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Theorem lsmdisj2r 15276
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 2408 . . . . 5  |-  (oppg `  G
)  =  (oppg `  G
)
2 lsmcntz.p . . . . 5  |-  .(+)  =  (
LSSum `  G )
31, 2oppglsm 15235 . . . 4  |-  ( U ( LSSum `  (oppg
`  G ) ) S )  =  ( S  .(+)  U )
43ineq2i 3503 . . 3  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( T  i^i  ( S  .(+)  U ) )
5 incom 3497 . . 3  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
64, 5eqtri 2428 . 2  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( ( S  .(+)  U )  i^i  T )
7 eqid 2408 . . 3  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
8 lsmcntz.u . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
91oppgsubg 15118 . . . 4  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
108, 9syl6eleq 2498 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  (oppg `  G ) ) )
11 lsmcntz.t . . . 4  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
1211, 9syl6eleq 2498 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  (oppg `  G ) ) )
13 lsmcntz.s . . . 4  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
1413, 9syl6eleq 2498 . . 3  |-  ( ph  ->  S  e.  (SubGrp `  (oppg `  G ) ) )
15 lsmdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
161, 15oppgid 15111 . . 3  |-  .0.  =  ( 0g `  (oppg `  G
) )
171, 2oppglsm 15235 . . . . . 6  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
1817ineq1i 3502 . . . . 5  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( ( T  .(+)  U )  i^i  S )
19 incom 3497 . . . . 5  |-  ( ( T  .(+)  U )  i^i  S )  =  ( S  i^i  ( T 
.(+)  U ) )
2018, 19eqtri 2428 . . . 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 2452 . . 3  |-  ( ph  ->  ( ( U (
LSSum `  (oppg
`  G ) ) T )  i^i  S
)  =  {  .0.  } )
23 incom 3497 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
24 lsmdisj2r.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
2523, 24syl5eqr 2454 . . 3  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
267, 10, 12, 14, 16, 22, 25lsmdisj2 15273 . 2  |-  ( ph  ->  ( T  i^i  ( U ( LSSum `  (oppg `  G
) ) S ) )  =  {  .0.  } )
276, 26syl5eqr 2454 1  |-  ( ph  ->  ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    i^i cin 3283   {csn 3778   ` cfv 5417  (class class class)co 6044   0gc0g 13682  SubGrpcsubg 14897  oppgcoppg 15100   LSSumclsm 15227
This theorem is referenced by:  lsmdisj3r  15277  lsmdisj2b  15279
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-0g 13686  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-subg 14900  df-oppg 15101  df-lsm 15229
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