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Theorem dprdfinv 17587
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 17572 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 17 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2429 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( invg `  G )
64, 5grpinvf 16661 . . . . 5  |-  ( G  e.  Grp  ->  N : ( Base `  G
) --> ( Base `  G
) )
73, 6syl 17 . . . 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 17580 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 6074 . . . 4  |-  ( ( N : ( Base `  G ) --> ( Base `  G )  /\  F : I --> ( Base `  G ) )  -> 
( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
137, 11, 12syl2anc 665 . . 3  |-  ( ph  ->  ( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
141, 9dprdf2 17574 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfcl 17581 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 16777 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
) )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
1815, 16, 17syl2anc 665 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
191, 9dprddomcld 17568 . . . . . 6  |-  ( ph  ->  I  e.  _V )
20 mptexg 6150 . . . . . 6  |-  ( I  e.  _V  ->  (
x  e.  I  |->  ( N `  ( F `
 x ) ) )  e.  _V )
2119, 20syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  _V )
22 funmpt 5637 . . . . . 6  |-  Fun  (
x  e.  I  |->  ( N `  ( F `
 x ) ) )
2322a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( x  e.  I  |->  ( N `  ( F `  x ) ) ) )
248, 1, 9, 10dprdffsupp 17582 . . . . 5  |-  ( ph  ->  F finSupp  .0.  )
25 ssid 3489 . . . . . . . . . 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 5891 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
2927, 28eqeltri 2513 . . . . . . . . . 10  |-  .0.  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
3111, 26, 19, 30suppssr 6957 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( F `  x )  =  .0.  )
3231fveq2d 5885 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
3327, 5grpinvid 16668 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
343, 33syl 17 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
3534adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  .0.  )  =  .0.  )
3632, 35eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
3736, 19suppss2 6960 . . . . 5  |-  ( ph  ->  ( ( x  e.  I  |->  ( N `  ( F `  x ) ) ) supp  .0.  )  C_  ( F supp  .0.  )
)
38 fsuppsssupp 7905 . . . . 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 1265 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) ) finSupp  .0.  )
408, 1, 9, 18, 39dprdwd 17578 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
4113, 40eqeltrd 2517 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
42 eqid 2429 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
438, 1, 9, 10, 42dprdfcntz 17583 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
444, 27, 42, 5, 3, 19, 11, 43, 24gsumzinv 17513 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4541, 44jca 534 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 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    \ cdif 3439    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    o. ccom 4858   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   supp csupp 6925   X_cixp 7530   finSupp cfsupp 7889   Basecbs 15084   0gc0g 15297    gsumg cgsu 15298   Grpcgrp 16620   invgcminusg 16621  SubGrpcsubg 16762  Cntzccntz 16920   DProd cdprd 17560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-subg 16765  df-ghm 16832  df-gim 16874  df-cntz 16922  df-oppg 16948  df-cmn 17367  df-dprd 17562
This theorem is referenced by:  dprdfsub  17589
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