Theorem coeeq 26579

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 26575	. . 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 18	. 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 7469	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑁 + 1)))
8	7	imaeq2d 6065	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐴 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑁 + 1))))
9	8	eqeq1d 2765	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}))
10		oveq2 7454	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
11	10	sumeq1d 15899	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))
12	11	mpteq2dv 5203	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
13	12	eqeq2d 2774	. . . . . 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 644	. . . . 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 3580	. . . 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 850	. . 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 11311	. . . . . 6 ⊢ ℂ ∈ V
19		nn0ex 12642	. . . . . 6 ⊢ ℕ0 ∈ V
20	18, 19	elmap 8930	. . . . 5 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
21	17, 20	sylibr 237	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
22		coeeu 26577	. . . . 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 18	. . . 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 6060	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑛 + 1))))
25	24	eqeq1d 2765	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0}))
26		fveq1 6910	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
27	26	oveq1d 7461	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑧↑𝑘)))
28	27	sumeq2sdv 15902	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))
29	28	mpteq2dv 5203	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))
30	29	eqeq2d 2774	. . . . . . 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 644	. . . . . 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 3190	. . . . 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 7428	. . . 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 596	. . 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 2798	1 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  ∃!wreu 3367  {csn 4588   ↦ cmpt 5190   “ cima 5659  ⟶wf 6557  ‘cfv 6561  ℩crio 7402  (class class class)co 7446   ↑_m cmap 8878  ℂcc 11228  0cc0 11230  1c1 11231   + caddc 11233   · cmul 11235  ℕ₀cn0 12636  ℤ_≥cuz 12995  ...cfz 13669  ↑cexp 14234  Σcsu 15885  Polycply 26535  coeffccoe 26537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-of 7720  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-map 8880  df-pm 8881  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-inf 9465  df-oi 9534  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-z 12724  df-uz 12996  df-rp 13151  df-fz 13670  df-fzo 13819  df-fl 13962  df-seq 14175  df-exp 14235  df-hash 14505  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-clim 15687  df-rlim 15688  df-sum 15886  df-0p 26024  df-ply 26539  df-coe 26541

This theorem is used by:  dgrlem  26581  coeidlem  26589  coeeq2  26594  dgreq  26596  coeaddlem  26601  coemullem  26602  coe1termlem  26610  coecj  26630  basellem2  27442  aacllem  51251