Theorem coeeq 23787

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 23783	. . 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		oveq1 6556	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
8	7	fveq2d 6107	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑁 + 1)))
9	8	imaeq2d 5385	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐴 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑁 + 1))))
10	9	eqeq1d 2612	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}))
11		oveq2 6557	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
12	11	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))
13	12	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
14	13	eqeq2d 2620	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))))
15	10, 14	anbi12d 743	. . . . 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))))))
16	15	rspcev 3282	. . . 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)))))
17	4, 5, 6, 16	syl12anc 1316	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))))
18		coeeq.3	. . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
19		cnex 9896	. . . . . 6 ⊢ ℂ ∈ V
20		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
21	19, 20	elmap 7772	. . . . 5 ⊢ (𝐴 ∈ (ℂ ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
22	18, 21	sylibr 223	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑𝑚 ℕ0))
23		coeeu 23785	. . . . 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)))))
24	1, 23	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)))))
25		imaeq1 5380	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑛 + 1))))
26	25	eqeq1d 2612	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0}))
27		fveq1 6102	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
28	27	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑧↑𝑘)))
29	28	sumeq2sdv 14282	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))
30	29	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))
31	30	eqeq2d 2620	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))))
32	26, 31	anbi12d 743	. . . . . 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))))))
33	32	rexbidv 3034	. . . . 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))))))
34	33	riota2 6533	. . . 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))
35	22, 24, 34	syl2anc 691	. . 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))
36	17, 35	mpbid 221	. 2 ⊢ (𝜑 → (℩𝑎 ∈ (ℂ ↑𝑚 ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) = 𝐴)
37	3, 36	eqtrd 2644	1 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∃!wreu 2898  {csn 4125   ↦ cmpt 4643   “ cima 5041  ⟶wf 5800  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ↑_𝑚 cmap 7744  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744  coeffccoe 23746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750

This theorem is referenced by:  dgrlem  23789  coeidlem  23797  coeeq2  23802  dgreq  23804  coeaddlem  23809  coemullem  23810  coe1termlem  23818  coecj  23838  basellem2  24608  aacllem  42356