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Theorem plyun0 24714
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 10621 . . . . . . 7 0 ∈ ℂ
2 snssi 4733 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
31, 2ax-mp 5 . . . . . 6 {0} ⊆ ℂ
43biantru 530 . . . . 5 (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5 unss 4157 . . . . 5 ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
64, 5bitr2i 277 . . . 4 ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7 unass 4139 . . . . . . . 8 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8 unidm 4125 . . . . . . . . 9 ({0} ∪ {0}) = {0}
98uneq2i 4133 . . . . . . . 8 (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
107, 9eqtri 2841 . . . . . . 7 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
1110oveq1i 7155 . . . . . 6 (((𝑆 ∪ {0}) ∪ {0}) ↑m0) = ((𝑆 ∪ {0}) ↑m0)
1211rexeqi 3412 . . . . 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 3244 . . . 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 626 . . 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 24712 . . 3 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 elply 24712 . . 3 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
1714, 15, 163bitr4i 304 . 2 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
1817eqriv 2815 1 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  wrex 3136  cun 3931  wss 3933  {csn 4557  cmpt 5137  cfv 6348  (class class class)co 7145  m cmap 8395  cc 10523  0cc0 10525   · cmul 10530  0cn0 11885  ...cfz 12880  cexp 13417  Σcsu 15030  Polycply 24701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-nn 11627  df-n0 11886  df-ply 24705
This theorem is referenced by:  elplyd  24719  ply1term  24721  ply0  24725  plyaddlem  24732  plymullem  24733  plyco  24758  plycj  24794
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