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Theorem dprdfinv 15608
Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
dprdfinv.b  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
dprdfinv  |-  ( ph  ->  ( ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) ) )
Distinct variable groups:    h, F    h, i, G    h, I,
i    h, N    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    N( i)    W( h, i)    .0. ( i)

Proof of Theorem dprdfinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
2 dprdgrp 15594 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2442 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( inv g `  G )
64, 5grpinvf 14880 . . . . 5  |-  ( G  e.  Grp  ->  N : ( Base `  G
) --> ( Base `  G
) )
73, 6syl 16 . . . 4  |-  ( ph  ->  N : ( Base `  G ) --> ( Base `  G ) )
8 eldprdi.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
9 eldprdi.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
10 eldprdi.3 . . . . 5  |-  ( ph  ->  F  e.  W )
118, 1, 9, 10, 4dprdff 15601 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 5933 . . . 4  |-  ( ( N : ( Base `  G ) --> ( Base `  G )  /\  F : I --> ( Base `  G ) )  -> 
( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
137, 11, 12syl2anc 644 . . 3  |-  ( ph  ->  ( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
141, 9dprdf2 15596 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 5899 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfcl 15602 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 14984 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
) )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
1815, 16, 17syl2anc 644 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
198, 1, 9, 10dprdffi 15603 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
20 ssid 3353 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2211, 21suppssr 5893 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
2322fveq2d 5761 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
24 eldprdi.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
2524, 5grpinvid 14887 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
263, 25syl 16 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
2726adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  .0.  )  =  .0.  )
2823, 27eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
2928suppss2 6329 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) ) )
30 ssfi 7358 . . . . 5  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( x  e.  I  |->  ( N `  ( F `  x )
) ) " ( _V  \  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
3119, 29, 30syl2anc 644 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
328, 1, 9, 18, 31dprdwd 15600 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
3313, 32eqeltrd 2516 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
34 eqid 2442 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
35 reldmdprd 15589 . . . . . 6  |-  Rel  dom DProd
3635brrelex2i 4948 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
37 dmexg 5159 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
381, 36, 373syl 19 . . . 4  |-  ( ph  ->  dom  S  e.  _V )
399, 38eqeltrrd 2517 . . 3  |-  ( ph  ->  I  e.  _V )
408, 1, 9, 10, 34dprdfcntz 15604 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
414, 24, 34, 5, 3, 39, 11, 40, 19gsumzinv 15571 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4233, 41jca 520 1  |-  ( ph  ->  ( ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   {crab 2715   _Vcvv 2962    \ cdif 3303    C_ wss 3306   {csn 3838   class class class wbr 4237    e. cmpt 4291   `'ccnv 4906   dom cdm 4907   "cima 4910    o. ccom 4911   -->wf 5479   ` cfv 5483  (class class class)co 6110   X_cixp 7092   Fincfn 7138   Basecbs 13500   0gc0g 13754    gsumg cgsu 13755   Grpcgrp 14716   inv gcminusg 14717  SubGrpcsubg 14969  Cntzccntz 15145   DProd cdprd 15585
This theorem is referenced by:  dprdfsub  15610
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-tpos 6508  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-fzo 11167  df-seq 11355  df-hash 11650  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-0g 13758  df-gsum 13759  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-mhm 14769  df-submnd 14770  df-grp 14843  df-minusg 14844  df-subg 14972  df-ghm 15035  df-gim 15077  df-cntz 15147  df-oppg 15173  df-cmn 15445  df-dprd 15587
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