Theorem plyval 15485

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 15483	. 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 3353	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 ∪ {0}) = (𝑆 ∪ {0}))
3	2	oveq1d 6038	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑥 ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
4	3	rexeqdv 2736	. . . 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 2532	. . 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 2348	. 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 8161	. . . 4 ⊢ ℂ ∈ V
8	7	elpw2 4248	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
9	8	biimpri 133	. 2 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ 𝒫 ℂ)
10		nn0ex 9413	. . 3 ⊢ ℕ0 ∈ V
11		fnmap 6829	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
12	7	ssex 4227	. . . . . . 7 ⊢ (𝑆 ⊆ ℂ → 𝑆 ∈ V)
13		c0ex 8178	. . . . . . . 8 ⊢ 0 ∈ V
14	13	snex 4277	. . . . . . 7 ⊢ {0} ∈ V
15		unexg 4542	. . . . . . 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 6056	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
19	11, 16, 17, 18	mp3an2i 1378	. . . . 5 ⊢ (𝑆 ⊆ ℂ → ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ∈ V)
20		abrexexg 6285	. . . . 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 2605	. . 3 ⊢ (𝑆 ⊆ ℂ → ∀𝑛 ∈ ℕ0 {𝑓 ∣ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))} ∈ V)
23		abrexex2g 6287	. . 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 5743	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 1397   ∈ wcel 2201  {cab 2216  ∀wral 2509  ∃wrex 2510  Vcvv 2801   ∪ cun 3197   ⊆ wss 3199  𝒫 cpw 3653  {csn 3670   ↦ cmpt 4151   × cxp 4725   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023   ↑_𝑚 cmap 6822  ℂcc 8035  0cc0 8037   · cmul 8042  ℕ₀cn0 9407  ...cfz 10248  ↑cexp 10806  Σcsu 11936  Polycply 15481

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-i2m1 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-map 6824  df-inn 9149  df-n0 9408  df-ply 15483

This theorem is referenced by:  elply  15487  plyss  15491