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Mirrors > Home > MPE Home > Th. List > plycn | Structured version Visualization version GIF version |
Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
Ref | Expression |
---|---|
plycn | ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . . 4 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | eqid 2651 | . . . 4 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
3 | 1, 2 | coeid 24039 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)))) |
4 | eqid 2651 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | cnfldtopon 22633 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
7 | fzfid 12812 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (0...(deg‘𝐹)) ∈ Fin) | |
8 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
9 | 1 | coef3 24033 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ) |
10 | elfznn0 12471 | . . . . . . 7 ⊢ (𝑘 ∈ (0...(deg‘𝐹)) → 𝑘 ∈ ℕ0) | |
11 | ffvelrn 6397 | . . . . . . 7 ⊢ (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ) | |
12 | 9, 10, 11 | syl2an 493 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ) |
13 | 8, 8, 12 | cnmptc 21513 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
14 | 10 | adantl 481 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝑘 ∈ ℕ0) |
15 | 4 | expcn 22722 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
17 | 4 | mulcn 22717 | . . . . . 6 ⊢ · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
19 | 8, 13, 16, 18 | cnmpt12f 21517 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝑧 ∈ ℂ ↦ (((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
20 | 4, 6, 7, 19 | fsumcn 22720 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
21 | 3, 20 | eqeltrd 2730 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
22 | 4 | cncfcn1 22760 | . 2 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
23 | 21, 22 | syl6eleqr 2741 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2030 ↦ cmpt 4762 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 · cmul 9979 ℕ0cn0 11330 ...cfz 12364 ↑cexp 12900 Σcsu 14460 TopOpenctopn 16129 ℂfldccnfld 19794 TopOnctopon 20763 Cn ccn 21076 ×t ctx 21411 –cn→ccncf 22726 Polycply 23985 coeffccoe 23987 degcdgr 23988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cn 21079 df-cnp 21080 df-tx 21413 df-hmeo 21606 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-0p 23482 df-ply 23989 df-coe 23991 df-dgr 23992 |
This theorem is referenced by: plycpn 24089 taylthlem2 24173 ftalem3 24846 signsply0 30756 |
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