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Theorem dprdf11 16604
Description: Two group sums over a direct product that give the same value are equal as functions. (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 )
dprdf11.4  |-  ( ph  ->  H  e.  W )
Assertion
Ref Expression
dprdf11  |-  ( ph  ->  ( ( G  gsumg  F )  =  ( G  gsumg  H )  <-> 
F  =  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)    W( h, i)    .0. ( i)

Proof of Theorem dprdf11
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldprdi.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
2 eldprdi.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
3 eldprdi.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
4 eldprdi.3 . . . . 5  |-  ( ph  ->  F  e.  W )
5 eqid 2450 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 16587 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5643 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 16 . . 3  |-  ( ph  ->  F  Fn  I )
9 dprdf11.4 . . . . 5  |-  ( ph  ->  H  e.  W )
101, 2, 3, 9, 5dprdff 16587 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5643 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 16 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5882 . . 3  |-  ( ( F  Fn  I  /\  H  Fn  I )  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
148, 12, 13syl2anc 661 . 2  |-  ( ph  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
15 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
16 eqid 2450 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsub 16602 . . . . 5  |-  ( ph  ->  ( ( F  oF ( -g `  G
) H )  e.  W  /\  ( G 
gsumg  ( F  oF
( -g `  G ) H ) )  =  ( ( G  gsumg  F ) ( -g `  G
) ( G  gsumg  H ) ) ) )
1817simpld 459 . . . 4  |-  ( ph  ->  ( F  oF ( -g `  G
) H )  e.  W )
1915, 1, 2, 3, 18dprdfeq0 16603 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  oF ( -g `  G ) H ) )  =  .0.  <->  ( F  oF ( -g `  G ) H )  =  ( x  e.  I  |->  .0.  ) )
)
2017simprd 463 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  oF ( -g `  G
) H ) )  =  ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) ) )
2120eqeq1d 2452 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  oF ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
22 reldmdprd 16570 . . . . . . . . 9  |-  Rel  dom DProd
2322brrelex2i 4964 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 6595 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 20 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2537 . . . . . 6  |-  ( ph  ->  I  e.  _V )
27 fvex 5785 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
29 fvex 5785 . . . . . . 7  |-  ( H `
 x )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
316feqmptd 5829 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
3210feqmptd 5829 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
3326, 28, 30, 31, 32offval2 6422 . . . . 5  |-  ( ph  ->  ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3433eqeq1d 2452 . . . 4  |-  ( ph  ->  ( ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G
) ( H `  x ) ) )  =  ( x  e.  I  |->  .0.  ) )
)
35 ovex 6201 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3635rgenw 2869 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
37 mpteqb 5873 . . . . . 6  |-  ( A. x  e.  I  (
( F `  x
) ( -g `  G
) ( H `  x ) )  e. 
_V  ->  ( ( x  e.  I  |->  ( ( F `  x ) ( -g `  G
) ( H `  x ) ) )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  =  .0.  ) )
3836, 37ax-mp 5 . . . . 5  |-  ( ( x  e.  I  |->  ( ( F `  x
) ( -g `  G
) ( H `  x ) ) )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  =  .0.  )
39 dprdgrp 16580 . . . . . . . . 9  |-  ( G dom DProd  S  ->  G  e. 
Grp )
402, 39syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
4140adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
426ffvelrnda 5928 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
4310ffvelrnda 5928 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
445, 15, 16grpsubeq0 15700 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( F `  x )  e.  ( Base `  G
)  /\  ( H `  x )  e.  (
Base `  G )
)  ->  ( (
( F `  x
) ( -g `  G
) ( H `  x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4541, 42, 43, 44syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( F `  x ) ( -g `  G ) ( H `
 x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4645ralbidva 2811 . . . . 5  |-  ( ph  ->  ( A. x  e.  I  ( ( F `
 x ) (
-g `  G )
( H `  x
) )  =  .0.  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4738, 46syl5bb 257 . . . 4  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( F `
 x ) (
-g `  G )
( H `  x
) ) )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4834, 47bitrd 253 . . 3  |-  ( ph  ->  ( ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4919, 21, 483bitr3d 283 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
505dprdssv 16597 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
5115, 1, 2, 3, 4eldprdi 16599 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
5250, 51sseldi 3438 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5315, 1, 2, 3, 9eldprdi 16599 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5450, 53sseldi 3438 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
555, 15, 16grpsubeq0 15700 . . 3  |-  ( ( G  e.  Grp  /\  ( G  gsumg  F )  e.  (
Base `  G )  /\  ( G  gsumg  H )  e.  (
Base `  G )
)  ->  ( (
( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  ( G  gsumg  F )  =  ( G  gsumg  H ) ) )
5640, 52, 54, 55syl3anc 1219 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  ( G  gsumg  F )  =  ( G  gsumg  H ) ) )
5714, 49, 563bitr2rd 282 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  ( G  gsumg  H )  <-> 
F  =  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   {crab 2796   _Vcvv 3054   class class class wbr 4376    |-> cmpt 4434   dom cdm 4924    Fn wfn 5497   -->wf 5498   ` cfv 5502  (class class class)co 6176    oFcof 6404   X_cixp 7349   finSupp cfsupp 7707   Basecbs 14262   0gc0g 14466    gsumg cgsu 14467   Grpcgrp 15498   -gcsg 15501   DProd cdprd 16566
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-of 6406  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-tpos 6831  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-0g 14468  df-gsum 14469  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-sbg 15635  df-mulg 15636  df-subg 15766  df-ghm 15833  df-gim 15875  df-cntz 15923  df-oppg 15949  df-cmn 16369  df-dprd 16568
This theorem is referenced by:  dmdprdsplitlem  16625  dpjeq  16649
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