Theorem plyval 15052

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 15050	. 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 3311	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
3	2	oveq1d 5940	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
4	3	rexeqdv 2700	. . . 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 2498	. . 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 2314	. 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 8020	. . . 4 ⊢ ℂ ∈ V
8	7	elpw2 4191	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
9	8	biimpri 133	. 2 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ 𝒫 ℂ)
10		nn0ex 9272	. . 3 ⊢ ℕ0 ∈ V
11		fnmap 6723	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
12	7	ssex 4171	. . . . . . 7 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ V)
13		c0ex 8037	. . . . . . . 8 ⊢ 0 ∈ V
14	13	snex 4219	. . . . . . 7 ⊢ {0} ∈ V
15		unexg 4479	. . . . . . 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 5958	. . . . . 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 6184	. . . . 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 2571	. . 3 ⊢ (𝑆 ⊆ ℂ → ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
23		abrexex2g 6186	. . 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 5658	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 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  𝒫 cpw 3606  {csn 3623   ↦ cmpt 4095   × cxp 4662   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925   ↑_𝑚 cmap 6716  ℂcc 7894  0cc0 7896   · cmul 7901  ℕ₀cn0 9266  ...cfz 10100  ↑cexp 10647  Σcsu 11535  Polycply 15048

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-i2m1 8001

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-inn 9008  df-n0 9267  df-ply 15050

This theorem is referenced by:  elply  15054  plyss  15058