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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 21325 | . . 3 β’ βfld β Ring | |
8 | cnfldbas 21290 | . . . 4 β’ β = (Baseββfld) | |
9 | 8 | subrgid 20519 | . . 3 β’ (βfld β Ring β β β (SubRingββfld)) |
10 | 7, 9 | mp1i 13 | . 2 β’ (π β β β (SubRingββfld)) |
11 | cnex 11227 | . . . . . . . 8 β’ β β V | |
12 | 11 | a1i 11 | . . . . . . 7 β’ (π β β β V) |
13 | elpm2r 8870 | . . . . . . 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 837 | . . . . . 6 β’ (π β πΉ β (β βpm π)) |
15 | dvnbss 25878 | . . . . . 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 1368 | . . . . 5 β’ (π β dom ((π Dπ πΉ)βπ) β dom πΉ) |
17 | 2, 16 | fssdmd 6746 | . . . 4 β’ (π β dom ((π Dπ πΉ)βπ) β π΄) |
18 | recnprss 25853 | . . . . . 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 3992 | . . . 4 β’ (π β π΄ β β) |
21 | 17, 20 | sstrd 3992 | . . 3 β’ (π β dom ((π Dπ πΉ)βπ) β β) |
22 | 21, 5 | sseldd 3983 | . 2 β’ (π β π΅ β β) |
23 | 1 | adantr 479 | . . . . 5 β’ ((π β§ π β (0...π)) β π β {β, β}) |
24 | 14 | adantr 479 | . . . . 5 β’ ((π β§ π β (0...π)) β πΉ β (β βpm π)) |
25 | elfznn0 13634 | . . . . . 6 β’ (π β (0...π) β π β β0) | |
26 | 25 | adantl 480 | . . . . 5 β’ ((π β§ π β (0...π)) β π β β0) |
27 | dvnf 25877 | . . . . 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 1368 | . . . 4 β’ ((π β§ π β (0...π)) β ((π Dπ πΉ)βπ):dom ((π Dπ πΉ)βπ)βΆβ) |
29 | simpr 483 | . . . . . 6 β’ ((π β§ π β (0...π)) β π β (0...π)) | |
30 | dvn2bss 25880 | . . . . . 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 1368 | . . . . 5 β’ ((π β§ π β (0...π)) β dom ((π Dπ πΉ)βπ) β dom ((π Dπ πΉ)βπ)) |
32 | 5 | adantr 479 | . . . . 5 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
33 | 31, 32 | sseldd 3983 | . . . 4 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
34 | 28, 33 | ffvelcdmd 7100 | . . 3 β’ ((π β§ π β (0...π)) β (((π Dπ πΉ)βπ)βπ΅) β β) |
35 | 26 | faccld 14283 | . . . 4 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
36 | 35 | nncnd 12266 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
37 | 35 | nnne0d 12300 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β 0) |
38 | 34, 36, 37 | divcld 12028 | . 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 26322 | 1 β’ (π β (π β (Polyββ) β§ (degβπ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 β wss 3949 {cpr 4634 class class class wbr 5152 dom cdm 5682 βΆwf 6549 βcfv 6553 (class class class)co 7426 βpm cpm 8852 βcc 11144 βcr 11145 0cc0 11146 β€ cle 11287 β0cn0 12510 ...cfz 13524 !cfa 14272 Ringcrg 20180 SubRingcsubrg 20513 βfldccnfld 21286 Dπ cdvn 25813 Polycply 26138 degcdgr 26141 Tayl ctayl 26307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20490 df-subrg 20515 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cnp 23152 df-haus 23239 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-tsms 24051 df-xms 24246 df-ms 24247 df-0p 25619 df-limc 25815 df-dv 25816 df-dvn 25817 df-ply 26142 df-idp 26143 df-coe 26144 df-dgr 26145 df-tayl 26309 |
This theorem is referenced by: (None) |
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