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Theorem lsmdisj2b 16183
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 3541 . . . 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 465 . . . . 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 465 . . . . 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 465 . . . . 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 3541 . . . . . 6  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
11 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  .(+)  U )  i^i  T )  =  {  .0.  } )
1210, 11syl5eq 2485 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
13 simprr 756 . . . . 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 16180 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( T  .(+)  U )  i^i  S )  =  {  .0.  } )
151, 14syl5eq 2485 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
16 incom 3541 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
172, 6, 8, 4, 9, 11lsmdisj 16176 . . . . 5  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  (
( S  i^i  T
)  =  {  .0.  }  /\  ( U  i^i  T )  =  {  .0.  } ) )
1817simprd 463 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( U  i^i  T )  =  {  .0.  } )
1916, 18syl5eq 2485 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  U )  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )  ->  ( T  i^i  U )  =  {  .0.  } )
2015, 19jca 532 . 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 465 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  S  e.  (SubGrp `  G ) )
223adantr 465 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  T  e.  (SubGrp `  G ) )
237adantr 465 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  U  e.  (SubGrp `  G ) )
24 simprl 755 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  } )
25 simprr 756 . . . 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 16180 . . 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 16179 . . . 4  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
) )
2827simprd 463 . . 3  |-  ( (
ph  /\  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  U )  =  {  .0.  } )
2926, 28jca 532 . 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 828 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3325   {csn 3875   ` cfv 5416  (class class class)co 6089   0gc0g 14376  SubGrpcsubg 15673   LSSumclsm 16131
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-mnd 15413  df-submnd 15463  df-grp 15543  df-minusg 15544  df-subg 15676  df-oppg 15859  df-lsm 16133
This theorem is referenced by:  lsmdisj3b  16185
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