Theorem plyun0 26105

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 11222	. . . . . . 7 ⊢ 0 ∈ ℂ
2		snssi 4807	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
4	3	biantru 529	. . . . 5 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ))
5		unss 4180	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ)
6	4, 5	bitr2i 276	. . . 4 ⊢ ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ)
7		unass 4162	. . . . . . . 8 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0}))
8		unidm 4148	. . . . . . . . 9 ⊢ ({0} ∪ {0}) = {0}
9	8	uneq2i 4156	. . . . . . . 8 ⊢ (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0})
10	7, 9	eqtri 2755	. . . . . . 7 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0})
11	10	oveq1i 7424	. . . . . 6 ⊢ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0) = ((𝑆 ∪ {0}) ↑m ℕ0)
12	11	rexeqi 3319	. . . . 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 3089	. . . 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 626	. . 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 26103	. . 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 26103	. . 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 303	. 2 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆))
18	17	eqriv 2724	1 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3065   ∪ cun 3942   ⊆ wss 3944  {csn 4624   ↦ cmpt 5225  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8834  ℂcc 11122  0cc0 11124   · cmul 11129  ℕ₀cn0 12488  ...cfz 13502  ↑cexp 14044  Σcsu 15650  Polycply 26092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-mulcl 11186  ax-i2m1 11192

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-nn 12229  df-n0 12489  df-ply 26096

This theorem is referenced by:  elplyd  26110  ply1term  26112  ply0  26116  plyaddlem  26123  plymullem  26124  plyco  26149  plycj  26186