15696   LSSumclsm 16154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-subg 15699  df-lsm 16156
This theorem is referenced by:  lsmdisj3a  16207
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