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Theorem dprdfsub 17589
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 2429 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 17580 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfadd.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdff 17580 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2429 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2429 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
13 dprdfsub.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 16660 . . . . . 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 665 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( invg `  G
) `  ( H `  k ) ) ) )
1615mpteq2dva 4512 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) ) )
172, 3dprddomcld 17568 . . . . 5  |-  ( ph  ->  I  e.  _V )
186feqmptd 5934 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
199feqmptd 5934 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2017, 7, 10, 18, 19offval2 6562 . . . 4  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
21 fvex 5891 . . . . . 6  |-  ( ( invg `  G
) `  ( H `  k ) )  e. 
_V
2221a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( invg `  G ) `  ( H `  k )
)  e.  _V )
23 dprdgrp 17572 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
242, 23syl 17 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
255, 12, 24grpinvf1o 16675 . . . . . . . 8  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
26 f1of 5831 . . . . . . . 8  |-  ( ( invg `  G
) : ( Base `  G ) -1-1-onto-> ( Base `  G
)  ->  ( invg `  G ) : ( Base `  G
) --> ( Base `  G
) )
2725, 26syl 17 . . . . . . 7  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) --> ( Base `  G )
)
2827feqmptd 5934 . . . . . 6  |-  ( ph  ->  ( invg `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( invg `  G
) `  x )
) )
29 fveq2 5881 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( invg `  G ) `  x
)  =  ( ( invg `  G
) `  ( H `  k ) ) )
3010, 19, 28, 29fmptco 6071 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( invg `  G
) `  ( H `  k ) ) ) )
3117, 7, 22, 18, 30offval2 6562 . . . 4  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  =  ( k  e.  I  |->  ( ( F `
 k ) ( +g  `  G ) ( ( invg `  G ) `  ( H `  k )
) ) ) )
3216, 20, 313eqtr4d 2480 . . 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 17587 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  G )  o.  H )  e.  W  /\  ( G 
gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) ) )
3534simpld 460 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  e.  W )
3633, 1, 2, 3, 4, 35, 11dprdfadd 17588 . . . 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 460 . . 3  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  e.  W )
3832, 37eqeltrd 2517 . 2  |-  ( ph  ->  ( F  oF 
.-  H )  e.  W )
3932oveq2d 6321 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( G  gsumg  ( F  oF ( +g  `  G ) ( ( invg `  G
)  o.  H ) ) ) )
4034simprd 464 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) )
4140oveq2d 6321 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( invg `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
4236simprd 464 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( invg `  G )  o.  H
) ) ) )
435dprdssv 17584 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4433, 1, 2, 3, 4eldprdi 17586 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4543, 44sseldi 3468 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
4633, 1, 2, 3, 8eldprdi 17586 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
4743, 46sseldi 3468 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
485, 11, 12, 13grpsubval 16660 . . . . 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 665 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
5041, 42, 493eqtr4d 2480 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5139, 50eqtrd 2470 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
5238, 51jca 534 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 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    o. ccom 4858   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    oFcof 6543   X_cixp 7530   finSupp cfsupp 7889   Basecbs 15084   +g cplusg 15152   0gc0g 15297    gsumg cgsu 15298   Grpcgrp 16620   invgcminusg 16621   -gcsg 16622   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-of 6545  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-sbg 16626  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:  dprdfeq0  17590  dprdf11  17591  dprdsubg  17592
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