MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdf11OLD Structured version   Unicode version

Theorem dprdf11OLD 17196
Description: Two group sums over a direct product that give the same value are equal as functions. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdf11 17189 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
dprdf11OLD.4  |-  ( ph  ->  H  e.  W )
Assertion
Ref Expression
dprdf11OLD  |-  ( 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 dprdf11OLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2 eldprdiOLD.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
3 eldprdiOLD.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
4 eldprdiOLD.3 . . . . 5  |-  ( ph  ->  F  e.  W )
5 eqid 2457 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
61, 2, 3, 4, 5dprdffOLD 17178 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
7 ffn 5737 . . . 4  |-  ( F : I --> ( Base `  G )  ->  F  Fn  I )
86, 7syl 16 . . 3  |-  ( ph  ->  F  Fn  I )
9 dprdf11OLD.4 . . . . 5  |-  ( ph  ->  H  e.  W )
101, 2, 3, 9, 5dprdffOLD 17178 . . . 4  |-  ( ph  ->  H : I --> ( Base `  G ) )
11 ffn 5737 . . . 4  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
1210, 11syl 16 . . 3  |-  ( ph  ->  H  Fn  I )
13 eqfnfv 5982 . . 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 eldprdiOLD.0 . . . 4  |-  .0.  =  ( 0g `  G )
16 eqid 2457 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
1715, 1, 2, 3, 4, 9, 16dprdfsubOLD 17194 . . . . 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, 18dprdfeq0OLD 17195 . . 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 2459 . . 3  |-  ( ph  ->  ( ( G  gsumg  ( F  oF ( -g `  G ) H ) )  =  .0.  <->  ( ( G  gsumg  F ) ( -g `  G ) ( G 
gsumg  H ) )  =  .0.  ) )
22 reldmdprd 17154 . . . . . . . . 9  |-  Rel  dom DProd
2322brrelex2i 5050 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
24 dmexg 6730 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
252, 23, 243syl 20 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
263, 25eqeltrrd 2546 . . . . . 6  |-  ( ph  ->  I  e.  _V )
27 fvex 5882 . . . . . . 7  |-  ( F `
 x )  e. 
_V
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
29 fvex 5882 . . . . . . 7  |-  ( H `
 x )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  _V )
316feqmptd 5926 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
3210feqmptd 5926 . . . . . 6  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
3326, 28, 30, 31, 32offval2 6555 . . . . 5  |-  ( ph  ->  ( F  oF ( -g `  G
) H )  =  ( x  e.  I  |->  ( ( F `  x ) ( -g `  G ) ( H `
 x ) ) ) )
3433eqeq1d 2459 . . . 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 6324 . . . . . . 7  |-  ( ( F `  x ) ( -g `  G
) ( H `  x ) )  e. 
_V
3635rgenw 2818 . . . . . 6  |-  A. x  e.  I  ( ( F `  x )
( -g `  G ) ( H `  x
) )  e.  _V
37 mpteqb 5971 . . . . . 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 17164 . . . . . . . . 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 6032 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  G
) )
4310ffvelrnda 6032 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( Base `  G
) )
445, 15, 16grpsubeq0 16250 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( ( F `  x ) ( -g `  G ) ( H `
 x ) )  =  .0.  <->  ( F `  x )  =  ( H `  x ) ) )
4645ralbidva 2893 . . . . 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 17182 . . . 4  |-  ( G DProd 
S )  C_  ( Base `  G )
5115, 1, 2, 3, 4eldprdiOLD 17191 . . . 4  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
5250, 51sseldi 3497 . . 3  |-  ( ph  ->  ( G  gsumg  F )  e.  (
Base `  G )
)
5315, 1, 2, 3, 9eldprdiOLD 17191 . . . 4  |-  ( ph  ->  ( G  gsumg  H )  e.  ( G DProd  S ) )
5450, 53sseldi 3497 . . 3  |-  ( ph  ->  ( G  gsumg  H )  e.  (
Base `  G )
)
555, 15, 16grpsubeq0 16250 . . 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 1228 . 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 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   X_cixp 7488   Fincfn 7535   Basecbs 14643   0gc0g 14856    gsumg cgsu 14857   Grpcgrp 16179   -gcsg 16181   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-ghm 16391  df-gim 16433  df-cntz 16481  df-oppg 16507  df-cmn 16926  df-dprd 17152
This theorem is referenced by:  dmdprdsplitlemOLD  17211  dpjeqOLD  17241
  Copyright terms: Public domain W3C validator