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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 21274 | . . 3 β’ βfld β Ring | |
8 | cnfldbas 21239 | . . . 4 β’ β = (Baseββfld) | |
9 | 8 | subrgid 20472 | . . 3 β’ (βfld β Ring β β β (SubRingββfld)) |
10 | 7, 9 | mp1i 13 | . 2 β’ (π β β β (SubRingββfld)) |
11 | cnex 11190 | . . . . . . . 8 β’ β β V | |
12 | 11 | a1i 11 | . . . . . . 7 β’ (π β β β V) |
13 | elpm2r 8838 | . . . . . . 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 836 | . . . . . 6 β’ (π β πΉ β (β βpm π)) |
15 | dvnbss 25808 | . . . . . 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 6729 | . . . 4 β’ (π β dom ((π Dπ πΉ)βπ) β π΄) |
18 | recnprss 25783 | . . . . . 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 3987 | . . . 4 β’ (π β π΄ β β) |
21 | 17, 20 | sstrd 3987 | . . 3 β’ (π β dom ((π Dπ πΉ)βπ) β β) |
22 | 21, 5 | sseldd 3978 | . 2 β’ (π β π΅ β β) |
23 | 1 | adantr 480 | . . . . 5 β’ ((π β§ π β (0...π)) β π β {β, β}) |
24 | 14 | adantr 480 | . . . . 5 β’ ((π β§ π β (0...π)) β πΉ β (β βpm π)) |
25 | elfznn0 13597 | . . . . . 6 β’ (π β (0...π) β π β β0) | |
26 | 25 | adantl 481 | . . . . 5 β’ ((π β§ π β (0...π)) β π β β0) |
27 | dvnf 25807 | . . . . 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 484 | . . . . . 6 β’ ((π β§ π β (0...π)) β π β (0...π)) | |
30 | dvn2bss 25810 | . . . . . 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 480 | . . . . 5 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
33 | 31, 32 | sseldd 3978 | . . . 4 β’ ((π β§ π β (0...π)) β π΅ β dom ((π Dπ πΉ)βπ)) |
34 | 28, 33 | ffvelcdmd 7080 | . . 3 β’ ((π β§ π β (0...π)) β (((π Dπ πΉ)βπ)βπ΅) β β) |
35 | 26 | faccld 14246 | . . . 4 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
36 | 35 | nncnd 12229 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β β) |
37 | 35 | nnne0d 12263 | . . 3 β’ ((π β§ π β (0...π)) β (!βπ) β 0) |
38 | 34, 36, 37 | divcld 11991 | . 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 26252 | 1 β’ (π β (π β (Polyββ) β§ (degβπ) β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 {cpr 4625 class class class wbr 5141 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 βpm cpm 8820 βcc 11107 βcr 11108 0cc0 11109 β€ cle 11250 β0cn0 12473 ...cfz 13487 !cfa 14235 Ringcrg 20135 SubRingcsubrg 20466 βfldccnfld 21235 Dπ cdvn 25743 Polycply 26068 degcdgr 26071 Tayl ctayl 26237 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-fac 14236 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-subg 19047 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-subrng 20443 df-subrg 20468 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-fbas 21232 df-fg 21233 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-lp 22990 df-perf 22991 df-cnp 23082 df-haus 23169 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-tsms 23981 df-xms 24176 df-ms 24177 df-0p 25549 df-limc 25745 df-dv 25746 df-dvn 25747 df-ply 26072 df-idp 26073 df-coe 26074 df-dgr 26075 df-tayl 26239 |
This theorem is referenced by: (None) |
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