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Theorem dprdfsubOLD 17286
Description: Take the difference of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdfsub 17279 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
dprdfaddOLD.4  |-  ( ph  ->  H  e.  W )
dprdfsubOLD.b  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
dprdfsubOLD  |-  ( 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 dprdfsubOLD
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.w . . . . . . . 8  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2 eldprdiOLD.1 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
3 eldprdiOLD.2 . . . . . . . 8  |-  ( ph  ->  dom  S  =  I )
4 eldprdiOLD.3 . . . . . . . 8  |-  ( ph  ->  F  e.  W )
5 eqid 2457 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdffOLD 17270 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfaddOLD.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdffOLD 17270 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2457 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2457 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
13 dprdfsubOLD.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 16311 . . . . . 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 4543 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) ) )
17 reldmdprd 17246 . . . . . . . 8  |-  Rel  dom DProd
1817brrelex2i 5050 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
19 dmexg 6730 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
202, 18, 193syl 20 . . . . . 6  |-  ( ph  ->  dom  S  e.  _V )
213, 20eqeltrrd 2546 . . . . 5  |-  ( ph  ->  I  e.  _V )
226feqmptd 5926 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
239feqmptd 5926 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2421, 7, 10, 22, 23offval2 6555 . . . 4  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
25 fvex 5882 . . . . . 6  |-  ( ( invg `  G
) `  ( H `  k ) )  e. 
_V
2625a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( invg `  G ) `  ( H `  k )
)  e.  _V )
27 dprdgrp 17256 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
282, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
295, 12, 28grpinvf1o 16326 . . . . . . . 8  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
30 f1of 5822 . . . . . . . 8  |-  ( ( invg `  G
) : ( Base `  G ) -1-1-onto-> ( Base `  G
)  ->  ( invg `  G ) : ( Base `  G
) --> ( Base `  G
) )
3129, 30syl 16 . . . . . . 7  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) --> ( Base `  G )
)
3231feqmptd 5926 . . . . . 6  |-  ( ph  ->  ( invg `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( invg `  G
) `  x )
) )
33 fveq2 5872 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( invg `  G ) `  x
)  =  ( ( invg `  G
) `  ( H `  k ) ) )
3410, 23, 32, 33fmptco 6065 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( invg `  G
) `  ( H `  k ) ) ) )
3521, 7, 26, 22, 34offval2 6555 . . . 4  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  =  ( k  e.  I  |->  ( ( F `
 k ) ( +g  `  G ) ( ( invg `  G ) `  ( H `  k )
) ) ) )
3616, 24, 353eqtr4d 2508 . . 3  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )
37 eldprdiOLD.0 . . . . 5  |-  .0.  =  ( 0g `  G )
3837, 1, 2, 3, 8, 12dprdfinvOLD 17284 . . . . . 6  |-  ( ph  ->  ( ( ( invg `  G )  o.  H )  e.  W  /\  ( G 
gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) ) )
3938simpld 459 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  e.  W )
4037, 1, 2, 3, 4, 39, 11dprdfaddOLD 17285 . . . 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
) ) ) ) )
4140simpld 459 . . 3  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  e.  W )
4236, 41eqeltrd 2545 . 2  |-  ( ph  ->  ( F  oF 
.-  H )  e.  W )
4336oveq2d 6312 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( G  gsumg  ( F  oF ( +g  `  G ) ( ( invg `  G
)  o.  H ) ) ) )
4438simprd 463 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( ( invg `  G )  o.  H
) )  =  ( ( invg `  G ) `  ( G  gsumg  H ) ) )
4544oveq2d 6312 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) ( +g  `  G
) ( G  gsumg  ( ( invg `  G
)  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
4640simprd 463 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( G 
gsumg  ( ( invg `  G )  o.  H
) ) ) )
475dprdssv 17274 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4837, 1, 2, 3, 4eldprdiOLD 17283 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4947, 48sseldi 3497 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5037, 1, 2, 3, 8eldprdiOLD 17283 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5147, 50sseldi 3497 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
525, 11, 12, 13grpsubval 16311 . . . . 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 ) ) ) )
5349, 51, 52syl2anc 661 . . . 4  |-  ( ph  ->  ( ( G  gsumg  F ) 
.-  ( G  gsumg  H ) )  =  ( ( G  gsumg  F ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  H ) ) ) )
5445, 46, 533eqtr4d 2508 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5543, 54eqtrd 2498 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF 
.-  H ) )  =  ( ( G 
gsumg  F )  .-  ( G  gsumg  H ) ) )
5642, 55jca 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011    o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    oFcof 6537   X_cixp 7488   Fincfn 7535   Basecbs 14735   +g cplusg 14803   0gc0g 14948    gsumg cgsu 14949   Grpcgrp 16271   invgcminusg 16272   -gcsg 16273   DProd cdprd 17242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-0g 14950  df-gsum 14951  df-mre 15094  df-mrc 15095  df-acs 15097  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-mhm 16184  df-submnd 16185  df-grp 16275  df-minusg 16276  df-sbg 16277  df-subg 16416  df-ghm 16483  df-gim 16525  df-cntz 16573  df-oppg 16599  df-cmn 17018  df-dprd 17244
This theorem is referenced by:  dprdfeq0OLD  17287  dprdf11OLD  17288
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