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

Theorem plybss 25637
Description: Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plybss (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

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 25631 . . 3 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21mptrcl 6993 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
32elpwid 4605 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {cab 2708  wrex 3069  cun 3942  wss 3944  𝒫 cpw 4596  {csn 4622  cmpt 5224  cfv 6532  (class class class)co 7393  m cmap 8803  cc 11090  0cc0 11092   · cmul 11097  0cn0 12454  ...cfz 13466  cexp 14009  Σcsu 15614  Polycply 25627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fv 6540  df-ply 25631
This theorem is referenced by:  elply  25638  plyf  25641  plyssc  25643  plyaddlem  25658  plymullem  25659  plysub  25662  dgrlem  25672  coeidlem  25680  plyco  25684  plycj  25720  plyreres  25725  plydivlem3  25737  plydivlem4  25738  elmnc  41649
  Copyright terms: Public domain W3C validator