Theorem plyun0 26365

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 11204	. . . . . . 7 ⊢ 0 ∈ ℂ
2		snssi 4750	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
4	3	biantru 538	. . . . 5 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5		unss 4142	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
6	4, 5	bitr2i 279	. . . 4 ⊢ ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7		unass 4124	. . . . . . . 8 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8		unidm 4110	. . . . . . . . 9 ⊢ ({0} ∪ {0}) = {0}
9	8	uneq2i 4118	. . . . . . . 8 ⊢ (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
10	7, 9	eqtri 2785	. . . . . . 7 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
11	10	oveq1i 7422	. . . . . 6 ⊢ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0) = ((𝑆 ∪ {0}) ↑m ℕ0)
12	11	rexeqi 3321	. . . . 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 3111	. . . 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 639	. . 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 26363	. . 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 26363	. . 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 2759	1 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ∪ cun 3902   ⊆ wss 3904  {csn 4588   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  ℂcc 11104  0cc0 11106   · cmul 11111  ℕ₀cn0 12510  ...cfz 13541  ↑cexp 14104  Σcsu 15744  Polycply 26352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-mulcl 11168  ax-i2m1 11174

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-nn 12240  df-n0 12511  df-ply 26356

This theorem is used by:  elplyd  26370  ply1term  26372  ply0  26376  plyaddlem  26383  plymullem  26384  plyco  26409  plycj  26445  plycjOLD  26447