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Theorem plybss 20066
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 20060 . . . 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 5325 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5716 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3306 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3768 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    u. cun 3278    C_ wss 3280   ~Pcpw 3759   {csn 3774    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944   0cc0 8946    x. cmul 8951   NN0cn0 10177   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056
This theorem is referenced by:  elply  20067  plyf  20070  plyssc  20072  plyaddlem  20087  plymullem  20088  plysub  20091  dgrlem  20101  coeidlem  20109  plyco  20113  plycj  20148  plyreres  20153  plydivlem3  20165  plydivlem4  20166  elmnc  27209
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ply 20060
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