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Theorem dprdfsub 15508
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 2389 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15499 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 5811 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfadd.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdff 15499 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 5811 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2389 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2389 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
13 dprdfsub.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 14777 . . . . . 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 4238 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( inv g `  G ) `
 ( H `  k ) ) ) ) )
17 reldmdprd 15487 . . . . . . . 8  |-  Rel  dom DProd
1817brrelex2i 4861 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
19 dmexg 5072 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
202, 18, 193syl 19 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
213, 20eqeltrrd 2464 . . . . 5  |-  ( ph  ->  I  e.  _V )
226feqmptd 5720 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
239feqmptd 5720 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2421, 7, 10, 22, 23offval2 6263 . . . 4  |-  ( ph  ->  ( F  o F 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
25 fvex 5684 . . . . . 6  |-  ( ( inv g `  G
) `  ( H `  k ) )  e. 
_V
2625a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( inv g `  G ) `  ( H `  k )
)  e.  _V )
27 dprdgrp 15492 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
282, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
295, 12, 28grpinvf1o 14790 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
30 f1of 5616 . . . . . . . 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 5720 . . . . . 6  |-  ( ph  ->  ( inv g `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( inv g `  G
) `  x )
) )
33 fveq2 5670 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( inv g `  G ) `  x
)  =  ( ( inv g `  G
) `  ( H `  k ) ) )
3410, 23, 32, 33fmptco 5842 . . . . 5  |-  ( ph  ->  ( ( inv g `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( inv g `  G
) `  ( H `  k ) ) ) )
3521, 7, 26, 22, 34offval2 6263 . . . 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 2431 . . 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 15506 . . . . . 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 15507 . . . 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 2463 . 2  |-  ( ph  ->  ( F  o F 
.-  H )  e.  W )
4336oveq2d 6038 . . 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 6038 . . . 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 15503 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4837, 1, 2, 3, 4eldprdi 15505 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4947, 48sseldi 3291 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5037, 1, 2, 3, 8eldprdi 15505 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5147, 50sseldi 3291 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
525, 11, 12, 13grpsubval 14777 . . . . 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 2431 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  o F ( +g  `  G
) ( ( inv g `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5543, 54eqtrd 2421 . 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 1649    e. wcel 1717   {crab 2655   _Vcvv 2901    \ cdif 3262   {csn 3759   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   dom cdm 4820   "cima 4823    o. ccom 4824   -->wf 5392   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022    o Fcof 6244   X_cixp 7001   Fincfn 7047   Basecbs 13398   +g cplusg 13458   0gc0g 13652    gsumg cgsu 13653   Grpcgrp 14614   inv gcminusg 14615   -gcsg 14617   DProd cdprd 15483
This theorem is referenced by:  dprdfeq0  15509  dprdf11  15510  dprdsubg  15511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-ghm 14933  df-gim 14975  df-cntz 15045  df-oppg 15071  df-cmn 15343  df-dprd 15485
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