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Theorem dprdfsub 16875
Description: Take the difference of group sums over two families 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 )
dprdfadd.4  |-  ( ph  ->  H  e.  W )
dprdfsub.b  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
dprdfsub  |-  ( ph  ->  ( ( F  oF  .-  H )  e.  W  /\  ( G 
gsumg  ( F  oF  .-  H ) )  =  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) ) ) )
Distinct variable groups:    h, F    h, H    h, i, G   
h, I, i    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    H( i)    .- ( h, i)    W( h, i)    .0. ( i)

Proof of Theorem dprdfsub
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldprdi.w . . . . . . . 8  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
2 eldprdi.1 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
3 eldprdi.2 . . . . . . . 8  |-  ( ph  ->  dom  S  =  I )
4 eldprdi.3 . . . . . . . 8  |-  ( ph  ->  F  e.  W )
5 eqid 2467 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 16860 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 6022 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfadd.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdff 16860 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 6022 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2467 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2467 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
13 dprdfsub.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 15907 . . . . . 6  |-  ( ( ( F `  k
)  e.  ( Base `  G )  /\  ( H `  k )  e.  ( Base `  G
) )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( invg `  G
) `  ( H `  k ) ) ) )
157, 10, 14syl2anc 661 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( invg `  G
) `  ( H `  k ) ) ) )
1615mpteq2dva 4533 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) ) )
172, 3dprddomcld 16847 . . . . 5  |-  ( ph  ->  I  e.  _V )
186feqmptd 5921 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
199feqmptd 5921 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2017, 7, 10, 18, 19offval2 6541 . . . 4  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
21 fvex 5876 . . . . . 6  |-  ( ( invg `  G
) `  ( H `  k ) )  e. 
_V
2221a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( invg `  G ) `  ( H `  k )
)  e.  _V )
23 dprdgrp 16853 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
242, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
255, 12, 24grpinvf1o 15922 . . . . . . . 8  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
26 f1of 5816 . . . . . . . 8  |-  ( ( invg `  G
) : ( Base `  G ) -1-1-onto-> ( Base `  G
)  ->  ( invg `  G ) : ( Base `  G
) --> ( Base `  G
) )
2725, 26syl 16 . . . . . . 7  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) --> ( Base `  G )
)
2827feqmptd 5921 . . . . . 6  |-  ( ph  ->  ( invg `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( invg `  G
) `  x )
) )
29 fveq2 5866 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( invg `  G ) `  x
)  =  ( ( invg `  G
) `  ( H `  k ) ) )
3010, 19, 28, 29fmptco 6055 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( invg `  G
) `  ( H `  k ) ) ) )
3117, 7, 22, 18, 30offval2 6541 . . . 4  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  =  ( k  e.  I  |->  ( ( F `
 k ) ( +g  `  G ) ( ( invg `  G ) `  ( H `  k )
) ) ) )
3216, 20, 313eqtr4d 2518 . . 3  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )
33 eldprdi.0 . . . . 5  |-  .0.  =  ( 0g `  G )
3433, 1, 2, 3, 8, 12dprdfinv 16873 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  G )  o.  H )  e.  W  /\  ( G 
gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) ) )
3534simpld 459 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  e.  W )
3633, 1, 2, 3, 4, 35, 11dprdfadd 16874 . . . 4  |-  ( ph  ->  ( ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  e.  W  /\  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( invg `  G )  o.  H
) ) ) ) )
3736simpld 459 . . 3  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  e.  W )
3832, 37eqeltrd 2555 . 2  |-  ( ph  ->  ( F  oF 
.-  H )  e.  W )
3932oveq2d 6301 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( G  gsumg  ( F  oF ( +g  `  G ) ( ( invg `  G
)  o.  H ) ) ) )
4034simprd 463 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) )
4140oveq2d 6301 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( invg `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
4236simprd 463 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( invg `  G )  o.  H
) ) ) )
435dprdssv 16870 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4433, 1, 2, 3, 4eldprdi 16872 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4543, 44sseldi 3502 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
4633, 1, 2, 3, 8eldprdi 16872 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
4743, 46sseldi 3502 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
485, 11, 12, 13grpsubval 15907 . . . . 5  |-  ( ( ( G  gsumg  F )  e.  (
Base `  G )  /\  ( G  gsumg  H )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  H ) ) ) )
4945, 47, 48syl2anc 661 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
5041, 42, 493eqtr4d 2518 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5139, 50eqtrd 2508 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
5238, 51jca 532 1  |-  ( ph  ->  ( ( F  oF  .-  H )  e.  W  /\  ( G 
gsumg  ( F  oF  .-  H ) )  =  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999    o. ccom 5003   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285    oFcof 6523   X_cixp 7470   finSupp cfsupp 7830   Basecbs 14493   +g cplusg 14558   0gc0g 14698    gsumg cgsu 14699   Grpcgrp 15730   invgcminusg 15731   -gcsg 15733   DProd cdprd 16839
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-0g 14700  df-gsum 14701  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-mhm 15789  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-ghm 16079  df-gim 16121  df-cntz 16169  df-oppg 16195  df-cmn 16615  df-dprd 16841
This theorem is referenced by:  dprdfeq0  16876  dprdf11  16877  dprdsubg  16885
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