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Theorem lsmdisj2a 16822
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 463 . . . 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 463 . . . 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 463 . . . 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 754 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  (
( S  .(+)  T )  i^i  U )  =  {  .0.  } )
10 simprr 755 . . . 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 16817 . . 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 16816 . . . 4  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  (
( S  i^i  U
)  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } ) )
1312simpld 457 . . 3  |-  ( (
ph  /\  ( (
( S  .(+)  T )  i^i  U )  =  {  .0.  }  /\  ( S  i^i  T )  =  {  .0.  }
) )  ->  ( S  i^i  U )  =  {  .0.  } )
1411, 13jca 530 . 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 3605 . . . 4  |-  ( ( S  .(+)  T )  i^i  U )  =  ( U  i^i  ( S 
.(+)  T ) )
162adantr 463 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  S  e.  (SubGrp `  G ) )
176adantr 463 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  U  e.  (SubGrp `  G ) )
184adantr 463 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  T  e.  (SubGrp `  G ) )
19 incom 3605 . . . . . 6  |-  ( ( S  .(+)  U )  i^i  T )  =  ( T  i^i  ( S 
.(+)  U ) )
20 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
2119, 20syl5eq 2435 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  .(+)  U )  i^i 
T )  =  {  .0.  } )
22 simprr 755 . . . . 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 16817 . . . 4  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( U  i^i  ( S  .(+)  T ) )  =  {  .0.  } )
2415, 23syl5eq 2435 . . 3  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( S  .(+)  T )  i^i 
U )  =  {  .0.  } )
25 incom 3605 . . . 4  |-  ( S  i^i  T )  =  ( T  i^i  S
)
261, 18, 16, 17, 8, 20lsmdisjr 16819 . . . . 5  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( ( T  i^i  S )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  }
) )
2726simpld 457 . . . 4  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( T  i^i  S )  =  {  .0.  } )
2825, 27syl5eq 2435 . . 3  |-  ( (
ph  /\  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } ) )  ->  ( S  i^i  T )  =  {  .0.  } )
2924, 28jca 530 . 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 830 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 367    = wceq 1399    e. wcel 1826    i^i cin 3388   {csn 3944   ` cfv 5496  (class class class)co 6196   0gc0g 14847  SubGrpcsubg 16312   LSSumclsm 16771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-grp 16174  df-minusg 16175  df-subg 16315  df-lsm 16773
This theorem is referenced by:  lsmdisj3a  16824
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