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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 13002 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | 3 | nn0cnd 11958 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | elfznn0 13001 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ0) | |
6 | 1, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
7 | 6 | nn0cnd 11958 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
8 | 4, 7 | npcand 11001 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁) |
9 | 8 | oveq1d 7171 | . . . . . . 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 2874 | . . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇) |
12 | 11 | oveq2d 7172 | . . . . 5 ⊢ (𝜑 → (ℂ D𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D𝑛 𝑇)) |
13 | 12 | fveq1d 6672 | . . . 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 12940 | . . . . . 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 6674 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
21 | 20 | dmeqd 5774 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
22 | 19, 21 | eleqtrrd 2916 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀))) |
23 | 14, 15, 16, 6, 18, 22 | dvntaylp 24959 | . . . 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 2858 | . . 3 ⊢ (𝜑 → ((ℂ D𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
25 | 24 | fveq1d 6672 | . 2 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵)) |
26 | cnex 10618 | . . . . . . 7 ⊢ ℂ ∈ V | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℂ ∈ V) |
28 | elpm2r 8424 | . . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
29 | 27, 14, 15, 16, 28 | syl22anc 836 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
30 | dvnf 24524 | . . . . 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 1367 | . . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑀):dom ((𝑆 D𝑛 𝐹)‘𝑀)⟶ℂ) |
32 | dvnbss 24525 | . . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) | |
33 | 14, 29, 6, 32 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ dom 𝐹) |
34 | 15, 33 | fssdmd 6529 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝐴) |
35 | 34, 16 | sstrd 3977 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑀) ⊆ 𝑆) |
36 | 18 | orcd 869 | . . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ0 ∨ (𝑁 − 𝑀) = +∞)) |
37 | dvnadd 24526 | . . . . . . . . 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 836 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀)))) |
39 | 7, 4 | pncan3d 11000 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁) |
40 | 39 | fveq2d 6674 | . . . . . . . 8 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
41 | 38, 40 | eqtrd 2856 | . . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
42 | 41 | dmeqd 5774 | . . . . . 6 ⊢ (𝜑 → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
43 | 19, 42 | eleqtrrd 2916 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀))) |
44 | 14, 31, 35, 18, 43 | taylplem1 24951 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑘)) |
45 | eqid 2821 | . . . 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 24950 | . . 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 498 | . 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
48 | 25, 47 | eqtrd 2856 | 1 ⊢ (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℂcc 10535 ℝcr 10536 0cc0 10537 + caddc 10540 +∞cpnf 10672 − cmin 10870 ℕ0cn0 11898 ...cfz 12893 D𝑛 cdvn 24462 Tayl ctayl 24941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-dvn 24466 df-tayl 24943 |
This theorem is referenced by: taylthlem1 24961 taylthlem2 24962 |
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