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Mirrors > Home > MPE Home > Th. List > sinasin | Structured version Visualization version GIF version |
Description: The arcsine function is an inverse to sin. This is the main property that justifies the notation arcsin or sin↑-1. Because sin is not an injection, the other converse identity asinsin 25472 is only true under limited circumstances. (Contributed by Mario Carneiro, 1-Apr-2015.) |
Ref | Expression |
---|---|
sinasin | ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asincl 25453 | . . 3 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | |
2 | sinval 15477 | . . 3 ⊢ ((arcsin‘𝐴) ∈ ℂ → (sin‘(arcsin‘𝐴)) = (((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) / (2 · i))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = (((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) / (2 · i))) |
4 | ax-icn 10598 | . . . . . 6 ⊢ i ∈ ℂ | |
5 | mulcl 10623 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
6 | 4, 5 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
7 | 6 | negcld 10986 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(i · 𝐴) ∈ ℂ) |
8 | ax-1cn 10597 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | sqcl 13487 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
10 | subcl 10887 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
11 | 8, 9, 10 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) |
12 | 11 | sqrtcld 14799 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
13 | 6, 7, 12 | pnpcan2d 11037 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) − (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) = ((i · 𝐴) − -(i · 𝐴))) |
14 | efiasin 25468 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | |
15 | mulneg12 11080 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) | |
16 | 4, 1, 15 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) |
17 | asinneg 25466 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴)) | |
18 | 17 | oveq2d 7174 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘-𝐴)) = (i · -(arcsin‘𝐴))) |
19 | 16, 18 | eqtr4d 2861 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · (arcsin‘-𝐴))) |
20 | 19 | fveq2d 6676 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (exp‘(i · (arcsin‘-𝐴)))) |
21 | negcl 10888 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
22 | efiasin 25468 | . . . . . . 7 ⊢ (-𝐴 ∈ ℂ → (exp‘(i · (arcsin‘-𝐴))) = ((i · -𝐴) + (√‘(1 − (-𝐴↑2))))) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘-𝐴))) = ((i · -𝐴) + (√‘(1 − (-𝐴↑2))))) |
24 | mulneg2 11079 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
25 | 4, 24 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = -(i · 𝐴)) |
26 | sqneg 13485 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | |
27 | 26 | oveq2d 7174 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (1 − (-𝐴↑2)) = (1 − (𝐴↑2))) |
28 | 27 | fveq2d 6676 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (-𝐴↑2))) = (√‘(1 − (𝐴↑2)))) |
29 | 25, 28 | oveq12d 7176 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · -𝐴) + (√‘(1 − (-𝐴↑2)))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
30 | 20, 23, 29 | 3eqtrd 2862 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
31 | 14, 30 | oveq12d 7176 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) − (-(i · 𝐴) + (√‘(1 − (𝐴↑2)))))) |
32 | 6 | 2timesd 11883 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) |
33 | 2cn 11715 | . . . . . 6 ⊢ 2 ∈ ℂ | |
34 | mulass 10627 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((2 · i) · 𝐴) = (2 · (i · 𝐴))) | |
35 | 33, 4, 34 | mp3an12 1447 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((2 · i) · 𝐴) = (2 · (i · 𝐴))) |
36 | 6, 6 | subnegd 11006 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) |
37 | 32, 35, 36 | 3eqtr4d 2868 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((2 · i) · 𝐴) = ((i · 𝐴) − -(i · 𝐴))) |
38 | 13, 31, 37 | 3eqtr4d 2868 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = ((2 · i) · 𝐴)) |
39 | mulcl 10623 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (i · (arcsin‘𝐴)) ∈ ℂ) | |
40 | 4, 1, 39 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘𝐴)) ∈ ℂ) |
41 | efcl 15438 | . . . . . 6 ⊢ ((i · (arcsin‘𝐴)) ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) ∈ ℂ) | |
42 | 40, 41 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) ∈ ℂ) |
43 | negicn 10889 | . . . . . . 7 ⊢ -i ∈ ℂ | |
44 | mulcl 10623 | . . . . . . 7 ⊢ ((-i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (-i · (arcsin‘𝐴)) ∈ ℂ) | |
45 | 43, 1, 44 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) ∈ ℂ) |
46 | efcl 15438 | . . . . . 6 ⊢ ((-i · (arcsin‘𝐴)) ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) ∈ ℂ) | |
47 | 45, 46 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) ∈ ℂ) |
48 | 42, 47 | subcld 10999 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) ∈ ℂ) |
49 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
50 | 2mulicn 11863 | . . . . 5 ⊢ (2 · i) ∈ ℂ | |
51 | 50 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · i) ∈ ℂ) |
52 | 2muline0 11864 | . . . . 5 ⊢ (2 · i) ≠ 0 | |
53 | 52 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · i) ≠ 0) |
54 | 48, 49, 51, 53 | divmul2d 11451 | . . 3 ⊢ (𝐴 ∈ ℂ → ((((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) / (2 · i)) = 𝐴 ↔ ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = ((2 · i) · 𝐴))) |
55 | 38, 54 | mpbird 259 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) / (2 · i)) = 𝐴) |
56 | 3, 55 | eqtrd 2858 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 ici 10541 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 / cdiv 11299 2c2 11695 ↑cexp 13432 √csqrt 14594 expce 15417 sincsin 15419 arcsincasin 25442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-asin 25445 |
This theorem is referenced by: cosacos 25470 asinsinb 25477 |
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