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Theorem coeeq 25732
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 (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜π‘†))
coeeq.2 (πœ‘ β†’ 𝑁 ∈ β„•0)
coeeq.3 (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)
coeeq.4 (πœ‘ β†’ (𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0})
coeeq.5 (πœ‘ β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
Assertion
Ref Expression
coeeq (πœ‘ β†’ (coeffβ€˜πΉ) = 𝐴)
Distinct variable groups:   𝑧,π‘˜,𝐴   π‘˜,𝑁,𝑧
Allowed substitution hints:   πœ‘(𝑧,π‘˜)   𝑆(𝑧,π‘˜)   𝐹(𝑧,π‘˜)
Proof of Theorem coeeq
Dummy variables π‘Ž 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coeeq.1 . . 3 (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜π‘†))
2 coeval 25728 . . 3 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜πΉ) = (β„©π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
31, 2syl 17 . 2 (πœ‘ β†’ (coeffβ€˜πΉ) = (β„©π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
4 coeeq.2 . . . 4 (πœ‘ β†’ 𝑁 ∈ β„•0)
5 coeeq.4 . . . 4 (πœ‘ β†’ (𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0})
6 coeeq.5 . . . 4 (πœ‘ β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
7 fvoveq1 7428 . . . . . . . 8 (𝑛 = 𝑁 β†’ (β„€β‰₯β€˜(𝑛 + 1)) = (β„€β‰₯β€˜(𝑁 + 1)))
87imaeq2d 6057 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = (𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))))
98eqeq1d 2734 . . . . . 6 (𝑛 = 𝑁 β†’ ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ↔ (𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0}))
10 oveq2 7413 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (0...𝑛) = (0...𝑁))
1110sumeq1d 15643 . . . . . . . 8 (𝑛 = 𝑁 β†’ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)) = Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
1211mpteq2dv 5249 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
1312eqeq2d 2743 . . . . . 6 (𝑛 = 𝑁 β†’ (𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ↔ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
149, 13anbi12d 631 . . . . 5 (𝑛 = 𝑁 β†’ (((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ↔ ((𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
1514rspcev 3612 . . . 4 ((𝑁 ∈ β„•0 ∧ ((𝐴 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))) β†’ βˆƒπ‘› ∈ β„•0 ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
164, 5, 6, 15syl12anc 835 . . 3 (πœ‘ β†’ βˆƒπ‘› ∈ β„•0 ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
17 coeeq.3 . . . . 5 (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)
18 cnex 11187 . . . . . 6 β„‚ ∈ V
19 nn0ex 12474 . . . . . 6 β„•0 ∈ V
2018, 19elmap 8861 . . . . 5 (𝐴 ∈ (β„‚ ↑m β„•0) ↔ 𝐴:β„•0βŸΆβ„‚)
2117, 20sylibr 233 . . . 4 (πœ‘ β†’ 𝐴 ∈ (β„‚ ↑m β„•0))
22 coeeu 25730 . . . . 5 (𝐹 ∈ (Polyβ€˜π‘†) β†’ βˆƒ!π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
231, 22syl 17 . . . 4 (πœ‘ β†’ βˆƒ!π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
24 imaeq1 6052 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = (𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))))
2524eqeq1d 2734 . . . . . . 7 (π‘Ž = 𝐴 β†’ ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ↔ (𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0}))
26 fveq1 6887 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ (π‘Žβ€˜π‘˜) = (π΄β€˜π‘˜))
2726oveq1d 7420 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)) = ((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
2827sumeq2sdv 15646 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)) = Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
2928mpteq2dv 5249 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
3029eqeq2d 2743 . . . . . . 7 (π‘Ž = 𝐴 β†’ (𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ↔ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
3125, 30anbi12d 631 . . . . . 6 (π‘Ž = 𝐴 β†’ (((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ↔ ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
3231rexbidv 3178 . . . . 5 (π‘Ž = 𝐴 β†’ (βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ↔ βˆƒπ‘› ∈ β„•0 ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
3332riota2 7387 . . . 4 ((𝐴 ∈ (β„‚ ↑m β„•0) ∧ βˆƒ!π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) β†’ (βˆƒπ‘› ∈ β„•0 ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ↔ (β„©π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = 𝐴))
3421, 23, 33syl2anc 584 . . 3 (πœ‘ β†’ (βˆƒπ‘› ∈ β„•0 ((𝐴 β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ↔ (β„©π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = 𝐴))
3516, 34mpbid 231 . 2 (πœ‘ β†’ (β„©π‘Ž ∈ (β„‚ ↑m β„•0)βˆƒπ‘› ∈ β„•0 ((π‘Ž β€œ (β„€β‰₯β€˜(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = 𝐴)
363, 35eqtrd 2772 1 (πœ‘ β†’ (coeffβ€˜πΉ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  βˆƒ!wreu 3374  {csn 4627   ↦ cmpt 5230   β€œ cima 5678  βŸΆwf 6536  β€˜cfv 6540  β„©crio 7360  (class class class)co 7405   ↑m cmap 8816  β„‚cc 11104  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111  β„•0cn0 12468  β„€β‰₯cuz 12818  ...cfz 13480  β†‘cexp 14023  Ξ£csu 15628  Polycply 25689  coeffccoe 25691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-0p 25178  df-ply 25693  df-coe 25695
This theorem is referenced by:  dgrlem  25734  coeidlem  25742  coeeq2  25747  dgreq  25749  coeaddlem  25754  coemullem  25755  coe1termlem  25763  coecj  25783  basellem2  26575  aacllem  47801
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