Theorem plyval 15371

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 15369	. 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 3331	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
3	2	oveq1d 5989	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
4	3	rexeqdv 2715	. . . 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 2511	. . 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 2327	. 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 8091	. . . 4 ⊢ ℂ ∈ V
8	7	elpw2 4220	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
9	8	biimpri 133	. 2 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ 𝒫 ℂ)
10		nn0ex 9343	. . 3 ⊢ ℕ0 ∈ V
11		fnmap 6772	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
12	7	ssex 4200	. . . . . . 7 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ V)
13		c0ex 8108	. . . . . . . 8 ⊢ 0 ∈ V
14	13	snex 4248	. . . . . . 7 ⊢ {0} ∈ V
15		unexg 4511	. . . . . . 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 6007	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
19	11, 16, 17, 18	mp3an2i 1357	. . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
20		abrexexg 6233	. . . . 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 2584	. . 3 ⊢ (𝑆 ⊆ ℂ → ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
23		abrexex2g 6235	. . 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 5701	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 1375   ∈ wcel 2180  {cab 2195  ∀wral 2488  ∃wrex 2489  Vcvv 2779   ∪ cun 3175   ⊆ wss 3177  𝒫 cpw 3629  {csn 3646   ↦ cmpt 4124   × cxp 4694   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974   ↑_𝑚 cmap 6765  ℂcc 7965  0cc0 7967   · cmul 7972  ℕ₀cn0 9337  ...cfz 10172  ↑cexp 10727  Σcsu 11830  Polycply 15367

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-i2m1 8072

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-map 6767  df-inn 9079  df-n0 9338  df-ply 15369

This theorem is referenced by:  elply  15373  plyss  15377