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Theorem lsmdisj2r 15322
Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-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 )
lsmdisjr.i  |-  ( ph  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }
)
lsmdisj2r.i  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
Assertion
Ref Expression
lsmdisj2r  |-  ( ph  ->  ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }
)

Proof of Theorem lsmdisj2r
StepHypRef Expression
1 eqid 2438 . . . . 5  |-  (oppg `  G
)  =  (oppg `  G
)
2 lsmcntz.p . . . . 5  |-  .(+)  =  (
LSSum `  G )
31, 2oppglsm 15281 . . . 4  |-  ( U ( LSSum `  (oppg
`  G ) ) S )  =  ( S  .(+)  U )
43ineq2i 3541 . . 3  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( T  i^i  ( S  .(+)  U ) )
5 incom 3535 . . 3  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
64, 5eqtri 2458 . 2  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( ( S  .(+)  U )  i^i  T )
7 eqid 2438 . . 3  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
8 lsmcntz.u . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
91oppgsubg 15164 . . . 4  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
108, 9syl6eleq 2528 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  (oppg `  G ) ) )
11 lsmcntz.t . . . 4  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
1211, 9syl6eleq 2528 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  (oppg `  G ) ) )
13 lsmcntz.s . . . 4  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
1413, 9syl6eleq 2528 . . 3  |-  ( ph  ->  S  e.  (SubGrp `  (oppg `  G ) ) )
15 lsmdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
161, 15oppgid 15157 . . 3  |-  .0.  =  ( 0g `  (oppg `  G
) )
171, 2oppglsm 15281 . . . . . 6  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
1817ineq1i 3540 . . . . 5  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( ( T  .(+)  U )  i^i  S )
19 incom 3535 . . . . 5  |-  ( ( T  .(+)  U )  i^i  S )  =  ( S  i^i  ( T 
.(+)  U ) )
2018, 19eqtri 2458 . . . 4  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( S  i^i  ( T  .(+)  U ) )
21 lsmdisjr.i . . . 4  |-  ( ph  ->  ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }
)
2220, 21syl5eq 2482 . . 3  |-  ( ph  ->  ( ( U (
LSSum `  (oppg
`  G ) ) T )  i^i  S
)  =  {  .0.  } )
23 incom 3535 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
24 lsmdisj2r.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
2523, 24syl5eqr 2484 . . 3  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
267, 10, 12, 14, 16, 22, 25lsmdisj2 15319 . 2  |-  ( ph  ->  ( T  i^i  ( U ( LSSum `  (oppg `  G
) ) S ) )  =  {  .0.  } )
276, 26syl5eqr 2484 1  |-  ( ph  ->  ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    i^i cin 3321   {csn 3816   ` cfv 5457  (class class class)co 6084   0gc0g 13728  SubGrpcsubg 14943  oppgcoppg 15146   LSSumclsm 15273
This theorem is referenced by:  lsmdisj3r  15323  lsmdisj2b  15325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-subg 14946  df-oppg 15147  df-lsm 15275
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