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Theorem dprdfinvOLD 17284
Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdfinv 17277 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
dprdfinvOLD.b  |-  N  =  ( invg `  G )
Assertion
Ref Expression
dprdfinvOLD  |-  ( 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 dprdfinvOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
2 dprdgrp 17256 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2457 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinvOLD.b . . . . . 6  |-  N  =  ( invg `  G )
64, 5grpinvf 16312 . . . . 5  |-  ( G  e.  Grp  ->  N : ( Base `  G
) --> ( Base `  G
) )
73, 6syl 16 . . . 4  |-  ( ph  ->  N : ( Base `  G ) --> ( Base `  G ) )
8 eldprdiOLD.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
9 eldprdiOLD.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
10 eldprdiOLD.3 . . . . 5  |-  ( ph  ->  F  e.  W )
118, 1, 9, 10, 4dprdffOLD 17270 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 6068 . . . 4  |-  ( ( N : ( Base `  G ) --> ( Base `  G )  /\  F : I --> ( Base `  G ) )  -> 
( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
137, 11, 12syl2anc 661 . . 3  |-  ( ph  ->  ( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
141, 9dprdf2 17258 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 6032 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfclOLD 17271 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 16428 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
) )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
1815, 16, 17syl2anc 661 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
198, 1, 9, 10dprdffiOLD 17272 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
20 ssid 3518 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2211, 21suppssrOLD 6022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
2322fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
24 eldprdiOLD.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
2524, 5grpinvid 16319 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
263, 25syl 16 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
2726adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  .0.  )  =  .0.  )
2823, 27eqtrd 2498 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
2928suppss2OLD 6529 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) ) )
30 ssfi 7759 . . . . 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 661 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
328, 1, 9, 18, 31dprdwdOLD 17269 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
3313, 32eqeltrd 2545 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
34 eqid 2457 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
35 reldmdprd 17246 . . . . . 6  |-  Rel  dom DProd
3635brrelex2i 5050 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
37 dmexg 6730 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
381, 36, 373syl 20 . . . 4  |-  ( ph  ->  dom  S  e.  _V )
399, 38eqeltrrd 2546 . . 3  |-  ( ph  ->  I  e.  _V )
408, 1, 9, 10, 34dprdfcntzOLD 17273 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
414, 24, 34, 5, 3, 39, 11, 40, 19gsumzinvOLD 17188 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4233, 41jca 532 1  |-  ( ph  ->  ( ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468    C_ wss 3471   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   Basecbs 14735   0gc0g 14948    gsumg cgsu 14949   Grpcgrp 16271   invgcminusg 16272  SubGrpcsubg 16413  Cntzccntz 16571   DProd cdprd 17242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-0g 14950  df-gsum 14951  df-mre 15094  df-mrc 15095  df-acs 15097  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-mhm 16184  df-submnd 16185  df-grp 16275  df-minusg 16276  df-subg 16416  df-ghm 16483  df-gim 16525  df-cntz 16573  df-oppg 16599  df-cmn 17018  df-dprd 17244
This theorem is referenced by:  dprdfsubOLD  17286
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