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Theorem dprdf11 15581
Description: Two group sums over a direct product that give the same value are equal as functions. (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 )
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 " ( _V  \  {  .0.  } ) )  e. 
Fin }
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 2436 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15570 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5591 . . . 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 15570 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5591 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 16 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5827 . . 3  |-  ( ( F  Fn  I  /\  H  Fn  I )  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
148, 12, 13syl2anc 643 . 2  |-  ( ph  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
15 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
16 eqid 2436 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsub 15579 . . . . 5  |-  ( ph  ->  ( ( F  o F ( -g `  G
) H )  e.  W  /\  ( G 
gsumg  ( F  o F
( -g `  G ) H ) )  =  ( ( G  gsumg  F ) ( -g `  G
) ( G  gsumg  H ) ) ) )
1817simpld 446 . . . 4  |-  ( ph  ->  ( F  o F ( -g `  G
) H )  e.  W )
1915, 1, 2, 3, 18dprdfeq0 15580 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  o F ( -g `  G ) H ) )  =  .0.  <->  ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  ) ) )
2017simprd 450 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  o F ( -g `  G
) H ) )  =  ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) ) )
2120eqeq1d 2444 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  o F ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
22 reldmdprd 15558 . . . . . . . . 9  |-  Rel  dom DProd
2322brrelex2i 4919 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 5130 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 19 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2511 . . . . . 6  |-  ( ph  ->  I  e.  _V )
27 fvex 5742 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
29 fvex 5742 . . . . . . 7  |-  ( H `
 x )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
316feqmptd 5779 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
3210feqmptd 5779 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
3326, 28, 30, 31, 32offval2 6322 . . . . 5  |-  ( ph  ->  ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3433eqeq1d 2444 . . . 4  |-  ( ph  ->  ( ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G
) ( H `  x ) ) )  =  ( x  e.  I  |->  .0.  ) )
)
35 ovex 6106 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3635rgenw 2773 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
37 mpteqb 5819 . . . . . 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 8 . . . . 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 15563 . . . . . . . . 9  |-  ( G dom DProd  S  ->  G  e. 
Grp )
402, 39syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
4140adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
426ffvelrnda 5870 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
4310ffvelrnda 5870 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
445, 15, 16grpsubeq0 14875 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( F `  x ) ( -g `  G ) ( H `
 x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4645ralbidva 2721 . . . . 5  |-  ( ph  ->  ( A. x  e.  I  ( ( F `
 x ) (
-g `  G )
( H `  x
) )  =  .0.  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4738, 46syl5bb 249 . . . 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 245 . . 3  |-  ( ph  ->  ( ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4919, 21, 483bitr3d 275 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
505dprdssv 15574 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
5115, 1, 2, 3, 4eldprdi 15576 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
5250, 51sseldi 3346 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5315, 1, 2, 3, 9eldprdi 15576 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5450, 53sseldi 3346 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
555, 15, 16grpsubeq0 14875 . . 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 1184 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  ( G  gsumg  F )  =  ( G  gsumg  H ) ) )
5714, 49, 563bitr2rd 274 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  ( G  gsumg  H )  <-> 
F  =  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317   {csn 3814   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   dom cdm 4878   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   X_cixp 7063   Fincfn 7109   Basecbs 13469   0gc0g 13723    gsumg cgsu 13724   Grpcgrp 14685   -gcsg 14688   DProd cdprd 15554
This theorem is referenced by:  dmdprdsplitlem  15595  dpjeq  15617
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-gim 15046  df-cntz 15116  df-oppg 15142  df-cmn 15414  df-dprd 15556
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