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Theorem dprdf11 15260
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 2285 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdff 15249 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5391 . . . 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 15249 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5391 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 15 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5624 . . 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 2285 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsub 15258 . . . . 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 15259 . . 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 2293 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  o F ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
22 reldmdprd 15237 . . . . . . . . 9  |-  Rel  dom DProd
2322brrelex2i 4732 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 4941 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 18 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2360 . . . . . 6  |-  ( ph  ->  I  e.  _V )
27 fvex 5541 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2827a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
29 fvex 5541 . . . . . . 7  |-  ( H `
 x )  e. 
_V
3029a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
316feqmptd 5577 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
3210feqmptd 5577 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
3326, 28, 30, 31, 32offval2 6097 . . . . 5  |-  ( ph  ->  ( F  o F ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3433eqeq1d 2293 . . . 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 5885 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3635rgenw 2612 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
37 mpteqb 5616 . . . . . 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 15242 . . . . . . . . 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 5665 . . . . . . . 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 5665 . . . . . . . 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 14554 . . . . . . 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 2561 . . . . 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 15253 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
5315, 1, 2, 3, 4eldprdi 15255 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
5452, 53sseldi 3180 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5515, 1, 2, 3, 9eldprdi 15255 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5652, 55sseldi 3180 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
575, 15, 16grpsubeq0 14554 . . 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 1625    e. wcel 1686   A.wral 2545   {crab 2549   _Vcvv 2790    \ cdif 3151   {csn 3642   class class class wbr 4025    e. cmpt 4079   `'ccnv 4690   dom cdm 4691   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860    o Fcof 6078   X_cixp 6819   Fincfn 6865   Basecbs 13150   0gc0g 13402    gsumg cgsu 13403   Grpcgrp 14364   -gcsg 14367   DProd cdprd 15233
This theorem is referenced by:  dmdprdsplitlem  15274  dpjeq  15296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-gsum 13407  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-ghm 14683  df-gim 14725  df-cntz 14795  df-oppg 14821  df-cmn 15093  df-dprd 15235
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