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Theorem plyun0 23757
 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 9911 . . . . . . 7 0 ∈ ℂ
2 snssi 4280 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
31, 2ax-mp 5 . . . . . 6 {0} ⊆ ℂ
43biantru 525 . . . . 5 (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5 unss 3749 . . . . 5 ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
64, 5bitr2i 264 . . . 4 ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7 unass 3732 . . . . . . . 8 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8 unidm 3718 . . . . . . . . 9 ({0} ∪ {0}) = {0}
98uneq2i 3726 . . . . . . . 8 (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
107, 9eqtri 2632 . . . . . . 7 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
1110oveq1i 6559 . . . . . 6 (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0) = ((𝑆 ∪ {0}) ↑𝑚0)
1211rexeqi 3120 . . . . 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 3023 . . . 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 729 . . 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 23755 . . 3 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 elply 23755 . . 3 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
1714, 15, 163bitr4i 291 . 2 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
1817eqriv 2607 1 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℂcc 9813  0cc0 9815   · cmul 9820  ℕ0cn0 11169  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-ply 23748 This theorem is referenced by:  elplyd  23762  ply1term  23764  ply0  23768  plyaddlem  23775  plymullem  23776  plyco  23801  plycj  23837
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