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Theorem dprdfinv 16514
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 16494 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2443 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( invg `  G )
64, 5grpinvf 15587 . . . . 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 16501 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 5884 . . . 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 16496 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
1514ffvelrnda 5848 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
168, 1, 9, 10dprdfcl 16502 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
175subginvcl 15695 . . . . 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 16488 . . . . . 6  |-  ( ph  ->  I  e.  _V )
20 mptexg 5952 . . . . . 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 5459 . . . . . 6  |-  Fun  (
x  e.  I  |->  ( N `  ( F `
 x ) ) )
2322a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( x  e.  I  |->  ( N `  ( F `  x ) ) ) )
248, 1, 9, 10dprdffsupp 16503 . . . . 5  |-  ( ph  ->  F finSupp  .0.  )
25 ssid 3380 . . . . . . . . . 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 5706 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
2927, 28eqeltri 2513 . . . . . . . . . 10  |-  .0.  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
3111, 26, 19, 30suppssr 6725 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( F `  x )  =  .0.  )
3231fveq2d 5700 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
3327, 5grpinvid 15594 . . . . . . . . 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 2475 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( F supp  .0.  ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
3736, 19suppss2 6728 . . . . 5  |-  ( ph  ->  ( ( x  e.  I  |->  ( N `  ( F `  x ) ) ) supp  .0.  )  C_  ( F supp  .0.  )
)
38 fsuppsssupp 7641 . . . . 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 1219 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) ) finSupp  .0.  )
408, 1, 9, 18, 39dprdwd 16500 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
4113, 40eqeltrd 2517 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
42 eqid 2443 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
438, 1, 9, 10, 42dprdfcntz 16504 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
444, 27, 42, 5, 3, 19, 11, 43, 24gsumzinv 16447 . 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 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    \ cdif 3330    C_ wss 3333   class class class wbr 4297    e. cmpt 4355   dom cdm 4845    o. ccom 4849   Fun wfun 5417   -->wf 5419   ` cfv 5423  (class class class)co 6096   supp csupp 6695   X_cixp 7268   finSupp cfsupp 7625   Basecbs 14179   0gc0g 14383    gsumg cgsu 14384   Grpcgrp 15415   invgcminusg 15416  SubGrpcsubg 15680  Cntzccntz 15838   DProd cdprd 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-subg 15683  df-ghm 15750  df-gim 15792  df-cntz 15840  df-oppg 15866  df-cmn 16284  df-dprd 16482
This theorem is referenced by:  dprdfsub  16516
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