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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 26935 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 26916 | . . 3 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | |
| 2 | sinval 16158 | . . 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 11214 | . . . . . 6 ⊢ i ∈ ℂ | |
| 5 | mulcl 11239 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
| 6 | 4, 5 | mpan 690 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) | 
| 7 | 6 | negcld 11607 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(i · 𝐴) ∈ ℂ) | 
| 8 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 9 | sqcl 14158 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
| 10 | subcl 11507 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
| 11 | 8, 9, 10 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) | 
| 12 | 11 | sqrtcld 15476 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) | 
| 13 | 6, 7, 12 | pnpcan2d 11658 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) − (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) = ((i · 𝐴) − -(i · 𝐴))) | 
| 14 | efiasin 26931 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | |
| 15 | mulneg12 11701 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) | |
| 16 | 4, 1, 15 | sylancr 587 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) | 
| 17 | asinneg 26929 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴)) | |
| 18 | 17 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘-𝐴)) = (i · -(arcsin‘𝐴))) | 
| 19 | 16, 18 | eqtr4d 2780 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · (arcsin‘-𝐴))) | 
| 20 | 19 | fveq2d 6910 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (exp‘(i · (arcsin‘-𝐴)))) | 
| 21 | negcl 11508 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
| 22 | efiasin 26931 | . . . . . . 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 11700 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
| 25 | 4, 24 | mpan 690 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = -(i · 𝐴)) | 
| 26 | sqneg 14156 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | |
| 27 | 26 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (1 − (-𝐴↑2)) = (1 − (𝐴↑2))) | 
| 28 | 27 | fveq2d 6910 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (-𝐴↑2))) = (√‘(1 − (𝐴↑2)))) | 
| 29 | 25, 28 | oveq12d 7449 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · -𝐴) + (√‘(1 − (-𝐴↑2)))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) | 
| 30 | 20, 23, 29 | 3eqtrd 2781 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) | 
| 31 | 14, 30 | oveq12d 7449 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = (((i · 𝐴) + (√‘(1 − (𝐴↑2)))) − (-(i · 𝐴) + (√‘(1 − (𝐴↑2)))))) | 
| 32 | 6 | 2timesd 12509 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) | 
| 33 | 2cn 12341 | . . . . . 6 ⊢ 2 ∈ ℂ | |
| 34 | mulass 11243 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((2 · i) · 𝐴) = (2 · (i · 𝐴))) | |
| 35 | 33, 4, 34 | mp3an12 1453 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((2 · i) · 𝐴) = (2 · (i · 𝐴))) | 
| 36 | 6, 6 | subnegd 11627 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) | 
| 37 | 32, 35, 36 | 3eqtr4d 2787 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((2 · i) · 𝐴) = ((i · 𝐴) − -(i · 𝐴))) | 
| 38 | 13, 31, 37 | 3eqtr4d 2787 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) = ((2 · i) · 𝐴)) | 
| 39 | mulcl 11239 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (i · (arcsin‘𝐴)) ∈ ℂ) | |
| 40 | 4, 1, 39 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘𝐴)) ∈ ℂ) | 
| 41 | efcl 16118 | . . . . . 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 11509 | . . . . . . 7 ⊢ -i ∈ ℂ | |
| 44 | mulcl 11239 | . . . . . . 7 ⊢ ((-i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (-i · (arcsin‘𝐴)) ∈ ℂ) | |
| 45 | 43, 1, 44 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) ∈ ℂ) | 
| 46 | efcl 16118 | . . . . . 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 11620 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) ∈ ℂ) | 
| 49 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
| 50 | 2mulicn 12489 | . . . . 5 ⊢ (2 · i) ∈ ℂ | |
| 51 | 50 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · i) ∈ ℂ) | 
| 52 | 2muline0 12490 | . . . . 5 ⊢ (2 · i) ≠ 0 | |
| 53 | 52 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · i) ≠ 0) | 
| 54 | 48, 49, 51, 53 | divmul2d 12076 | . . 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 257 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (arcsin‘𝐴))) − (exp‘(-i · (arcsin‘𝐴)))) / (2 · i)) = 𝐴) | 
| 56 | 3, 55 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘(arcsin‘𝐴)) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 ↑cexp 14102 √csqrt 15272 expce 16097 sincsin 16099 arcsincasin 26905 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-asin 26908 | 
| This theorem is referenced by: cosacos 26933 asinsinb 26940 | 
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