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

Theorem plybss 19982
Description: Reverse closure of the parameter  S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plybss  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )

Proof of Theorem plybss
Dummy variables  k 
a  n  z  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ply 19976 . . . 4  |- Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) } )
21dmmptss 5308 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5699 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3291 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3753 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {cab 2375   E.wrex 2652    u. cun 3263    C_ wss 3265   ~Pcpw 3744   {csn 3759    e. cmpt 4209   dom cdm 4820   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   CCcc 8923   0cc0 8925    x. cmul 8930   NN0cn0 10155   ...cfz 10977   ^cexp 11311   sum_csu 12408  Polycply 19972
This theorem is referenced by:  elply  19983  plyf  19986  plyssc  19988  plyaddlem  20003  plymullem  20004  plysub  20007  dgrlem  20017  coeidlem  20025  plyco  20029  plycj  20064  plyreres  20069  plydivlem3  20081  plydivlem4  20082  elmnc  27012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-xp 4826  df-rel 4827  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fv 5404  df-ply 19976
  Copyright terms: Public domain W3C validator