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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 25473 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 25454 | . . 3 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | |
2 | sinval 15478 | . . 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 10599 | . . . . . 6 ⊢ i ∈ ℂ | |
5 | mulcl 10624 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
6 | 4, 5 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
7 | 6 | negcld 10987 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(i · 𝐴) ∈ ℂ) |
8 | ax-1cn 10598 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | sqcl 13487 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
10 | subcl 10888 | . . . . . . 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 14800 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
13 | 6, 7, 12 | pnpcan2d 11038 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) − (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) = ((i · 𝐴) − -(i · 𝐴))) |
14 | efiasin 25469 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | |
15 | mulneg12 11081 | . . . . . . . . 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 25467 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴)) | |
18 | 17 | oveq2d 7175 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘-𝐴)) = (i · -(arcsin‘𝐴))) |
19 | 16, 18 | eqtr4d 2862 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · (arcsin‘-𝐴))) |
20 | 19 | fveq2d 6677 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (exp‘(i · (arcsin‘-𝐴)))) |
21 | negcl 10889 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
22 | efiasin 25469 | . . . . . . 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 11080 | . . . . . . . 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 7175 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (1 − (-𝐴↑2)) = (1 − (𝐴↑2))) |
28 | 27 | fveq2d 6677 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (-𝐴↑2))) = (√‘(1 − (𝐴↑2)))) |
29 | 25, 28 | oveq12d 7177 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · -𝐴) + (√‘(1 − (-𝐴↑2)))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
30 | 20, 23, 29 | 3eqtrd 2863 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
31 | 14, 30 | oveq12d 7177 | . . . 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 10628 | . . . . . 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 11007 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) |
37 | 32, 35, 36 | 3eqtr4d 2869 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((2 · i) · 𝐴) = ((i · 𝐴) − -(i · 𝐴))) |
38 | 13, 31, 37 | 3eqtr4d 2869 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = ((2 · i) · 𝐴)) |
39 | mulcl 10624 | . . . . . . 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 15439 | . . . . . 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 10890 | . . . . . . 7 ⊢ -i ∈ ℂ | |
44 | mulcl 10624 | . . . . . . 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 15439 | . . . . . 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 11000 | . . . 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 11452 | . . 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 2859 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 ici 10542 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 2c2 11695 ↑cexp 13432 √csqrt 14595 expce 15418 sincsin 15420 arcsincasin 25443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-asin 25446 |
This theorem is referenced by: cosacos 25471 asinsinb 25478 |
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