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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvsinexp | Structured version Visualization version GIF version |
Description: The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
dvsinexp.5 | β’ (π β π β β) |
Ref | Expression |
---|---|
dvsinexp | β’ (π β (β D (π₯ β β β¦ ((sinβπ₯)βπ))) = (π₯ β β β¦ ((π Β· ((sinβπ₯)β(π β 1))) Β· (cosβπ₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnelprrecn 11202 | . . 3 β’ β β {β, β} | |
2 | 1 | a1i 11 | . 2 β’ (π β β β {β, β}) |
3 | sinf 16071 | . . . 4 β’ sin:ββΆβ | |
4 | 3 | a1i 11 | . . 3 β’ (π β sin:ββΆβ) |
5 | 4 | ffvelcdmda 7079 | . 2 β’ ((π β§ π₯ β β) β (sinβπ₯) β β) |
6 | cosf 16072 | . . . 4 β’ cos:ββΆβ | |
7 | 6 | a1i 11 | . . 3 β’ (π β cos:ββΆβ) |
8 | 7 | ffvelcdmda 7079 | . 2 β’ ((π β§ π₯ β β) β (cosβπ₯) β β) |
9 | simpr 484 | . . 3 β’ ((π β§ π¦ β β) β π¦ β β) | |
10 | dvsinexp.5 | . . . . 5 β’ (π β π β β) | |
11 | 10 | nnnn0d 12533 | . . . 4 β’ (π β π β β0) |
12 | 11 | adantr 480 | . . 3 β’ ((π β§ π¦ β β) β π β β0) |
13 | 9, 12 | expcld 14113 | . 2 β’ ((π β§ π¦ β β) β (π¦βπ) β β) |
14 | 10 | nncnd 12229 | . . . 4 β’ (π β π β β) |
15 | 14 | adantr 480 | . . 3 β’ ((π β§ π¦ β β) β π β β) |
16 | nnm1nn0 12514 | . . . . . 6 β’ (π β β β (π β 1) β β0) | |
17 | 10, 16 | syl 17 | . . . . 5 β’ (π β (π β 1) β β0) |
18 | 17 | adantr 480 | . . . 4 β’ ((π β§ π¦ β β) β (π β 1) β β0) |
19 | 9, 18 | expcld 14113 | . . 3 β’ ((π β§ π¦ β β) β (π¦β(π β 1)) β β) |
20 | 15, 19 | mulcld 11235 | . 2 β’ ((π β§ π¦ β β) β (π Β· (π¦β(π β 1))) β β) |
21 | dvsin 25864 | . . 3 β’ (β D sin) = cos | |
22 | 4 | feqmptd 6953 | . . . 4 β’ (π β sin = (π₯ β β β¦ (sinβπ₯))) |
23 | 22 | oveq2d 7420 | . . 3 β’ (π β (β D sin) = (β D (π₯ β β β¦ (sinβπ₯)))) |
24 | 7 | feqmptd 6953 | . . 3 β’ (π β cos = (π₯ β β β¦ (cosβπ₯))) |
25 | 21, 23, 24 | 3eqtr3a 2790 | . 2 β’ (π β (β D (π₯ β β β¦ (sinβπ₯))) = (π₯ β β β¦ (cosβπ₯))) |
26 | dvexp 25835 | . . 3 β’ (π β β β (β D (π¦ β β β¦ (π¦βπ))) = (π¦ β β β¦ (π Β· (π¦β(π β 1))))) | |
27 | 10, 26 | syl 17 | . 2 β’ (π β (β D (π¦ β β β¦ (π¦βπ))) = (π¦ β β β¦ (π Β· (π¦β(π β 1))))) |
28 | oveq1 7411 | . 2 β’ (π¦ = (sinβπ₯) β (π¦βπ) = ((sinβπ₯)βπ)) | |
29 | oveq1 7411 | . . 3 β’ (π¦ = (sinβπ₯) β (π¦β(π β 1)) = ((sinβπ₯)β(π β 1))) | |
30 | 29 | oveq2d 7420 | . 2 β’ (π¦ = (sinβπ₯) β (π Β· (π¦β(π β 1))) = (π Β· ((sinβπ₯)β(π β 1)))) |
31 | 2, 2, 5, 8, 13, 20, 25, 27, 28, 30 | dvmptco 25854 | 1 β’ (π β (β D (π₯ β β β¦ ((sinβπ₯)βπ))) = (π₯ β β β¦ ((π Β· ((sinβπ₯)β(π β 1))) Β· (cosβπ₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {cpr 4625 β¦ cmpt 5224 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 1c1 11110 Β· cmul 11114 β cmin 11445 βcn 12213 β0cn0 12473 βcexp 14029 sincsin 16010 cosccos 16011 D cdv 25742 |
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-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 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-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15017 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-limsup 15418 df-clim 15435 df-rlim 15436 df-sum 15636 df-ef 16014 df-sin 16016 df-cos 16017 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-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-mulg 18993 df-cntz 19230 df-cmn 19699 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-cn 23081 df-cnp 23082 df-haus 23169 df-tx 23416 df-hmeo 23609 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-xms 24176 df-ms 24177 df-tms 24178 df-cncf 24748 df-limc 25745 df-dv 25746 |
This theorem is referenced by: itgsinexplem1 45224 |
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