Theorem coeid 26163

Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypotheses

Ref	Expression
dgrub.1	⊢ 𝐴 = (coeff‘𝐹)
dgrub.2	⊢ 𝑁 = (deg‘𝐹)

Assertion

Ref	Expression
coeid	⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))

Distinct variable groups:   𝑧,𝑘,𝐴   𝑘,𝐹   𝑆,𝑘,𝑧   𝑘,𝑁,𝑧   𝑧,𝐹

Proof of Theorem coeid

Dummy variables 𝑎 𝑛 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply2 26121	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))))
2	1	simprbi 496	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚)))))
3		dgrub.1	. . . . 5 ⊢ 𝐴 = (coeff‘𝐹)
4		dgrub.2	. . . . 5 ⊢ 𝑁 = (deg‘𝐹)
5		simpll 766	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝐹 ∈ (Poly‘𝑆))
6		simplrl 776	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝑛 ∈ ℕ0)
7		simplrr 777	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
8		simprl 770	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0})
9		simprr 772	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))
10		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (𝑎‘𝑚) = (𝑎‘𝑘))
11		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (𝑥↑𝑚) = (𝑥↑𝑘))
12	10, 11	oveq12d 7359	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → ((𝑎‘𝑚) · (𝑥↑𝑚)) = ((𝑎‘𝑘) · (𝑥↑𝑘)))
13	12	cbvsumv 15595	. . . . . . . 8 ⊢ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))
14		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥↑𝑘) = (𝑧↑𝑘))
15	14	oveq2d 7357	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝑧↑𝑘)))
16	15	sumeq2sdv 15602	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
17	13, 16	eqtrid 2777	. . . . . . 7 ⊢ (𝑥 = 𝑧 → Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
18	17	cbvmptv 5193	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
19	9, 18	eqtrdi 2781	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
20	3, 4, 5, 6, 7, 8, 19	coeidlem 26162	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
21	20	ex 412	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
22	21	rexlimdvva 3187	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑚 ∈ (0...𝑛)((𝑎‘𝑚) · (𝑥↑𝑚)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
23	2, 22	mpd 15	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃wrex 3054   ∪ cun 3898   ⊆ wss 3900  {csn 4574   ↦ cmpt 5170   “ cima 5617  ‘cfv 6477  (class class class)co 7341   ↑_m cmap 8745  ℂcc 10996  0cc0 10998  1c1 10999   + caddc 11001   · cmul 11003  ℕ₀cn0 12373  ℤ_≥cuz 12724  ...cfz 13399  ↑cexp 13960  Σcsu 15585  Polycply 26109  coeffccoe 26111  degcdgr 26112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-div 11767  df-nn 12118  df-2 12180  df-3 12181  df-n0 12374  df-z 12461  df-uz 12725  df-rp 12883  df-fz 13400  df-fzo 13547  df-fl 13688  df-seq 13901  df-exp 13961  df-hash 14230  df-cj 14998  df-re 14999  df-im 15000  df-sqrt 15134  df-abs 15135  df-clim 15387  df-rlim 15388  df-sum 15586  df-0p 25591  df-ply 26113  df-coe 26115  df-dgr 26116

This theorem is referenced by:  coeid2  26164  plyco  26166  0dgrb  26171  coeaddlem  26174  coemullem  26175  coe11  26178  plycn  26186  plycnOLD  26187  plycjlem  26202