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

Theorem dprdfinv 15347
Description: Take the inverse of a group sum over a family of elements of disjoint subgroups. (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 )
dprdfinv.b  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
dprdfinv  |-  ( ph  ->  ( ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) ) )
Distinct variable groups:    h, F    h, i, G    h, I,
i    h, N    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    N( i)    W( h, i)    .0. ( i)

Proof of Theorem dprdfinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
2 dprdgrp 15333 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31, 2syl 15 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2358 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
5 dprdfinv.b . . . . . 6  |-  N  =  ( inv g `  G )
64, 5grpinvf 14619 . . . . 5  |-  ( G  e.  Grp  ->  N : ( Base `  G
) --> ( Base `  G
) )
73, 6syl 15 . . . 4  |-  ( ph  ->  N : ( Base `  G ) --> ( Base `  G ) )
8 eldprdi.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
9 eldprdi.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
10 eldprdi.3 . . . . 5  |-  ( ph  ->  F  e.  W )
118, 1, 9, 10, 4dprdff 15340 . . . 4  |-  ( ph  ->  F : I --> ( Base `  G ) )
12 fcompt 5774 . . . 4  |-  ( ( N : ( Base `  G ) --> ( Base `  G )  /\  F : I --> ( Base `  G ) )  -> 
( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
137, 11, 12syl2anc 642 . . 3  |-  ( ph  ->  ( N  o.  F
)  =  ( x  e.  I  |->  ( N `
 ( F `  x ) ) ) )
141, 9dprdf2 15335 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
15 ffvelrn 5743 . . . . . 6  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
1614, 15sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
178, 1, 9, 10dprdfcl 15341 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
185subginvcl 14723 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
) )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
1916, 17, 18syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( N `  ( F `  x ) )  e.  ( S `  x
) )
208, 1, 9, 10dprdffi 15342 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
21 ssid 3273 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2221a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2311, 22suppssr 5739 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
2423fveq2d 5609 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  ( N `
 .0.  ) )
25 eldprdi.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
2625, 5grpinvid 14626 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
273, 26syl 15 . . . . . . . 8  |-  ( ph  ->  ( N `  .0.  )  =  .0.  )
2827adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  .0.  )  =  .0.  )
2924, 28eqtrd 2390 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( N `  ( F `  x ) )  =  .0.  )
3029suppss2 6157 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) ) )
31 ssfi 7168 . . . . 5  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( x  e.  I  |->  ( N `  ( F `  x )
) ) " ( _V  \  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
3220, 30, 31syl2anc 642 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( N `
 ( F `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
338, 1, 9, 19, 32dprdwd 15339 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( N `  ( F `  x )
) )  e.  W
)
3413, 33eqeltrd 2432 . 2  |-  ( ph  ->  ( N  o.  F
)  e.  W )
35 eqid 2358 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
36 reldmdprd 15328 . . . . . 6  |-  Rel  dom DProd
3736brrelex2i 4809 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
38 dmexg 5018 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
391, 37, 383syl 18 . . . 4  |-  ( ph  ->  dom  S  e.  _V )
409, 39eqeltrrd 2433 . . 3  |-  ( ph  ->  I  e.  _V )
418, 1, 9, 10, 35dprdfcntz 15343 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
424, 25, 35, 5, 3, 40, 11, 41, 20gsumzinv 15310 . 2  |-  ( ph  ->  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) )
4334, 42jca 518 1  |-  ( ph  ->  ( ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F
) )  =  ( N `  ( G 
gsumg  F ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {crab 2623   _Vcvv 2864    \ cdif 3225    C_ wss 3228   {csn 3716   class class class wbr 4102    e. cmpt 4156   `'ccnv 4767   dom cdm 4768   "cima 4771    o. ccom 4772   -->wf 5330   ` cfv 5334  (class class class)co 5942   X_cixp 6902   Fincfn 6948   Basecbs 13239   0gc0g 13493    gsumg cgsu 13494   Grpcgrp 14455   inv gcminusg 14456  SubGrpcsubg 14708  Cntzccntz 14884   DProd cdprd 15324
This theorem is referenced by:  dprdfsub  15349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-fzo 10960  df-seq 11136  df-hash 11428  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-0g 13497  df-gsum 13498  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-subg 14711  df-ghm 14774  df-gim 14816  df-cntz 14886  df-oppg 14912  df-cmn 15184  df-dprd 15326
  Copyright terms: Public domain W3C validator