Theorem plyun0 26542

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.

Step	Hyp	Ref	Expression
1		0cn 11322	. . . . . . 7 ⊢ 0 ∈ ℂ
2		snssi 4746	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
4	3	biantru 539	. . . . 5 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5		unss 4191	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
6	4, 5	bitr2i 279	. . . 4 ⊢ ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7		unass 4173	. . . . . . . 8 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8		unidm 4159	. . . . . . . . 9 ⊢ ({0} ∪ {0}) = {0}
9	8	uneq2i 4167	. . . . . . . 8 ⊢ (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
10	7, 9	eqtri 2786	. . . . . . 7 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
11	10	oveq1i 7450	. . . . . 6 ⊢ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0) = ((𝑆 ∪ {0}) ↑m ℕ0)
12	11	rexeqi 3322	. . . . 5 ⊢ (∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
13	12	rexbii 3113	. . . 4 ⊢ (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
14	6, 13	anbi12i 640	. . 3 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
15		elply 26540	. . 3 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
16		elply 26540	. . 3 ⊢ (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
17	14, 15, 16	3bitr4i 306	. 2 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
18	17	eqriv 2760	1 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∪ cun 4152  {csn 4588   ↦ cmpt 5189  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  ℂcc 11222  0cc0 11224   · cmul 11229  ℕ₀cn0 12630  ...cfz 13663  ↑cexp 14228  Σcsu 15879  Polycply 26529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-mulcl 11286  ax-i2m1 11292

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-nn 12360  df-n0 12631  df-ply 26533

This theorem is used by:  elplyd  26547  ply1term  26549  ply0  26553  plyaddlem  26561  plymullem  26562  plyco  26587  plycj  26623