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Theorem dprdf11 17176
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 2382 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 17159 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5639 . . . 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 17159 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5639 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 16 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5883 . . 3  |-  ( ( F  Fn  I  /\  H  Fn  I )  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
148, 12, 13syl2anc 659 . 2  |-  ( ph  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
15 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
16 eqid 2382 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsub 17174 . . . . 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 457 . . . 4  |-  ( ph  ->  ( F  oF ( -g `  G
) H )  e.  W )
1915, 1, 2, 3, 18dprdfeq0 17175 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  oF ( -g `  G ) H ) )  =  .0.  <->  ( F  oF ( -g `  G ) H )  =  ( x  e.  I  |->  .0.  ) )
)
2017simprd 461 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  oF ( -g `  G
) H ) )  =  ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) ) )
2120eqeq1d 2384 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  oF ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
222, 3dprddomcld 17145 . . . . . 6  |-  ( ph  ->  I  e.  _V )
23 fvex 5784 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2423a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
25 fvex 5784 . . . . . . 7  |-  ( H `
 x )  e. 
_V
2625a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
276feqmptd 5827 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
2810feqmptd 5827 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
2922, 24, 26, 27, 28offval2 6455 . . . . 5  |-  ( ph  ->  ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3029eqeq1d 2384 . . . 4  |-  ( ph  ->  ( ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G
) ( H `  x ) ) )  =  ( x  e.  I  |->  .0.  ) )
)
31 ovex 6224 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3231rgenw 2743 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
33 mpteqb 5872 . . . . . 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.  ) )
3432, 33ax-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.  )
35 dprdgrp 17151 . . . . . . . . 9  |-  ( G dom DProd  S  ->  G  e. 
Grp )
362, 35syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
3736adantr 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
386ffvelrnda 5933 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
3910ffvelrnda 5933 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
405, 15, 16grpsubeq0 16241 . . . . . . 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 ) ) )
4137, 38, 39, 40syl3anc 1226 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( F `  x ) ( -g `  G ) ( H `
 x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4241ralbidva 2818 . . . . 5  |-  ( ph  ->  ( A. x  e.  I  ( ( F `
 x ) (
-g `  G )
( H `  x
) )  =  .0.  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4334, 42syl5bb 257 . . . 4  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( F `
 x ) (
-g `  G )
( H `  x
) ) )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4430, 43bitrd 253 . . 3  |-  ( ph  ->  ( ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4519, 21, 443bitr3d 283 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
465dprdssv 17169 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
4715, 1, 2, 3, 4eldprdi 17171 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
4846, 47sseldi 3415 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
4915, 1, 2, 3, 9eldprdi 17171 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5046, 49sseldi 3415 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
515, 15, 16grpsubeq0 16241 . . 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 ) ) )
5236, 48, 50, 51syl3anc 1226 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  ( G  gsumg  F )  =  ( G  gsumg  H ) ) )
5314, 45, 523bitr2rd 282 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  ( G  gsumg  H )  <-> 
F  =  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   {crab 2736   _Vcvv 3034   class class class wbr 4367    |-> cmpt 4425   dom cdm 4913    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196    oFcof 6437   X_cixp 7388   finSupp cfsupp 7744   Basecbs 14634   0gc0g 14847    gsumg cgsu 14848   Grpcgrp 16170   -gcsg 16172   DProd cdprd 17137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-tpos 6873  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011  df-hash 12308  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-gsum 14850  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mulg 16177  df-subg 16315  df-ghm 16382  df-gim 16424  df-cntz 16472  df-oppg 16498  df-cmn 16917  df-dprd 17139
This theorem is referenced by:  dmdprdsplitlem  17197  dpjeq  17221
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