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Theorem plybss 26248
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 26242 . . 3 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21mptrcl 7025 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
32elpwid 4614 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  cun 3961  wss 3963  𝒫 cpw 4605  {csn 4631  cmpt 5231  cfv 6563  (class class class)co 7431  m cmap 8865  cc 11151  0cc0 11153   · cmul 11158  0cn0 12524  ...cfz 13544  cexp 14099  Σcsu 15719  Polycply 26238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ply 26242
This theorem is referenced by:  elply  26249  plyf  26252  plyssc  26254  plyaddlem  26269  plymullem  26270  plysub  26273  dgrlem  26283  coeidlem  26291  plyco  26295  plycj  26332  plycjOLD  26334  plyreres  26339  plydivlem3  26352  plydivlem4  26353  elmnc  43125
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