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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 12434 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | 3 | nn0cnd 11353 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | elfznn0 12433 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
6 | 1, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
7 | 6 | nn0cnd 11353 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
8 | 4, 7 | npcand 10396 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁) |
9 | 8 | oveq1d 6665 | . . . . . . 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 | syl6eqr 2674 | . . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇) |
12 | 11 | oveq2d 6666 | . . . . 5 ⊢ (𝜑 → (ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D𝑛 𝑇)) |
13 | 12 | fveq1d 6193 | . . . 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 12373 | . . . . . 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 6195 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
21 | 20 | dmeqd 5326 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
22 | 19, 21 | eleqtrrd 2704 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀))) |
23 | 14, 15, 16, 6, 18, 22 | dvntaylp 24125 | . . . 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 2658 | . . 3 ⊢ (𝜑 → ((ℂ D𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
25 | 24 | fveq1d 6193 | . 2 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵)) |
26 | cnex 10017 | . . . . . . 7 ⊢ ℂ ∈ V | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℂ ∈ V) |
28 | elpm2r 7875 | . . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
29 | 27, 14, 15, 16, 28 | syl22anc 1327 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
30 | dvnf 23690 | . . . . 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 1326 | . . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) |
32 | dvnbss 23691 | . . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) | |
33 | 14, 29, 6, 32 | syl3anc 1326 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) |
34 | fdm 6051 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
35 | 15, 34 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
36 | 33, 35 | sseqtrd 3641 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝐴) |
37 | 36, 16 | sstrd 3613 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆) |
38 | 18 | orcd 407 | . . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ0 ∨ (𝑁 − 𝑀) = +∞)) |
39 | dvnadd 23692 | . . . . . . . . 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)))) | |
40 | 14, 29, 6, 18, 39 | syl22anc 1327 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) |
41 | 7, 4 | pncan3d 10395 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
42 | 41 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
43 | 40, 42 | eqtrd 2656 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
44 | 43 | dmeqd 5326 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
45 | 19, 44 | eleqtrrd 2704 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀))) |
46 | 14, 31, 37, 18, 45 | taylplem1 24117 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑘)) |
47 | eqid 2622 | . . . 4 ⊢ ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) | |
48 | 14, 31, 37, 38, 46, 47 | tayl0 24116 | . . 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))) |
49 | 48 | simprd 479 | . 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
50 | 25, 49 | eqtrd 2656 | 1 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {cpr 4179 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑pm cpm 7858 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 +∞cpnf 10071 − cmin 10266 ℕ0cn0 11292 ...cfz 12326 D𝑛 cdvn 23628 Tayl ctayl 24107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-dvn 23632 df-tayl 24109 |
This theorem is referenced by: taylthlem1 24127 taylthlem2 24128 |
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