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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvasinbx | Structured version Visualization version GIF version |
Description: Derivative exercise: the derivative with respect to y of A x sin(By), given two constants π΄ and π΅. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvasinbx | β’ ((π΄ β β β§ π΅ β β) β (β D (π¦ β β β¦ (π΄ Β· (sinβ(π΅ Β· π¦))))) = (π¦ β β β¦ ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnelprrecn 11239 | . . . 4 β’ β β {β, β} | |
2 | 1 | a1i 11 | . . 3 β’ ((π΄ β β β§ π΅ β β) β β β {β, β}) |
3 | simpll 765 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β π΄ β β) | |
4 | 0cnd 11245 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β 0 β β) | |
5 | 1 | a1i 11 | . . . . 5 β’ (π΄ β β β β β {β, β}) |
6 | id 22 | . . . . 5 β’ (π΄ β β β π΄ β β) | |
7 | 5, 6 | dvmptc 25910 | . . . 4 β’ (π΄ β β β (β D (π¦ β β β¦ π΄)) = (π¦ β β β¦ 0)) |
8 | 7 | adantr 479 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (β D (π¦ β β β¦ π΄)) = (π¦ β β β¦ 0)) |
9 | mulcl 11230 | . . . . 5 β’ ((π΅ β β β§ π¦ β β) β (π΅ Β· π¦) β β) | |
10 | 9 | sincld 16114 | . . . 4 β’ ((π΅ β β β§ π¦ β β) β (sinβ(π΅ Β· π¦)) β β) |
11 | 10 | adantll 712 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (sinβ(π΅ Β· π¦)) β β) |
12 | simpl 481 | . . . . 5 β’ ((π΅ β β β§ π¦ β β) β π΅ β β) | |
13 | 9 | coscld 16115 | . . . . 5 β’ ((π΅ β β β§ π¦ β β) β (cosβ(π΅ Β· π¦)) β β) |
14 | 12, 13 | mulcld 11272 | . . . 4 β’ ((π΅ β β β§ π¦ β β) β (π΅ Β· (cosβ(π΅ Β· π¦))) β β) |
15 | 14 | adantll 712 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (π΅ Β· (cosβ(π΅ Β· π¦))) β β) |
16 | dvsinax 45330 | . . . 4 β’ (π΅ β β β (β D (π¦ β β β¦ (sinβ(π΅ Β· π¦)))) = (π¦ β β β¦ (π΅ Β· (cosβ(π΅ Β· π¦))))) | |
17 | 16 | adantl 480 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (β D (π¦ β β β¦ (sinβ(π΅ Β· π¦)))) = (π¦ β β β¦ (π΅ Β· (cosβ(π΅ Β· π¦))))) |
18 | 2, 3, 4, 8, 11, 15, 17 | dvmptmul 25913 | . 2 β’ ((π΄ β β β§ π΅ β β) β (β D (π¦ β β β¦ (π΄ Β· (sinβ(π΅ Β· π¦))))) = (π¦ β β β¦ ((0 Β· (sinβ(π΅ Β· π¦))) + ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄)))) |
19 | 11 | mul02d 11450 | . . . . 5 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (0 Β· (sinβ(π΅ Β· π¦))) = 0) |
20 | 12 | adantll 712 | . . . . . . 7 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β π΅ β β) |
21 | 13 | adantll 712 | . . . . . . 7 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (cosβ(π΅ Β· π¦)) β β) |
22 | 20, 21, 3 | mul32d 11462 | . . . . . 6 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄) = ((π΅ Β· π΄) Β· (cosβ(π΅ Β· π¦)))) |
23 | simpr 483 | . . . . . . . . 9 β’ ((π΄ β β β§ π΅ β β) β π΅ β β) | |
24 | simpl 481 | . . . . . . . . 9 β’ ((π΄ β β β§ π΅ β β) β π΄ β β) | |
25 | 23, 24 | mulcomd 11273 | . . . . . . . 8 β’ ((π΄ β β β§ π΅ β β) β (π΅ Β· π΄) = (π΄ Β· π΅)) |
26 | 25 | adantr 479 | . . . . . . 7 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (π΅ Β· π΄) = (π΄ Β· π΅)) |
27 | 26 | oveq1d 7441 | . . . . . 6 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((π΅ Β· π΄) Β· (cosβ(π΅ Β· π¦))) = ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦)))) |
28 | 22, 27 | eqtrd 2768 | . . . . 5 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄) = ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦)))) |
29 | 19, 28 | oveq12d 7444 | . . . 4 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((0 Β· (sinβ(π΅ Β· π¦))) + ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄)) = (0 + ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦))))) |
30 | 3, 20 | mulcld 11272 | . . . . . 6 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (π΄ Β· π΅) β β) |
31 | 30, 21 | mulcld 11272 | . . . . 5 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦))) β β) |
32 | 31 | addlidd 11453 | . . . 4 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β (0 + ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦)))) = ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦)))) |
33 | 29, 32 | eqtrd 2768 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ π¦ β β) β ((0 Β· (sinβ(π΅ Β· π¦))) + ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄)) = ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦)))) |
34 | 33 | mpteq2dva 5252 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π¦ β β β¦ ((0 Β· (sinβ(π΅ Β· π¦))) + ((π΅ Β· (cosβ(π΅ Β· π¦))) Β· π΄))) = (π¦ β β β¦ ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦))))) |
35 | 18, 34 | eqtrd 2768 | 1 β’ ((π΄ β β β§ π΅ β β) β (β D (π¦ β β β¦ (π΄ Β· (sinβ(π΅ Β· π¦))))) = (π¦ β β β¦ ((π΄ Β· π΅) Β· (cosβ(π΅ Β· π¦))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {cpr 4634 β¦ cmpt 5235 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 0cc0 11146 + caddc 11149 Β· cmul 11151 sincsin 16047 cosccos 16048 D cdv 25812 |
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-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 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-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 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-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 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-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 |
This theorem is referenced by: dirkercncflem2 45521 fourierdlem57 45580 fourierdlem58 45581 fourierdlem62 45585 |
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