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Mirrors > Home > MPE Home > Th. List > reasinsin | Structured version Visualization version GIF version |
Description: The arcsine function composed with sin is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
reasinsin | ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neghalfpire 25045 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ | |
2 | 1 | rexri 10693 | . . . . 5 ⊢ -(π / 2) ∈ ℝ* |
3 | halfpire 25044 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
4 | 3 | rexri 10693 | . . . . 5 ⊢ (π / 2) ∈ ℝ* |
5 | pirp 25041 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
6 | rphalfcl 12410 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
8 | rpgt0 12395 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
10 | lt0neg2 11141 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ → (0 < (π / 2) ↔ -(π / 2) < 0)) | |
11 | 3, 10 | ax-mp 5 | . . . . . . . 8 ⊢ (0 < (π / 2) ↔ -(π / 2) < 0) |
12 | 9, 11 | mpbi 232 | . . . . . . 7 ⊢ -(π / 2) < 0 |
13 | 0re 10637 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
14 | 1, 13, 3 | lttri 10760 | . . . . . . 7 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2)) |
15 | 12, 9, 14 | mp2an 690 | . . . . . 6 ⊢ -(π / 2) < (π / 2) |
16 | 1, 3, 15 | ltleii 10757 | . . . . 5 ⊢ -(π / 2) ≤ (π / 2) |
17 | prunioo 12861 | . . . . 5 ⊢ ((-(π / 2) ∈ ℝ* ∧ (π / 2) ∈ ℝ* ∧ -(π / 2) ≤ (π / 2)) → ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2))) | |
18 | 2, 4, 16, 17 | mp3an 1457 | . . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2)) |
19 | 18 | eleq2i 2904 | . . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ 𝐴 ∈ (-(π / 2)[,](π / 2))) |
20 | elun 4124 | . . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)})) | |
21 | 19, 20 | bitr3i 279 | . 2 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)})) |
22 | elioore 12762 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ) | |
23 | 22 | recnd 10663 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ) |
24 | 22 | rered 14577 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) = 𝐴) |
25 | id 22 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ (-(π / 2)(,)(π / 2))) | |
26 | 24, 25 | eqeltrd 2913 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
27 | asinsin 25464 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴) | |
28 | 23, 26, 27 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
29 | elpri 4582 | . . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2))) | |
30 | ax-1cn 10589 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
31 | asinneg 25458 | . . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1)) | |
32 | 30, 31 | ax-mp 5 | . . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1) |
33 | asin1 25466 | . . . . . . . 8 ⊢ (arcsin‘1) = (π / 2) | |
34 | 33 | negeqi 10873 | . . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2) |
35 | 32, 34 | eqtri 2844 | . . . . . 6 ⊢ (arcsin‘-1) = -(π / 2) |
36 | fveq2 6664 | . . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2))) | |
37 | 3 | recni 10649 | . . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ |
38 | sinneg 15493 | . . . . . . . . . 10 ⊢ ((π / 2) ∈ ℂ → (sin‘-(π / 2)) = -(sin‘(π / 2))) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . . 9 ⊢ (sin‘-(π / 2)) = -(sin‘(π / 2)) |
40 | sinhalfpi 25048 | . . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1 | |
41 | 40 | negeqi 10873 | . . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1 |
42 | 39, 41 | eqtri 2844 | . . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1 |
43 | 36, 42 | syl6eq 2872 | . . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1) |
44 | 43 | fveq2d 6668 | . . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘-1)) |
45 | id 22 | . . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2)) | |
46 | 35, 44, 45 | 3eqtr4a 2882 | . . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴) |
47 | fveq2 6664 | . . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2))) | |
48 | 47, 40 | syl6eq 2872 | . . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1) |
49 | 48 | fveq2d 6668 | . . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1)) |
50 | id 22 | . . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2)) | |
51 | 33, 49, 50 | 3eqtr4a 2882 | . . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴) |
52 | 46, 51 | jaoi 853 | . . . 4 ⊢ ((𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
53 | 29, 52 | syl 17 | . . 3 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (arcsin‘(sin‘𝐴)) = 𝐴) |
54 | 28, 53 | jaoi 853 | . 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴) |
55 | 21, 54 | sylbi 219 | 1 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 {cpr 4562 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 -cneg 10865 / cdiv 11291 2c2 11686 ℝ+crp 12383 (,)cioo 12732 [,]cicc 12735 ℜcre 14450 sincsin 15411 πcpi 15414 arcsincasin 25434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-asin 25437 |
This theorem is referenced by: asinrebnd 25473 |
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