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Theorem dprdfsub 16636
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 2454 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 16621 . . . . . . 7  |-  ( ph  ->  F : I --> ( Base `  G ) )
76ffvelrnda 5955 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
8 dprdfadd.4 . . . . . . . 8  |-  ( ph  ->  H  e.  W )
91, 2, 3, 8, 5dprdff 16621 . . . . . . 7  |-  ( ph  ->  H : I --> ( Base `  G ) )
109ffvelrnda 5955 . . . . . 6  |-  ( (
ph  /\  k  e.  I )  ->  ( H `  k )  e.  ( Base `  G
) )
11 eqid 2454 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2454 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
13 dprdfsub.b . . . . . . 7  |-  .-  =  ( -g `  G )
145, 11, 12, 13grpsubval 15703 . . . . . 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 4489 . . . 4  |-  ( ph  ->  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) )  =  ( k  e.  I  |->  ( ( F `  k
) ( +g  `  G
) ( ( invg `  G ) `
 ( H `  k ) ) ) ) )
172, 3dprddomcld 16608 . . . . 5  |-  ( ph  ->  I  e.  _V )
186feqmptd 5856 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  I  |->  ( F `
 k ) ) )
199feqmptd 5856 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  I  |->  ( H `
 k ) ) )
2017, 7, 10, 18, 19offval2 6449 . . . 4  |-  ( ph  ->  ( F  oF 
.-  H )  =  ( k  e.  I  |->  ( ( F `  k )  .-  ( H `  k )
) ) )
21 fvex 5812 . . . . . 6  |-  ( ( invg `  G
) `  ( H `  k ) )  e. 
_V
2221a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  (
( invg `  G ) `  ( H `  k )
)  e.  _V )
23 dprdgrp 16614 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
242, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
255, 12, 24grpinvf1o 15718 . . . . . . . 8  |-  ( ph  ->  ( invg `  G ) : (
Base `  G ) -1-1-onto-> ( Base `  G ) )
26 f1of 5752 . . . . . . . 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 5856 . . . . . 6  |-  ( ph  ->  ( invg `  G )  =  ( x  e.  ( Base `  G )  |->  ( ( invg `  G
) `  x )
) )
29 fveq2 5802 . . . . . 6  |-  ( x  =  ( H `  k )  ->  (
( invg `  G ) `  x
)  =  ( ( invg `  G
) `  ( H `  k ) ) )
3010, 19, 28, 29fmptco 5988 . . . . 5  |-  ( ph  ->  ( ( invg `  G )  o.  H
)  =  ( k  e.  I  |->  ( ( invg `  G
) `  ( H `  k ) ) ) )
3117, 7, 22, 18, 30offval2 6449 . . . 4  |-  ( ph  ->  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) )  =  ( k  e.  I  |->  ( ( F `
 k ) ( +g  `  G ) ( ( invg `  G ) `  ( H `  k )
) ) ) )
3216, 20, 313eqtr4d 2505 . . 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 16634 . . . . . 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 16635 . . . 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 2542 . 2  |-  ( ph  ->  ( F  oF 
.-  H )  e.  W )
3932oveq2d 6219 . . 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 6219 . . . 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 16631 . . . . . 6  |-  ( G DProd 
S )  C_  ( Base `  G )
4433, 1, 2, 3, 4eldprdi 16633 . . . . . 6  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4543, 44sseldi 3465 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
4633, 1, 2, 3, 8eldprdi 16633 . . . . . 6  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
4743, 46sseldi 3465 . . . . 5  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
485, 11, 12, 13grpsubval 15703 . . . . 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 2505 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  oF ( +g  `  G
) ( ( invg `  G )  o.  H ) ) )  =  ( ( G  gsumg  F )  .-  ( G  gsumg  H ) ) )
5139, 50eqtrd 2495 . 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 1370    e. wcel 1758   {crab 2803   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   dom cdm 4951    o. ccom 4955   -->wf 5525   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203    oFcof 6431   X_cixp 7376   finSupp cfsupp 7734   Basecbs 14295   +g cplusg 14360   0gc0g 14500    gsumg cgsu 14501   Grpcgrp 15532   invgcminusg 15533   -gcsg 15535   DProd cdprd 16600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-tpos 6858  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-0g 14502  df-gsum 14503  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-mhm 15586  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-ghm 15867  df-gim 15909  df-cntz 15957  df-oppg 15983  df-cmn 16403  df-dprd 16602
This theorem is referenced by:  dprdfeq0  16637  dprdf11  16638  dprdsubg  16646
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