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Theorem dprdfinv 15540
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 15526 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2412 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( inv g `  G )
64, 5grpinvf 14812 . . . . 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 15533 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 5871 . . . 4  |-  ( ( N : ( Base `  G ) --> ( Base `  G )  /\  F : I --> ( Base `  G ) )  -> 
( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
137, 11, 12syl2anc 643 . . 3  |-  ( ph  ->  ( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
141, 9dprdf2 15528 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 5837 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfcl 15534 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 14916 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
) )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
1815, 16, 17syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
198, 1, 9, 10dprdffi 15535 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
20 ssid 3335 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2211, 21suppssr 5831 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
2322fveq2d 5699 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
24 eldprdi.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
2524, 5grpinvid 14819 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
263, 25syl 16 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
2726adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  .0.  )  =  .0.  )
2823, 27eqtrd 2444 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
2928suppss2 6267 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) ) )
30 ssfi 7296 . . . . 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 643 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
328, 1, 9, 18, 31dprdwd 15532 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
3313, 32eqeltrd 2486 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
34 eqid 2412 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
35 reldmdprd 15521 . . . . . 6  |-  Rel  dom DProd
3635brrelex2i 4886 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
37 dmexg 5097 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
381, 36, 373syl 19 . . . 4  |-  ( ph  ->  dom  S  e.  _V )
399, 38eqeltrrd 2487 . . 3  |-  ( ph  ->  I  e.  _V )
408, 1, 9, 10, 34dprdfcntz 15536 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
414, 24, 34, 5, 3, 39, 11, 40, 19gsumzinv 15503 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4233, 41jca 519 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 359    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924    \ cdif 3285    C_ wss 3288   {csn 3782   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   dom cdm 4845   "cima 4848    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   X_cixp 7030   Fincfn 7076   Basecbs 13432   0gc0g 13686    gsumg cgsu 13687   Grpcgrp 14648   inv gcminusg 14649  SubGrpcsubg 14901  Cntzccntz 15077   DProd cdprd 15517
This theorem is referenced by:  dprdfsub  15542
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-subg 14904  df-ghm 14967  df-gim 15009  df-cntz 15079  df-oppg 15105  df-cmn 15377  df-dprd 15519
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