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Theorem dprdfsub 15571
Description: Take the difference of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
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  o F  .-  H )  e.  W  /\  ( G 
gsumg  ( F  o F  .-  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 " ( _V  \  {  .0.  } ) )  e. 
Fin }
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 2435 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15562 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfadd.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdff 15562 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2435 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2435 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
13 dprdfsub.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 14840 . . . . . 6  |-  ( ( ( F `  k
)  e.  ( Base `  G )  /\  ( H `  k )  e.  ( Base `  G
) )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( inv g `  G
) `  ( H `  k ) ) ) )
157, 10, 14syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( F `  k
)  .-  ( H `  k ) )  =  ( ( F `  k ) ( +g  `  G ) ( ( inv g `  G
) `  ( H `  k ) ) ) )
1615mpteq2dva 4287 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
17 reldmdprd 15550 . . . . . . . 8  |-  Rel  dom DProd
1817brrelex2i 4911 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
19 dmexg 5122 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
202, 18, 193syl 19 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
213, 20eqeltrrd 2510 . . . . 5  |-  ( ph  ->  I  e.  _V )
226feqmptd 5771 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
239feqmptd 5771 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2421, 7, 10, 22, 23offval2 6314 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
25 fvex 5734 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
2625a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
27 dprdgrp 15555 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
282, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
295, 12, 28grpinvf1o 14853 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
30 f1of 5666 . . . . . . . 8  |-  ( ( inv g `  G
) : ( Base `  G ) -1-1-onto-> ( Base `  G
)  ->  ( inv g `  G ) : ( Base `  G
) --> ( Base `  G
) )
3129, 30syl 16 . . . . . . 7  |-  ( ph  ->  ( inv g `  G ) : (
Base `  G ) --> ( Base `  G )
)
3231feqmptd 5771 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( inv g `  G
) `  x )
) )
33 fveq2 5720 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
3410, 23, 32, 33fmptco 5893 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
3521, 7, 26, 22, 34offval2 6314 . . . 4  |-  ( ph  ->  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) )  =  ( k  e.  I  |->  ( ( F `
 k ) ( +g  `  G ) ( ( inv g `  G ) `  ( H `  k )
) ) ) )
3616, 24, 353eqtr4d 2477 . . 3  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )
37 eldprdi.0 . . . . 5  |-  .0.  =  ( 0g `  G )
3837, 1, 2, 3, 8, 12dprdfinv 15569 . . . . . 6  |-  ( ph  ->  ( ( ( inv g `  G )  o.  H )  e.  W  /\  ( G 
gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) ) )
3938simpld 446 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  e.  W )
4037, 1, 2, 3, 4, 39, 11dprdfadd 15570 . . . 4  |-  ( ph  ->  ( ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) )  e.  W  /\  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( inv g `  G )  o.  H
) ) ) ) )
4140simpld 446 . . 3  |-  ( ph  ->  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) )  e.  W )
4236, 41eqeltrd 2509 . 2  |-  ( ph  ->  ( F  o F 
.-  H )  e.  W )
4336oveq2d 6089 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( G  gsumg  ( F  o F ( +g  `  G ) ( ( inv g `  G
)  o.  H ) ) ) )
4438simprd 450 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( ( inv g `  G )  o.  H
) )  =  ( ( inv g `  G ) `  ( G  gsumg  H ) ) )
4544oveq2d 6089 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( inv g `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
4640simprd 450 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( inv g `  G )  o.  H
) ) ) )
475dprdssv 15566 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4837, 1, 2, 3, 4eldprdi 15568 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4947, 48sseldi 3338 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5037, 1, 2, 3, 8eldprdi 15568 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5147, 50sseldi 3338 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
525, 11, 12, 13grpsubval 14840 . . . . 5  |-  ( ( ( G  gsumg  F )  e.  (
Base `  G )  /\  ( G  gsumg  H )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  H ) ) ) )
5349, 51, 52syl2anc 643 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  H ) ) ) )
5445, 46, 533eqtr4d 2477 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5543, 54eqtrd 2467 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
5642, 55jca 519 1  |-  ( ph  ->  ( ( F  o F  .-  H )  e.  W  /\  ( G 
gsumg  ( F  o F  .-  H ) )  =  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   "cima 4873    o. ccom 4874   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    o Fcof 6295   X_cixp 7055   Fincfn 7101   Basecbs 13461   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716   Grpcgrp 14677   inv gcminusg 14678   -gcsg 14680   DProd cdprd 15546
This theorem is referenced by:  dprdfeq0  15572  dprdf11  15573  dprdsubg  15574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-ghm 14996  df-gim 15038  df-cntz 15108  df-oppg 15134  df-cmn 15406  df-dprd 15548
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