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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) π. (Contributed by Mario Carneiro, 31-Dec-2016.) |
Ref | Expression |
---|---|
taylpfval.s | β’ (π β π β {β, β}) |
taylpfval.f | β’ (π β πΉ:π΄βΆβ) |
taylpfval.a | β’ (π β π΄ β π) |
taylpfval.n | β’ (π β π β β0) |
taylpfval.b | β’ (π β π΅ β dom ((π Dπ πΉ)βπ)) |
taylpfval.t | β’ π = (π(π Tayl πΉ)π΅) |
Ref | Expression |
---|---|
taylply | β’ (π β (π β (Polyββ) β§ (degβπ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | taylpfval.s | . 2 β’ (π β π β {β, β}) | |
2 | taylpfval.f | . 2 β’ (π β πΉ:π΄βΆβ) | |
3 | taylpfval.a | . 2 β’ (π β π΄ β π) | |
4 | taylpfval.n | . 2 β’ (π β π β β0) | |
5 | taylpfval.b | . 2 β’ (π β π΅ β dom ((π Dπ πΉ)βπ)) | |
6 | taylpfval.t | . 2 β’ π = (π(π Tayl πΉ)π΅) | |
7 | cnring 20835 | . . 3 β’ βfld β Ring | |
8 | cnfldbas 20816 | . . . 4 β’ β = (Baseββfld) | |
9 | 8 | subrgid 20240 | . . 3 β’ (βfld β Ring β β β (SubRingββfld)) |
10 | 7, 9 | mp1i 13 | . 2 β’ (π β β β (SubRingββfld)) |
11 | cnex 11139 | . . . . . . . 8 β’ β β V | |
12 | 11 | a1i 11 | . . . . . . 7 β’ (π β β β V) |
13 | elpm2r 8790 | . . . . . . 7 β’ (((β β V β§ π β {β, β}) β§ (πΉ:π΄βΆβ β§ π΄ β π)) β πΉ β (β βpm π)) | |
14 | 12, 1, 2, 3, 13 | syl22anc 838 | . . . . . 6 β’ (π β πΉ β (β βpm π)) |
15 | dvnbss 25308 | . . . . . 6 β’ ((π β {β, β} β§ πΉ β (β βpm π) β§ π β β0) β dom ((π Dπ πΉ)βπ) β dom πΉ) | |
16 | 1, 14, 4, 15 | syl3anc 1372 | . . . . 5 β’ (π β dom ((π Dπ πΉ)βπ) β dom πΉ) |
17 | 2, 16 | fssdmd 6692 | . . . 4 β’ (π β dom ((π Dπ πΉ)βπ) β π΄) |
18 | recnprss 25284 | . . . . . 6 β’ (π β {β, β} β π β β) | |
19 | 1, 18 | syl 17 | . . . . 5 β’ (π β π β β) |
20 | 3, 19 | sstrd 3959 | . . . 4 β’ (π β π΄ β β) |
21 | 17, 20 | sstrd 3959 | . . 3 β’ (π β dom ((π Dπ πΉ)βπ) β β) |
22 | 21, 5 | sseldd 3950 | . 2 β’ (π β π΅ β β) |
23 | 1 | adantr 482 | . . . . 5 β’ ((π β§ π β (0...π)) β π β {β, β}) |
24 | 14 | adantr 482 | . . . . 5 β’ ((π β§ π β (0...π)) β πΉ β (β βpm π)) |
25 | elfznn0 13541 | . . . . . 6 β’ (π β (0...π) β π β β0) | |
26 | 25 | adantl 483 | . . . . 5 β’ ((π β§ π β (0...π)) β π β β0) |
27 | dvnf 25307 | . . . . 5 β’ ((π β {β, β} β§ πΉ β (β βpm π) β§ π β β0) β ((π Dπ πΉ)βπ):dom ((π Dπ πΉ)βπ)βΆβ) | |
28 | 23, 24, 26, 27 | syl3anc 1372 | . . . 4 β’ ((π β§ π β (0...π)) β ((π Dπ πΉ)βπ):dom ((π Dπ πΉ)βπ)βΆβ) |
29 | simpr 486 | . . . . . 6 β’ ((π β§ π β (0...π)) β π β (0...π)) | |
30 | dvn2bss 25310 | . . . . . 6 β’ ((π β {β, β} β§ πΉ β (β βpm π) β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) | |
31 | 23, 24, 29, 30 | syl3anc 1372 | . . . . 5 β’ ((π β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) |
32 | 5 | adantr 482 | . . . . 5 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
33 | 31, 32 | sseldd 3950 | . . . 4 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
34 | 28, 33 | ffvelcdmd 7041 | . . 3 β’ ((π β§ π β (0...π)) β (((π Dπ πΉ)βπ)βπ΅) β β) |
35 | 26 | faccld 14191 | . . . 4 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
36 | 35 | nncnd 12176 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
37 | 35 | nnne0d 12210 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β 0) |
38 | 34, 36, 37 | divcld 11938 | . 2 β’ ((π β§ π β (0...π)) β ((((π Dπ πΉ)βπ)βπ΅) / (!βπ)) β β) |
39 | 1, 2, 3, 4, 5, 6, 10, 22, 38 | taylply2 25743 | 1 β’ (π β (π β (Polyββ) β§ (degβπ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β wss 3915 {cpr 4593 class class class wbr 5110 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 βpm cpm 8773 βcc 11056 βcr 11057 0cc0 11058 β€ cle 11197 β0cn0 12420 ...cfz 13431 !cfa 14180 Ringcrg 19971 SubRingcsubrg 20234 βfldccnfld 20812 Dπ cdvn 25244 Polycply 25561 degcdgr 25564 Tayl ctayl 25728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-fac 14181 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-subg 18932 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-subrg 20236 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cnp 22595 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 df-xms 23689 df-ms 23690 df-0p 25050 df-limc 25246 df-dv 25247 df-dvn 25248 df-ply 25565 df-idp 25566 df-coe 25567 df-dgr 25568 df-tayl 25730 |
This theorem is referenced by: (None) |
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