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Theorem plybss 24699
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 24693 . . 3 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21mptrcl 6772 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
32elpwid 4555 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  {cab 2803  wrex 3143  cun 3937  wss 3939  𝒫 cpw 4541  {csn 4563  cmpt 5142  cfv 6351  (class class class)co 7151  m cmap 8399  cc 10527  0cc0 10529   · cmul 10534  0cn0 11889  ...cfz 12885  cexp 13422  Σcsu 15035  Polycply 24689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fv 6359  df-ply 24693
This theorem is referenced by:  elply  24700  plyf  24703  plyssc  24705  plyaddlem  24720  plymullem  24721  plysub  24724  dgrlem  24734  coeidlem  24742  plyco  24746  plycj  24782  plyreres  24787  plydivlem3  24799  plydivlem4  24800  elmnc  39597
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