Theorem coeeq 24976

Description: If 𝐴 satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
coeeq.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
coeeq.2	⊢ (𝜑 → 𝑁 ∈ ℕ0)
coeeq.3	⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
coeeq.4	⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
coeeq.5	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))

Assertion

Ref	Expression
coeeq	⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

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

Allowed substitution hints:   𝜑(𝑧,𝑘)   𝑆(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem coeeq

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

Step	Hyp	Ref	Expression
1		coeeq.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
2		coeval 24972	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (coeff‘𝐹) = (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
4		coeeq.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
5		coeeq.4	. . . 4 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
6		coeeq.5	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
7		fvoveq1 7193	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑁 + 1)))
8	7	imaeq2d 5903	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐴 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑁 + 1))))
9	8	eqeq1d 2740	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}))
10		oveq2 7178	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
11	10	sumeq1d 15151	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))
12	11	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
13	12	eqeq2d 2749	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
14	9, 13	anbi12d 634	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))) ↔ ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))))
15	14	rspcev 3526	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))) → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))))
16	4, 5, 6, 15	syl12anc 836	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))))
17		coeeq.3	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
18		cnex 10696	. . . . . 6 ⊢ ℂ ∈ V
19		nn0ex 11982	. . . . . 6 ⊢ ℕ0 ∈ V
20	18, 19	elmap 8481	. . . . 5 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
21	17, 20	sylibr 237	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
22		coeeu 24974	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
23	1, 22	syl 17	. . . 4 ⊢ (𝜑 → ∃!𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
24		imaeq1 5898	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑛 + 1))))
25	24	eqeq1d 2740	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0}))
26		fveq1 6673	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
27	26	oveq1d 7185	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑧↑𝑘)))
28	27	sumeq2sdv 15154	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))
29	28	mpteq2dv 5126	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))
30	29	eqeq2d 2749	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))))
31	25, 30	anbi12d 634	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))))
32	31	rexbidv 3207	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))))
33	32	riota2 7153	. . . 4 ⊢ ((𝐴 ∈ (ℂ ↑m ℕ0) ∧ ∃!𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))) ↔ (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) = 𝐴))
34	21, 23, 33	syl2anc 587	. . 3 ⊢ (𝜑 → (∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))) ↔ (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) = 𝐴))
35	16, 34	mpbid 235	. 2 ⊢ (𝜑 → (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) = 𝐴)
36	3, 35	eqtrd 2773	1 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∃wrex 3054  ∃!wreu 3055  {csn 4516   ↦ cmpt 5110   “ cima 5528  ⟶wf 6335  ‘cfv 6339  ℩crio 7126  (class class class)co 7170   ↑_m cmap 8437  ℂcc 10613  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  ℕ₀cn0 11976  ℤ_≥cuz 12324  ...cfz 12981  ↑cexp 13521  Σcsu 15135  Polycply 24933  coeffccoe 24935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-fl 13253  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-rlim 14936  df-sum 15136  df-0p 24422  df-ply 24937  df-coe 24939

This theorem is referenced by:  dgrlem  24978  coeidlem  24986  coeeq2  24991  dgreq  24993  coeaddlem  24998  coemullem  24999  coe1termlem  25007  coecj  25027  basellem2  25819  aacllem  45958