Theorem coeeq 26189

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 26185	. . 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 7433	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑁 + 1)))
8	7	imaeq2d 6052	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐴 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑁 + 1))))
9	8	eqeq1d 2738	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}))
10		oveq2 7418	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
11	10	sumeq1d 15721	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))
12	11	mpteq2dv 5220	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
13	12	eqeq2d 2747	. . . . . 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 632	. . . . 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 3606	. . . 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 11215	. . . . . 6 ⊢ ℂ ∈ V
19		nn0ex 12512	. . . . . 6 ⊢ ℕ0 ∈ V
20	18, 19	elmap 8890	. . . . 5 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
21	17, 20	sylibr 234	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
22		coeeu 26187	. . . . 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 6047	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑛 + 1))))
25	24	eqeq1d 2738	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0}))
26		fveq1 6880	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
27	26	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑧↑𝑘)))
28	27	sumeq2sdv 15724	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))
29	28	mpteq2dv 5220	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))
30	29	eqeq2d 2747	. . . . . . 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 632	. . . . . 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 3165	. . . . 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 7392	. . . 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 584	. . 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 232	. 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 2771	1 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  ∃!wreu 3362  {csn 4606   ↦ cmpt 5206   “ cima 5662  ⟶wf 6532  ‘cfv 6536  ℩crio 7366  (class class class)co 7410   ↑_m cmap 8845  ℂcc 11132  0cc0 11134  1c1 11135   + caddc 11137   · cmul 11139  ℕ₀cn0 12506  ℤ_≥cuz 12857  ...cfz 13529  ↑cexp 14084  Σcsu 15707  Polycply 26146  coeffccoe 26148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-fl 13814  df-seq 14025  df-exp 14085  df-hash 14354  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-clim 15509  df-rlim 15510  df-sum 15708  df-0p 25628  df-ply 26150  df-coe 26152

This theorem is referenced by:  dgrlem  26191  coeidlem  26199  coeeq2  26204  dgreq  26206  coeaddlem  26211  coemullem  26212  coe1termlem  26220  coecj  26241  coecjOLD  26243  basellem2  27049  aacllem  49632