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Theorem coeeq 24976
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 24972 . . 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 fvoveq1 7193 . . . . . . . 8 (𝑛 = 𝑁 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑁 + 1)))
87imaeq2d 5903 . . . . . . 7 (𝑛 = 𝑁 → (𝐴 “ (ℤ‘(𝑛 + 1))) = (𝐴 “ (ℤ‘(𝑁 + 1))))
98eqeq1d 2740 . . . . . 6 (𝑛 = 𝑁 → ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ‘(𝑁 + 1))) = {0}))
10 oveq2 7178 . . . . . . . . 9 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
1110sumeq1d 15151 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘)))
1211mpteq2dv 5126 . . . . . . 7 (𝑛 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
1312eqeq2d 2749 . . . . . 6 (𝑛 = 𝑁 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘)))))
149, 13anbi12d 634 . . . . 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))))))
1514rspcev 3526 . . . 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)))))
164, 5, 6, 15syl12anc 836 . . 3 (𝜑 → ∃𝑛 ∈ ℕ0 ((𝐴 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))))
17 coeeq.3 . . . . 5 (𝜑𝐴:ℕ0⟶ℂ)
18 cnex 10696 . . . . . 6 ℂ ∈ V
19 nn0ex 11982 . . . . . 6 0 ∈ V
2018, 19elmap 8481 . . . . 5 (𝐴 ∈ (ℂ ↑m0) ↔ 𝐴:ℕ0⟶ℂ)
2117, 20sylibr 237 . . . 4 (𝜑𝐴 ∈ (ℂ ↑m0))
22 coeeu 24974 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
231, 22syl 17 . . . 4 (𝜑 → ∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
24 imaeq1 5898 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝐴 “ (ℤ‘(𝑛 + 1))))
2524eqeq1d 2740 . . . . . . 7 (𝑎 = 𝐴 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝐴 “ (ℤ‘(𝑛 + 1))) = {0}))
26 fveq1 6673 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑎𝑘) = (𝐴𝑘))
2726oveq1d 7185 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝐴𝑘) · (𝑧𝑘)))
2827sumeq2sdv 15154 . . . . . . . . 9 (𝑎 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))
2928mpteq2dv 5126 . . . . . . . 8 (𝑎 = 𝐴 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘))))
3029eqeq2d 2749 . . . . . . 7 (𝑎 = 𝐴 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝑧𝑘)))))
3125, 30anbi12d 634 . . . . . 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))))))
3231rexbidv 3207 . . . . 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))))))
3332riota2 7153 . . . 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))
3421, 23, 33syl2anc 587 . . 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))
3516, 34mpbid 235 . 2 (𝜑 → (𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) = 𝐴)
363, 35eqtrd 2773 1 (𝜑 → (coeff‘𝐹) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  ∃!wreu 3055  {csn 4516  cmpt 5110  cima 5528  wf 6335  cfv 6339  crio 7126  (class class class)co 7170  m cmap 8437  cc 10613  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  0cn0 11976  cuz 12324  ...cfz 12981  cexp 13521  Σcsu 15135  Polycply 24933  coeffccoe 24935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-fl 13253  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-rlim 14936  df-sum 15136  df-0p 24422  df-ply 24937  df-coe 24939
This theorem is referenced by:  dgrlem  24978  coeidlem  24986  coeeq2  24991  dgreq  24993  coeaddlem  24998  coemullem  24999  coe1termlem  25007  coecj  25027  basellem2  25819  aacllem  45958
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