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Theorem lsmdisj2r 17029
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 2404 . . . . 5  |-  (oppg `  G
)  =  (oppg `  G
)
2 lsmcntz.p . . . . 5  |-  .(+)  =  (
LSSum `  G )
31, 2oppglsm 16988 . . . 4  |-  ( U ( LSSum `  (oppg
`  G ) ) S )  =  ( S  .(+)  U )
43ineq2i 3640 . . 3  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( T  i^i  ( S  .(+)  U ) )
5 incom 3634 . . 3  |-  ( T  i^i  ( S  .(+)  U ) )  =  ( ( S  .(+)  U )  i^i  T )
64, 5eqtri 2433 . 2  |-  ( T  i^i  ( U (
LSSum `  (oppg
`  G ) ) S ) )  =  ( ( S  .(+)  U )  i^i  T )
7 eqid 2404 . . 3  |-  ( LSSum `  (oppg
`  G ) )  =  ( LSSum `  (oppg `  G
) )
8 lsmcntz.u . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
91oppgsubg 16724 . . . 4  |-  (SubGrp `  G )  =  (SubGrp `  (oppg
`  G ) )
108, 9syl6eleq 2502 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  (oppg `  G ) ) )
11 lsmcntz.t . . . 4  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
1211, 9syl6eleq 2502 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  (oppg `  G ) ) )
13 lsmcntz.s . . . 4  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
1413, 9syl6eleq 2502 . . 3  |-  ( ph  ->  S  e.  (SubGrp `  (oppg `  G ) ) )
15 lsmdisj.o . . . 4  |-  .0.  =  ( 0g `  G )
161, 15oppgid 16717 . . 3  |-  .0.  =  ( 0g `  (oppg `  G
) )
171, 2oppglsm 16988 . . . . . 6  |-  ( U ( LSSum `  (oppg
`  G ) ) T )  =  ( T  .(+)  U )
1817ineq1i 3639 . . . . 5  |-  ( ( U ( LSSum `  (oppg `  G
) ) T )  i^i  S )  =  ( ( T  .(+)  U )  i^i  S )
19 incom 3634 . . . . 5  |-  ( ( T  .(+)  U )  i^i  S )  =  ( S  i^i  ( T 
.(+)  U ) )
2018, 19eqtri 2433 . . . 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 2457 . . 3  |-  ( ph  ->  ( ( U (
LSSum `  (oppg
`  G ) ) T )  i^i  S
)  =  {  .0.  } )
23 incom 3634 . . . 4  |-  ( T  i^i  U )  =  ( U  i^i  T
)
24 lsmdisj2r.i . . . 4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
2523, 24syl5eqr 2459 . . 3  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
267, 10, 12, 14, 16, 22, 25lsmdisj2 17026 . 2  |-  ( ph  ->  ( T  i^i  ( U ( LSSum `  (oppg `  G
) ) S ) )  =  {  .0.  } )
276, 26syl5eqr 2459 1  |-  ( ph  ->  ( ( S  .(+)  U )  i^i  T )  =  {  .0.  }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844    i^i cin 3415   {csn 3974   ` cfv 5571  (class class class)co 6280   0gc0g 15056  SubGrpcsubg 16521  oppgcoppg 16706   LSSumclsm 16980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-grp 16383  df-minusg 16384  df-subg 16524  df-oppg 16707  df-lsm 16982
This theorem is referenced by:  lsmdisj3r  17030  lsmdisj2b  17032
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