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

Theorem plybss 20114
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 20108 . . . 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 5367 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5758 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3347 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3809 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707    u. cun 3319    C_ wss 3321   ~Pcpw 3800   {csn 3815    e. cmpt 4267   dom cdm 4879   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   CCcc 8989   0cc0 8991    x. cmul 8996   NN0cn0 10222   ...cfz 11044   ^cexp 11383   sum_csu 12480  Polycply 20104
This theorem is referenced by:  elply  20115  plyf  20118  plyssc  20120  plyaddlem  20135  plymullem  20136  plysub  20139  dgrlem  20149  coeidlem  20157  plyco  20161  plycj  20196  plyreres  20201  plydivlem3  20213  plydivlem4  20214  elmnc  27319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-ply 20108
  Copyright terms: Public domain W3C validator