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Theorem plyun0 26237
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 11254 . . . . . . 7 0 ∈ ℂ
2 snssi 4807 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
31, 2ax-mp 5 . . . . . 6 {0} ⊆ ℂ
43biantru 529 . . . . 5 (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5 unss 4189 . . . . 5 ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
64, 5bitr2i 276 . . . 4 ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7 unass 4171 . . . . . . . 8 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8 unidm 4156 . . . . . . . . 9 ({0} ∪ {0}) = {0}
98uneq2i 4164 . . . . . . . 8 (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
107, 9eqtri 2764 . . . . . . 7 ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
1110oveq1i 7442 . . . . . 6 (((𝑆 ∪ {0}) ∪ {0}) ↑m0) = ((𝑆 ∪ {0}) ↑m0)
1211rexeqi 3324 . . . . 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 3093 . . . 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 628 . . 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 26235 . . 3 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 elply 26235 . . 3 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
1714, 15, 163bitr4i 303 . 2 (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
1817eqriv 2733 1 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  wrex 3069  cun 3948  wss 3950  {csn 4625  cmpt 5224  cfv 6560  (class class class)co 7432  m cmap 8867  cc 11154  0cc0 11156   · cmul 11161  0cn0 12528  ...cfz 13548  cexp 14103  Σcsu 15723  Polycply 26224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-mulcl 11218  ax-i2m1 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-nn 12268  df-n0 12529  df-ply 26228
This theorem is referenced by:  elplyd  26242  ply1term  26244  ply0  26248  plyaddlem  26255  plymullem  26256  plyco  26281  plycj  26318  plycjOLD  26320
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