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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 2824 | . . . 4 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | eqid 2824 | . . . 4 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
3 | 1, 2 | coeid 24831 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)))) |
4 | eqid 2824 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | cnfldtopon 23394 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
7 | fzfid 13344 | . . . 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 24825 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ) |
10 | elfznn0 13003 | . . . . . . 7 ⊢ (𝑘 ∈ (0...(deg‘𝐹)) → 𝑘 ∈ ℕ0) | |
11 | ffvelrn 6852 | . . . . . . 7 ⊢ (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ) | |
12 | 9, 10, 11 | syl2an 597 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ) |
13 | 8, 8, 12 | cnmptc 22273 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘𝑘)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
14 | 10 | adantl 484 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ (0...(deg‘𝐹))) → 𝑘 ∈ ℕ0) |
15 | 4 | expcn 23483 | . . . . . 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 23478 | . . . . . 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 22277 | . . . 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 23481 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
21 | 3, 20 | eqeltrd 2916 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
22 | 4 | cncfcn1 23521 | . 2 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) |
23 | 21, 22 | eleqtrrdi 2927 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 · cmul 10545 ℕ0cn0 11900 ...cfz 12895 ↑cexp 13432 Σcsu 15045 TopOpenctopn 16698 ℂfldccnfld 20548 TopOnctopon 21521 Cn ccn 21835 ×t ctx 22171 –cn→ccncf 23487 Polycply 24777 coeffccoe 24779 degcdgr 24780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cn 21838 df-cnp 21839 df-tx 22173 df-hmeo 22366 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-0p 24274 df-ply 24781 df-coe 24783 df-dgr 24784 |
This theorem is referenced by: plycpn 24881 taylthlem2 24965 ftalem3 25655 signsply0 31825 |
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