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Theorem lsmdisj2b 15283
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 3501 . . . 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 452 . . . . 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 452 . . . . 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 452 . . . . 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 3501 . . . . . 6  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
11 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  .(+)  U )  i^i  T )  =  {  .0.  } )
1210, 11syl5eq 2456 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
13 simprr 734 . . . . 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 15280 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( T  .(+)  U )  i^i  S )  =  {  .0.  } )
151, 14syl5eq 2456 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
16 incom 3501 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
172, 6, 8, 4, 9, 11lsmdisj 15276 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  i^i  T
)  =  {  .0.  }  /\  ( U  i^i  T )  =  {  .0.  } ) )
1817simprd 450 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( U  i^i  T )  =  {  .0.  } )
1916, 18syl5eq 2456 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  U )  =  {  .0.  } )
2015, 19jca 519 . 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 452 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  S  e.  (SubGrp `  G ) )
223adantr 452 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  T  e.  (SubGrp `  G ) )
237adantr 452 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  U  e.  (SubGrp `  G ) )
24 simprl 733 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
25 simprr 734 . . . 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 15280 . . 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 15279 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )
2827simprd 450 . . 3  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  U )  =  {  .0.  } )
2926, 28jca 519 . 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 806 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3287   {csn 3782   ` cfv 5421  (class class class)co 6048   0gc0g 13686  SubGrpcsubg 14901   LSSumclsm 15231
This theorem is referenced by:  lsmdisj3b  15285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-subg 14904  df-oppg 15105  df-lsm 15233
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