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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 (𝜑𝐹 ∈ (Poly‘𝑆))
coeeq.2 (𝜑𝑁 ∈ ℕ0)
coeeq.3 (𝜑𝐴:ℕ0⟶ℂ)
coeeq.4 (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})
coeeq.5 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
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.
StepHypRef Expression
1 coeeq.1 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
2 coeval 23783 . . 3 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
31, 2syl 17 . 2 (𝜑 → (coeff‘𝐹) = (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
4 coeeq.2 . . . 4 (𝜑𝑁 ∈ ℕ0)
5 coeeq.4 . . . 4 (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})
6 coeeq.5 . . . 4 (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
7 oveq1 6556 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
87fveq2d 6107 . . . . . . . 8 (𝑛 = 𝑁 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑁 + 1)))
98imaeq2d 5385 . . . . . . 7 (𝑛 = 𝑁 → (𝐴 “ (ℤ‘(𝑛 + 1))) = (𝐴 “ (ℤ‘(𝑁 + 1))))
109eqeq1d 2612 . . . . . 6 (𝑛 = 𝑁 → ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ‘(𝑁 + 1))) = {0}))
11 oveq2 6557 . . . . . . . . 9 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
1211sumeq1d 14279 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘)))
1312mpteq2dv 4673 . . . . . . 7 (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
1413eqeq2d 2620 . . . . . 6 (𝑛 = 𝑁 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘)))))
1510, 14anbi12d 743 . . . . 5 (𝑛 = 𝑁 → (((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))) ↔ ((𝐴 “ (ℤ‘(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))))
1615rspcev 3282 . . . 4 ((𝑁 ∈ ℕ0 ∧ ((𝐴 “ (ℤ‘(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))) → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))))
174, 5, 6, 16syl12anc 1316 . . 3 (𝜑 → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))))
18 coeeq.3 . . . . 5 (𝜑𝐴:ℕ0⟶ℂ)
19 cnex 9896 . . . . . 6 ℂ ∈ V
20 nn0ex 11175 . . . . . 6 0 ∈ V
2119, 20elmap 7772 . . . . 5 (𝐴 ∈ (ℂ ↑𝑚0) ↔ 𝐴:ℕ0⟶ℂ)
2218, 21sylibr 223 . . . 4 (𝜑𝐴 ∈ (ℂ ↑𝑚0))
23 coeeu 23785 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
241, 23syl 17 . . . 4 (𝜑 → ∃!𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
25 imaeq1 5380 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝐴 “ (ℤ‘(𝑛 + 1))))
2625eqeq1d 2612 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ‘(𝑛 + 1))) = {0}))
27 fveq1 6102 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎𝑘) = (𝐴𝑘))
2827oveq1d 6564 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝐴𝑘) · (𝑧𝑘)))
2928sumeq2sdv 14282 . . . . . . . . 9 (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))
3029mpteq2dv 4673 . . . . . . . 8 (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))))
3130eqeq2d 2620 . . . . . . 7 (𝑎 = 𝐴 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))))
3226, 31anbi12d 743 . . . . . 6 (𝑎 = 𝐴 → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))))))
3332rexbidv 3034 . . . . 5 (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))))))
3433riota2 6533 . . . 4 ((𝐴 ∈ (ℂ ↑𝑚0) ∧ ∃!𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))) ↔ (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) = 𝐴))
3522, 24, 34syl2anc 691 . . 3 (𝜑 → (∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))) ↔ (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) = 𝐴))
3617, 35mpbid 221 . 2 (𝜑 → (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) = 𝐴)
373, 36eqtrd 2644 1 (𝜑 → (coeff‘𝐹) = 𝐴)
 Colors of variables: wff setvar class 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
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