Theorem plyval 15816

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 15814	. 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 3376	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
3	2	oveq1d 6094	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
4	3	rexeqdv 2756	. . . 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 2551	. . 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 2358	. 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 8297	. . . 4 ⊢ ℂ ∈ V
8	7	elpw2 4291	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
9	8	biimpri 133	. 2 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ 𝒫 ℂ)
10		nn0ex 9552	. . 3 ⊢ ℕ0 ∈ V
11		fnmap 6923	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
12	7	ssex 4268	. . . . . . 7 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ V)
13		c0ex 8314	. . . . . . . 8 ⊢ 0 ∈ V
14	13	snex 4320	. . . . . . 7 ⊢ {0} ∈ V
15		unexg 4587	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
16	12, 14, 15	sylancl 417	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → (𝑆 ∪ {0}) ∈ V)
17	10	a1i 9	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → ℕ0 ∈ V)
18		fnovex 6112	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
19	11, 16, 17, 18	mp3an2i 1383	. . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
20		abrexexg 6341	. . . . 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 2624	. . 3 ⊢ (𝑆 ⊆ ℂ → ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
23		abrexex2g 6343	. . 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 418	. 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 5796	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 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  𝒫 cpw 3688  {csn 3708   ↦ cmpt 4190   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ↑_𝑚 cmap 6916  ℂcc 8171  0cc0 8173   · cmul 8178  ℕ₀cn0 9546  ...cfz 10394  ↑cexp 10958  Σcsu 12102  Polycply 15812

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-i2m1 8278

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-inn 9288  df-n0 9547  df-ply 15814

This theorem is referenced by:  elply  15818  plyss  15822