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

Theorem plybss 19592
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 19586 . . . 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 5185 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5570 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3191 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
5 elpwi 3646 . 2  |-  ( S  e.  ~P CC  ->  S 
C_  CC )
64, 5syl 15 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    u. cun 3163    C_ wss 3165   ~Pcpw 3638   {csn 3653    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   0cc0 8753    x. cmul 8758   NN0cn0 9981   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582
This theorem is referenced by:  elply  19593  plyf  19596  plyssc  19598  plyaddlem  19613  plymullem  19614  plysub  19617  dgrlem  19627  coeidlem  19635  plyco  19639  plycj  19674  plyreres  19679  plydivlem3  19691  plydivlem4  19692  elmnc  27444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ply 19586
  Copyright terms: Public domain W3C validator