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Theorem lsmdisj2b 15207
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
lsmdisj2b  |-  ( ph  ->  ( ( ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } )  <-> 
( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) ) )

Proof of Theorem lsmdisj2b
StepHypRef Expression
1 incom 3449 . . . 4  |-  ( S  i^i  ( T  .(+)  U ) )  =  ( ( T  .(+)  U )  i^i  S )
2 lsmcntz.p . . . . 5  |-  .(+)  =  (
LSSum `  G )
3 lsmcntz.t . . . . . 6  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
43adantr 451 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  T  e.  (SubGrp `  G )
)
5 lsmcntz.s . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
65adantr 451 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  S  e.  (SubGrp `  G )
)
7 lsmcntz.u . . . . . 6  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
87adantr 451 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  U  e.  (SubGrp `  G )
)
9 lsmdisj.o . . . . 5  |-  .0.  =  ( 0g `  G )
10 incom 3449 . . . . . 6  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
11 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  .(+)  U )  i^i  T )  =  {  .0.  } )
1210, 11syl5eq 2410 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
13 simprr 733 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( S  i^i  U )  =  {  .0.  } )
142, 4, 6, 8, 9, 12, 13lsmdisj2r 15204 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( T  .(+)  U )  i^i  S )  =  {  .0.  } )
151, 14syl5eq 2410 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
16 incom 3449 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
172, 6, 8, 4, 9, 11lsmdisj 15200 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  i^i  T
)  =  {  .0.  }  /\  ( U  i^i  T )  =  {  .0.  } ) )
1817simprd 449 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( U  i^i  T )  =  {  .0.  } )
1916, 18syl5eq 2410 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  U )  =  {  .0.  } )
2015, 19jca 518 . 2  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U
)  =  {  .0.  } ) )
215adantr 451 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  S  e.  (SubGrp `  G ) )
223adantr 451 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  T  e.  (SubGrp `  G ) )
237adantr 451 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  U  e.  (SubGrp `  G ) )
24 simprl 732 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
25 simprr 733 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( T  i^i  U )  =  {  .0.  } )
262, 21, 22, 23, 9, 24, 25lsmdisj2r 15204 . . 3  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  .(+)  U )  i^i 
T )  =  {  .0.  } )
272, 21, 22, 23, 9, 24lsmdisjr 15203 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )
2827simprd 449 . . 3  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  U )  =  {  .0.  } )
2926, 28jca 518 . 2  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )
3020, 29impbida 805 1  |-  ( ph  ->  ( ( ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .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 1647    e. wcel 1715    i^i cin 3237   {csn 3729   ` cfv 5358  (class class class)co 5981   0gc0g 13610  SubGrpcsubg 14825   LSSumclsm 15155
This theorem is referenced by:  lsmdisj3b  15209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-tpos 6376  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-0g 13614  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-subg 14828  df-oppg 15029  df-lsm 15157
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