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Theorem dprdfinv 16861
Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
dprdfinv.b  |-  N  =  ( invg `  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 16841 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2467 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( invg `  G )
64, 5grpinvf 15904 . . . . 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 finSupp  .0.  }
9 eldprdi.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
10 eldprdi.3 . . . . 5  |-  ( ph  ->  F  e.  W )
118, 1, 9, 10, 4dprdff 16848 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 6057 . . . 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 16843 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 6021 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfcl 16849 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 16015 . . . . 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
) )
191, 9dprddomcld 16835 . . . . . 6  |-  ( ph  ->  I  e.  _V )
20 mptexg 6130 . . . . . 6  |-  ( I  e.  _V  ->  (
x  e.  I  |->  ( N `  ( F `
 x ) ) )  e.  _V )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  _V )
22 funmpt 5624 . . . . . 6  |-  Fun  (
x  e.  I  |->  ( N `  ( F `
 x ) ) )
2322a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( x  e.  I  |->  ( N `  ( F `  x ) ) ) )
248, 1, 9, 10dprdffsupp 16850 . . . . 5  |-  ( ph  ->  F finSupp  .0.  )
25 ssid 3523 . . . . . . . . . 10  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2625a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
27 eldprdi.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
28 fvex 5876 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
2927, 28eqeltri 2551 . . . . . . . . . 10  |-  .0.  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
3111, 26, 19, 30suppssr 6931 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( F `  x )  =  .0.  )
3231fveq2d 5870 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
3327, 5grpinvid 15911 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
343, 33syl 16 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
3534adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  .0.  )  =  .0.  )
3632, 35eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
3736, 19suppss2 6934 . . . . 5  |-  ( ph  ->  ( ( x  e.  I  |->  ( N `  ( F `  x ) ) ) supp  .0.  )  C_  ( F supp  .0.  )
)
38 fsuppsssupp 7845 . . . . 5  |-  ( ( ( ( x  e.  I  |->  ( N `  ( F `  x ) ) )  e.  _V  /\ 
Fun  ( x  e.  I  |->  ( N `  ( F `  x ) ) ) )  /\  ( F finSupp  .0.  /\  (
( x  e.  I  |->  ( N `  ( F `  x )
) ) supp  .0.  )  C_  ( F supp  .0.  )
) )  ->  (
x  e.  I  |->  ( N `  ( F `
 x ) ) ) finSupp  .0.  )
3921, 23, 24, 37, 38syl22anc 1229 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) ) finSupp  .0.  )
408, 1, 9, 18, 39dprdwd 16847 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
4113, 40eqeltrd 2555 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
42 eqid 2467 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
438, 1, 9, 10, 42dprdfcntz 16851 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
444, 27, 42, 5, 3, 19, 11, 43, 24gsumzinv 16772 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4541, 44jca 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 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    \ cdif 3473    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999    o. ccom 5003   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284   supp csupp 6901   X_cixp 7469   finSupp cfsupp 7829   Basecbs 14490   0gc0g 14695    gsumg cgsu 14696   Grpcgrp 15727   invgcminusg 15728  SubGrpcsubg 16000  Cntzccntz 16158   DProd cdprd 16827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-subg 16003  df-ghm 16070  df-gim 16112  df-cntz 16160  df-oppg 16186  df-cmn 16606  df-dprd 16829
This theorem is referenced by:  dprdfsub  16863
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