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Theorem plyun0 23870
Description: The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
plyun0 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)

Proof of Theorem plyun0
Dummy variables 𝑘 𝑎 𝑛 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 9983 . . . . . . 7 0 ∈ ℂ
2 snssi 4313 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
31, 2ax-mp 5 . . . . . 6 {0} ⊆ ℂ
43biantru 526 . . . . 5 (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5 unss 3770 . . . . 5 ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
64, 5bitr2i 265 . . . 4 ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7 unass 3753 . . . . . . . 8 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8 unidm 3739 . . . . . . . . 9 ({0} ∪ {0}) = {0}
98uneq2i 3747 . . . . . . . 8 (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
107, 9eqtri 2643 . . . . . . 7 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
1110oveq1i 6620 . . . . . 6 (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0) = ((𝑆 ∪ {0}) ↑𝑚0)
1211rexeqi 3135 . . . . 5 (∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
1312rexbii 3035 . . . 4 (∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
146, 13anbi12i 732 . . 3 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
15 elply 23868 . . 3 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 elply 23868 . . 3 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
1714, 15, 163bitr4i 292 . 2 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
1817eqriv 2618 1 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wrex 2908  cun 3557  wss 3559  {csn 4153  cmpt 4678  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  cc 9885  0cc0 9887   · cmul 9892  0cn0 11243  ...cfz 12275  cexp 12807  Σcsu 14357  Polycply 23857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-i2m1 9955  ax-1ne0 9956  ax-rrecex 9959  ax-cnre 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-nn 10972  df-n0 11244  df-ply 23861
This theorem is referenced by:  elplyd  23875  ply1term  23877  ply0  23881  plyaddlem  23888  plymullem  23889  plyco  23914  plycj  23950
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