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Theorem plybss 20074
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 20068 . . . 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 5333 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5724 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3314 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3776 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {cab 2398   E.wrex 2675    u. cun 3286    C_ wss 3288   ~Pcpw 3767   {csn 3782    e. cmpt 4234   dom cdm 4845   ` cfv 5421  (class class class)co 6048    ^m cmap 6985   CCcc 8952   0cc0 8954    x. cmul 8959   NN0cn0 10185   ...cfz 11007   ^cexp 11345   sum_csu 12442  Polycply 20064
This theorem is referenced by:  elply  20075  plyf  20078  plyssc  20080  plyaddlem  20095  plymullem  20096  plysub  20099  dgrlem  20109  coeidlem  20117  plyco  20121  plycj  20156  plyreres  20161  plydivlem3  20173  plydivlem4  20174  elmnc  27217
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fv 5429  df-ply 20068
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