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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 12303 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | 3 | nn0cnd 11230 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | elfznn0 12302 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
6 | 1, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
7 | 6 | nn0cnd 11230 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
8 | 4, 7 | npcand 10275 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁) |
9 | 8 | oveq1d 6564 | . . . . . . 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 2662 | . . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇) |
12 | 11 | oveq2d 6565 | . . . . 5 ⊢ (𝜑 → (ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D𝑛 𝑇)) |
13 | 12 | fveq1d 6105 | . . . 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 12244 | . . . . . 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 6107 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
21 | 20 | dmeqd 5248 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
22 | 19, 21 | eleqtrrd 2691 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀))) |
23 | 14, 15, 16, 6, 18, 22 | dvntaylp 23929 | . . . 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 2646 | . . 3 ⊢ (𝜑 → ((ℂ D𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
25 | 24 | fveq1d 6105 | . 2 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵)) |
26 | cnex 9896 | . . . . . . 7 ⊢ ℂ ∈ V | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℂ ∈ V) |
28 | elpm2r 7761 | . . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
29 | 27, 14, 15, 16, 28 | syl22anc 1319 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
30 | dvnf 23496 | . . . . 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 1318 | . . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) |
32 | dvnbss 23497 | . . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) | |
33 | 14, 29, 6, 32 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) |
34 | fdm 5964 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
35 | 15, 34 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
36 | 33, 35 | sseqtrd 3604 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝐴) |
37 | 36, 16 | sstrd 3578 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆) |
38 | 18 | orcd 406 | . . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ0 ∨ (𝑁 − 𝑀) = +∞)) |
39 | dvnadd 23498 | . . . . . . . . 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 1319 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) |
41 | 7, 4 | pncan3d 10274 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
42 | 41 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
43 | 40, 42 | eqtrd 2644 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
44 | 43 | dmeqd 5248 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
45 | 19, 44 | eleqtrrd 2691 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀))) |
46 | 14, 31, 37, 18, 45 | taylplem1 23921 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑘)) |
47 | eqid 2610 | . . . 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 23920 | . . 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 478 | . 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
50 | 25, 49 | eqtrd 2644 | 1 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {cpr 4127 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑pm cpm 7745 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 +∞cpnf 9950 − cmin 10145 ℕ0cn0 11169 ...cfz 12197 D𝑛 cdvn 23434 Tayl ctayl 23911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tsms 21740 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-dvn 23438 df-tayl 23913 |
This theorem is referenced by: taylthlem1 23931 taylthlem2 23932 |
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