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Theorem plybss 19576
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 19570 . . . 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 5169 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5554 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3178 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
5 elpwi 3633 . 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 1623    e. wcel 1684   {cab 2269   E.wrex 2544    u. cun 3150    C_ wss 3152   ~Pcpw 3625   {csn 3640    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737    x. cmul 8742   NN0cn0 9965   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566
This theorem is referenced by:  elply  19577  plyf  19580  plyssc  19582  plyaddlem  19597  plymullem  19598  plysub  19601  dgrlem  19611  coeidlem  19619  plyco  19623  plycj  19658  plyreres  19663  plydivlem3  19675  plydivlem4  19676  elmnc  27341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ply 19570
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