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Theorem dprdf11 15258
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 2283 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15247 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5389 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 15 . . 3  |-  ( ph  ->  F  Fn  I )
9 dprdf11.4 . . . . 5  |-  ( ph  ->  H  e.  W )
101, 2, 3, 9, 5dprdff 15247 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5389 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 15 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5622 . . 3  |-  ( ( F  Fn  I  /\  H  Fn  I )  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
148, 12, 13syl2anc 642 . 2  |-  ( ph  ->  ( F  =  H  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
15 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
16 eqid 2283 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsub 15256 . . . . 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 445 . . . 4  |-  ( ph  ->  ( F  o F ( -g `  G
) H )  e.  W )
1915, 1, 2, 3, 18dprdfeq0 15257 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  o F ( -g `  G ) H ) )  =  .0.  <->  ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  ) ) )
2017simprd 449 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  o F ( -g `  G
) H ) )  =  ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) ) )
2120eqeq1d 2291 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  o F ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
22 reldmdprd 15235 . . . . . . . . 9  |-  Rel  dom DProd
2322brrelex2i 4730 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 4939 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 18 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2358 . . . . . 6  |-  ( ph  ->  I  e.  _V )
27 fvex 5539 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2827a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
29 fvex 5539 . . . . . . 7  |-  ( H `
 x )  e. 
_V
3029a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
316feqmptd 5575 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
3210feqmptd 5575 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
3326, 28, 30, 31, 32offval2 6095 . . . . 5  |-  ( ph  ->  ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3433eqeq1d 2291 . . . 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 5883 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3635rgenw 2610 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
37 mpteqb 5614 . . . . . 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 15240 . . . . . . . . 9  |-  ( G dom DProd  S  ->  G  e. 
Grp )
402, 39syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
4140adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
42 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : I --> ( Base `  G )  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
436, 42sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
44 ffvelrn 5663 . . . . . . . 8  |-  ( ( H : I --> ( Base `  G )  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
4510, 44sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
465, 15, 16grpsubeq0 14552 . . . . . . 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 ) ) )
4741, 43, 45, 46syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( F `  x ) ( -g `  G ) ( H `
 x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4847ralbidva 2559 . . . . 5  |-  ( ph  ->  ( A. x  e.  I  ( ( F `
 x ) (
-g `  G )
( H `  x
) )  =  .0.  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
4938, 48syl5bb 248 . . . 4  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( F `
 x ) (
-g `  G )
( H `  x
) ) )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
5034, 49bitrd 244 . . 3  |-  ( ph  ->  ( ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  .0.  )  <->  A. x  e.  I  ( F `  x )  =  ( H `  x ) ) )
5119, 21, 503bitr3d 274 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  A. x  e.  I 
( F `  x
)  =  ( H `
 x ) ) )
525dprdssv 15251 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
5315, 1, 2, 3, 4eldprdi 15253 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
5452, 53sseldi 3178 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5515, 1, 2, 3, 9eldprdi 15253 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5652, 55sseldi 3178 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
575, 15, 16grpsubeq0 14552 . . 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 ) ) )
5840, 54, 56, 57syl3anc 1182 . 2  |-  ( ph  ->  ( ( ( G 
gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  <->  ( G  gsumg  F )  =  ( G  gsumg  H ) ) )
5914, 51, 583bitr2rd 273 1  |-  ( ph  ->  ( ( G  gsumg  F )  =  ( G  gsumg  H )  <-> 
F  =  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   X_cixp 6817   Fincfn 6863   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   Grpcgrp 14362   -gcsg 14365   DProd cdprd 15231
This theorem is referenced by:  dmdprdsplitlem  15272  dpjeq  15294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-cmn 15091  df-dprd 15233
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