Theorem coeeq 26274

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 26270	. . 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 7413	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑁 + 1)))
8	7	imaeq2d 6044	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐴 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑁 + 1))))
9	8	eqeq1d 2763	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}))
10		oveq2 7398	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
11	10	sumeq1d 15717	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))
12	11	mpteq2dv 5191	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
13	12	eqeq2d 2772	. . . . . 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 641	. . . . 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 847	. . 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 11147	. . . . . 6 ⊢ ℂ ∈ V
19		nn0ex 12480	. . . . . 6 ⊢ ℕ0 ∈ V
20	18, 19	elmap 8846	. . . . 5 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
21	17, 20	sylibr 236	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
22		coeeu 26272	. . . . 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 6039	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝐴 “ (ℤ≥‘(𝑛 + 1))))
25	24	eqeq1d 2763	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ≥‘(𝑛 + 1))) = {0}))
26		fveq1 6860	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
27	26	oveq1d 7405	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑧↑𝑘)))
28	27	sumeq2sdv 15720	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘)))
29	28	mpteq2dv 5191	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝑧↑𝑘))))
30	29	eqeq2d 2772	. . . . . . 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 641	. . . . . 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 3185	. . . . 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 7372	. . . 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 593	. . 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 234	. 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 2796	1 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ∃!wreu 3364  {csn 4579   ↦ cmpt 5178   “ cima 5646  ⟶wf 6511  ‘cfv 6515  ℩crio 7346  (class class class)co 7390   ↑_m cmap 8801  ℂcc 11064  0cc0 11066  1c1 11067   + caddc 11069   · cmul 11071  ℕ₀cn0 12474  ℤ_≥cuz 12832  ...cfz 13505  ↑cexp 14067  Σcsu 15703  Polycply 26231  coeffccoe 26233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-inf2 9589  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7654  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9381  df-inf 9382  df-oi 9451  df-card 9890  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-3 12274  df-n0 12475  df-z 12562  df-uz 12833  df-rp 12987  df-fz 13506  df-fzo 13653  df-fl 13795  df-seq 14008  df-exp 14068  df-hash 14337  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15505  df-rlim 15506  df-sum 15704  df-0p 25719  df-ply 26235  df-coe 26237

This theorem is referenced by:  dgrlem  26276  coeidlem  26284  coeeq2  26289  dgreq  26291  coeaddlem  26296  coemullem  26297  coe1termlem  26305  coecj  26325  coecjOLD  26327  basellem2  27133  aacllem  50382