Theorem plyval 14911

Description: Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)

Assertion

Ref	Expression
plyval	⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})

Distinct variable groups:   𝑆,𝑎,𝑓,𝑛   𝑘,𝑎,𝑧,𝑓,𝑛

Allowed substitution hints:   𝑆(𝑧,𝑘)

Proof of Theorem plyval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ply 14909	. 2 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
2		uneq1 3307	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
3	2	oveq1d 5934	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
4	3	rexeqdv 2697	. . . 4 ⊢ (𝑥 = 𝑆 → (∃𝑎 ∈ ((𝑥 ∪ {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)))))
5	4	rexbidv 2495	. . 3 ⊢ (𝑥 = 𝑆 → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {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)))))
6	5	abbidv 2311	. 2 ⊢ (𝑥 = 𝑆 → {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
7		cnex 7998	. . . 4 ⊢ ℂ ∈ V
8	7	elpw2 4187	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
9	8	biimpri 133	. 2 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ 𝒫 ℂ)
10		nn0ex 9249	. . 3 ⊢ ℕ0 ∈ V
11		fnmap 6711	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
12	7	ssex 4167	. . . . . . 7 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ V)
13		c0ex 8015	. . . . . . . 8 ⊢ 0 ∈ V
14	13	snex 4215	. . . . . . 7 ⊢ {0} ∈ V
15		unexg 4475	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
16	12, 14, 15	sylancl 413	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → (𝑆 ∪ {0}) ∈ V)
17	10	a1i 9	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → ℕ0 ∈ V)
18		fnovex 5952	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
19	11, 16, 17, 18	mp3an2i 1353	. . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
20		abrexexg 6172	. . . . 5 ⊢ (((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V → {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
21	19, 20	syl 14	. . . 4 ⊢ (𝑆 ⊆ ℂ → {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
22	21	ralrimivw 2568	. . 3 ⊢ (𝑆 ⊆ ℂ → ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
23		abrexex2g 6174	. . 3 ⊢ ((ℕ0 ∈ V ∧ ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V) → {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
24	10, 22, 23	sylancr 414	. 2 ⊢ (𝑆 ⊆ ℂ → {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
25	1, 6, 9, 24	fvmptd3 5652	1 ⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ∪ cun 3152   ⊆ wss 3154  𝒫 cpw 3602  {csn 3619   ↦ cmpt 4091   × cxp 4658   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919   ↑_𝑚 cmap 6704  ℂcc 7872  0cc0 7874   · cmul 7879  ℕ₀cn0 9243  ...cfz 10077  ↑cexp 10612  Σcsu 11499  Polycply 14907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-i2m1 7979

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-inn 8985  df-n0 9244  df-ply 14909

This theorem is referenced by:  elply  14913  plyss  14917