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Mirrors > Home > MPE Home > Th. List > taylply | Structured version Visualization version GIF version |
Description: The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.) |
Ref | Expression |
---|---|
taylpfval.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
taylpfval.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
taylpfval.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
taylpfval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
taylpfval.b | ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
taylpfval.t | ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) |
Ref | Expression |
---|---|
taylply | ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | taylpfval.s | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | taylpfval.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | taylpfval.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | taylpfval.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | taylpfval.b | . 2 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
6 | taylpfval.t | . 2 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | |
7 | cnring 19816 | . . 3 ⊢ ℂfld ∈ Ring | |
8 | cnfldbas 19798 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
9 | 8 | subrgid 18830 | . . 3 ⊢ (ℂfld ∈ Ring → ℂ ∈ (SubRing‘ℂfld)) |
10 | 7, 9 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂfld)) |
11 | cnex 10055 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℂ ∈ V) |
13 | elpm2r 7917 | . . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
14 | 12, 1, 2, 3, 13 | syl22anc 1367 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
15 | dvnbss 23736 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) | |
16 | 1, 14, 4, 15 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) |
17 | fdm 6089 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
18 | 2, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
19 | 16, 18 | sseqtrd 3674 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ 𝐴) |
20 | recnprss 23713 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
21 | 1, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
22 | 3, 21 | sstrd 3646 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
23 | 19, 22 | sstrd 3646 | . . 3 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ ℂ) |
24 | 23, 5 | sseldd 3637 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
25 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
26 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
27 | elfznn0 12471 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
28 | 27 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
29 | dvnf 23735 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) | |
30 | 25, 26, 28, 29 | syl3anc 1366 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
31 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁)) | |
32 | dvn2bss 23738 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | |
33 | 25, 26, 31, 32 | syl3anc 1366 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
34 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
35 | 33, 34 | sseldd 3637 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
36 | 30, 35 | ffvelrnd 6400 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
37 | faccl 13110 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
38 | 28, 37 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ) |
39 | 38 | nncnd 11074 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ) |
40 | 38 | nnne0d 11103 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0) |
41 | 36, 39, 40 | divcld 10839 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
42 | 1, 2, 3, 4, 5, 6, 10, 24, 41 | taylply2 24167 | 1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 {cpr 4212 class class class wbr 4685 dom cdm 5143 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↑pm cpm 7900 ℂcc 9972 ℝcr 9973 0cc0 9974 ≤ cle 10113 ℕcn 11058 ℕ0cn0 11330 ...cfz 12364 !cfa 13100 Ringcrg 18593 SubRingcsubrg 18824 ℂfldccnfld 19794 D𝑛 cdvn 23673 Polycply 23985 degcdgr 23988 Tayl ctayl 24152 |
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-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 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-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-fac 13101 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-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-subg 17638 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-subrg 18826 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cnp 21080 df-haus 21167 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-tsms 21977 df-xms 22172 df-ms 22173 df-0p 23482 df-limc 23675 df-dv 23676 df-dvn 23677 df-ply 23989 df-idp 23990 df-coe 23991 df-dgr 23992 df-tayl 24154 |
This theorem is referenced by: (None) |
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