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Mirrors > Home > MPE Home > Th. List > taylpf | Structured version Visualization version GIF version |
Description: The Taylor polynomial is a function on the complex numbers (even if the base set of the original function is the reals). (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 |
---|---|
taylpf | β’ (π β π:ββΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | taylpfval.s | . . 3 β’ (π β π β {β, β}) | |
2 | taylpfval.f | . . 3 β’ (π β πΉ:π΄βΆβ) | |
3 | taylpfval.a | . . 3 β’ (π β π΄ β π) | |
4 | taylpfval.n | . . 3 β’ (π β π β β0) | |
5 | taylpfval.b | . . 3 β’ (π β π΅ β dom ((π Dπ πΉ)βπ)) | |
6 | taylpfval.t | . . 3 β’ π = (π(π Tayl πΉ)π΅) | |
7 | 1, 2, 3, 4, 5, 6 | taylpfval 26312 | . 2 β’ (π β π = (π₯ β β β¦ Ξ£π β (0...π)(((((π Dπ πΉ)βπ)βπ΅) / (!βπ)) Β· ((π₯ β π΅)βπ)))) |
8 | fzfid 13971 | . . 3 β’ ((π β§ π₯ β β) β (0...π) β Fin) | |
9 | 1, 2, 3, 4, 5 | taylplem2 26311 | . . 3 β’ (((π β§ π₯ β β) β§ π β (0...π)) β (((((π Dπ πΉ)βπ)βπ΅) / (!βπ)) Β· ((π₯ β π΅)βπ)) β β) |
10 | 8, 9 | fsumcl 15712 | . 2 β’ ((π β§ π₯ β β) β Ξ£π β (0...π)(((((π Dπ πΉ)βπ)βπ΅) / (!βπ)) Β· ((π₯ β π΅)βπ)) β β) |
11 | 7, 10 | fmpt3d 7126 | 1 β’ (π β π:ββΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 {cpr 4631 dom cdm 5678 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcc 11137 βcr 11138 0cc0 11139 Β· cmul 11144 β cmin 11475 / cdiv 11902 β0cn0 12503 ...cfz 13517 βcexp 14059 !cfa 14265 Ξ£csu 15665 Dπ cdvn 25806 Tayl ctayl 26300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-fac 14266 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-ur 20122 df-ring 20175 df-cring 20176 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cnp 23145 df-haus 23232 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-tsms 24044 df-xms 24239 df-ms 24240 df-limc 25808 df-dv 25809 df-dvn 25810 df-tayl 26302 |
This theorem is referenced by: dvntaylp 26319 taylthlem1 26321 |
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