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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 20567 | . . 3 ⊢ ℂfld ∈ Ring | |
8 | cnfldbas 20549 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
9 | 8 | subrgid 19537 | . . 3 ⊢ (ℂfld ∈ Ring → ℂ ∈ (SubRing‘ℂfld)) |
10 | 7, 9 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂfld)) |
11 | cnex 10618 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℂ ∈ V) |
13 | elpm2r 8424 | . . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
14 | 12, 1, 2, 3, 13 | syl22anc 836 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
15 | dvnbss 24525 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) | |
16 | 1, 14, 4, 15 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) |
17 | 2, 16 | fssdmd 6529 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ 𝐴) |
18 | recnprss 24502 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
19 | 1, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
20 | 3, 19 | sstrd 3977 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
21 | 17, 20 | sstrd 3977 | . . 3 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ ℂ) |
22 | 21, 5 | sseldd 3968 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
23 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
24 | 14 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
25 | elfznn0 13001 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
27 | dvnf 24524 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) | |
28 | 23, 24, 26, 27 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
29 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁)) | |
30 | dvn2bss 24527 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | |
31 | 23, 24, 29, 30 | syl3anc 1367 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
32 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
33 | 31, 32 | sseldd 3968 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
34 | 28, 33 | ffvelrnd 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
35 | 26 | faccld 13645 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ) |
36 | 35 | nncnd 11654 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ) |
37 | 35 | nnne0d 11688 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0) |
38 | 34, 36, 37 | divcld 11416 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
39 | 1, 2, 3, 4, 5, 6, 10, 22, 38 | taylply2 24956 | 1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℂcc 10535 ℝcr 10536 0cc0 10537 ≤ cle 10676 ℕ0cn0 11898 ...cfz 12893 !cfa 13634 Ringcrg 19297 SubRingcsubrg 19531 ℂfldccnfld 20545 D𝑛 cdvn 24462 Polycply 24774 degcdgr 24777 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-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 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-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-fl 13163 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-rlim 14846 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-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 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-cnp 21836 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-xms 22930 df-ms 22931 df-0p 24271 df-limc 24464 df-dv 24465 df-dvn 24466 df-ply 24778 df-idp 24779 df-coe 24780 df-dgr 24781 df-tayl 24943 |
This theorem is referenced by: (None) |
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