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Mirrors > Home > MPE Home > Th. List > dvntaylp0 | Structured version Visualization version GIF version |
Description: The first 𝑁 derivatives of the Taylor polynomial at 𝐵 match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
dvntaylp0.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvntaylp0.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
dvntaylp0.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
dvntaylp0.m | ⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) |
dvntaylp0.b | ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
dvntaylp0.t | ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) |
Ref | Expression |
---|---|
dvntaylp0 | ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvntaylp0.m | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) | |
2 | elfz3nn0 13577 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | 3 | nn0cnd 12516 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | elfznn0 13576 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
6 | 1, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
7 | 6 | nn0cnd 12516 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
8 | 4, 7 | npcand 11557 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁) |
9 | 8 | oveq1d 7408 | . . . . . . 7 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵)) |
10 | dvntaylp0.t | . . . . . . 7 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | |
11 | 9, 10 | eqtr4di 2789 | . . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇) |
12 | 11 | oveq2d 7409 | . . . . 5 ⊢ (𝜑 → (ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D𝑛 𝑇)) |
13 | 12 | fveq1d 6880 | . . . 4 ⊢ (𝜑 → ((ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((ℂ D𝑛 𝑇)‘𝑀)) |
14 | dvntaylp0.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
15 | dvntaylp0.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
16 | dvntaylp0.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
17 | fznn0sub 13515 | . . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ0) | |
18 | 1, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ0) |
19 | dvntaylp0.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
20 | 8 | fveq2d 6882 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
21 | 20 | dmeqd 5897 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
22 | 19, 21 | eleqtrrd 2835 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀))) |
23 | 14, 15, 16, 6, 18, 22 | dvntaylp 25812 | . . . 4 ⊢ (𝜑 → ((ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
24 | 13, 23 | eqtr3d 2773 | . . 3 ⊢ (𝜑 → ((ℂ D𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
25 | 24 | fveq1d 6880 | . 2 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵)) |
26 | cnex 11173 | . . . . . . 7 ⊢ ℂ ∈ V | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℂ ∈ V) |
28 | elpm2r 8822 | . . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
29 | 27, 14, 15, 16, 28 | syl22anc 837 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
30 | dvnf 25373 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) | |
31 | 14, 29, 6, 30 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) |
32 | dvnbss 25374 | . . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) | |
33 | 14, 29, 6, 32 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) |
34 | 15, 33 | fssdmd 6723 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝐴) |
35 | 34, 16 | sstrd 3988 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆) |
36 | 18 | orcd 871 | . . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ0 ∨ (𝑁 − 𝑀) = +∞)) |
37 | dvnadd 25375 | . . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑀 ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) | |
38 | 14, 29, 6, 18, 37 | syl22anc 837 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) |
39 | 7, 4 | pncan3d 11556 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
40 | 39 | fveq2d 6882 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
41 | 38, 40 | eqtrd 2771 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
42 | 41 | dmeqd 5897 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
43 | 19, 42 | eleqtrrd 2835 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀))) |
44 | 14, 31, 35, 18, 43 | taylplem1 25804 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑘)) |
45 | eqid 2731 | . . . 4 ⊢ ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) | |
46 | 14, 31, 35, 36, 44, 45 | tayl0 25803 | . . 3 ⊢ (𝜑 → (𝐵 ∈ dom ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) ∧ (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵))) |
47 | 46 | simprd 496 | . 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
48 | 25, 47 | eqtrd 2771 | 1 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ⊆ wss 3944 {cpr 4624 dom cdm 5669 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ↑pm cpm 8804 ℂcc 11090 ℝcr 11091 0cc0 11092 + caddc 11095 +∞cpnf 11227 − cmin 11426 ℕ0cn0 12454 ...cfz 13466 D𝑛 cdvn 25310 Tayl ctayl 25794 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-xnn0 12527 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-icc 13313 df-fz 13467 df-fzo 13610 df-seq 13949 df-exp 14010 df-fac 14216 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-sum 15615 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-lp 22569 df-perf 22570 df-cn 22660 df-cnp 22661 df-haus 22748 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-tsms 23560 df-xms 23755 df-ms 23756 df-tms 23757 df-cncf 24323 df-limc 25312 df-dv 25313 df-dvn 25314 df-tayl 25796 |
This theorem is referenced by: taylthlem1 25814 taylthlem2 25815 |
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