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Mirrors > Home > MPE Home > Th. List > asinrebnd | Structured version Visualization version GIF version |
Description: Bounds on the arcsine function. (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
asinrebnd | ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resinf1o 25106 | . . . . . . . 8 ⊢ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) | |
2 | f1ocnv 6613 | . . . . . . . 8 ⊢ ((sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) → ◡(sin ↾ (-(π / 2)[,](π / 2))):(-1[,]1)–1-1-onto→(-(π / 2)[,](π / 2))) | |
3 | f1of 6601 | . . . . . . . 8 ⊢ (◡(sin ↾ (-(π / 2)[,](π / 2))):(-1[,]1)–1-1-onto→(-(π / 2)[,](π / 2)) → ◡(sin ↾ (-(π / 2)[,](π / 2))):(-1[,]1)⟶(-(π / 2)[,](π / 2))) | |
4 | 1, 2, 3 | mp2b 10 | . . . . . . 7 ⊢ ◡(sin ↾ (-(π / 2)[,](π / 2))):(-1[,]1)⟶(-(π / 2)[,](π / 2)) |
5 | 4 | ffvelrni 6836 | . . . . . 6 ⊢ (𝐴 ∈ (-1[,]1) → (◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴) ∈ (-(π / 2)[,](π / 2))) |
6 | 5 | fvresd 6676 | . . . . 5 ⊢ (𝐴 ∈ (-1[,]1) → ((sin ↾ (-(π / 2)[,](π / 2)))‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) = (sin‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴))) |
7 | f1ocnvfv2 7020 | . . . . . 6 ⊢ (((sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) ∧ 𝐴 ∈ (-1[,]1)) → ((sin ↾ (-(π / 2)[,](π / 2)))‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) = 𝐴) | |
8 | 1, 7 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ (-1[,]1) → ((sin ↾ (-(π / 2)[,](π / 2)))‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) = 𝐴) |
9 | 6, 8 | eqtr3d 2858 | . . . 4 ⊢ (𝐴 ∈ (-1[,]1) → (sin‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) = 𝐴) |
10 | 9 | fveq2d 6660 | . . 3 ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘(sin‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴))) = (arcsin‘𝐴)) |
11 | reasinsin 25460 | . . . 4 ⊢ ((◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴) ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴))) = (◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) | |
12 | 5, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘(sin‘(◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴))) = (◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) |
13 | 10, 12 | eqtr3d 2858 | . 2 ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) = (◡(sin ↾ (-(π / 2)[,](π / 2)))‘𝐴)) |
14 | 13, 5 | eqeltrd 2913 | 1 ⊢ (𝐴 ∈ (-1[,]1) → (arcsin‘𝐴) ∈ (-(π / 2)[,](π / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ◡ccnv 5540 ↾ cres 5543 ⟶wf 6337 –1-1-onto→wf1o 6340 ‘cfv 6341 (class class class)co 7142 1c1 10524 -cneg 10857 / cdiv 11283 2c2 11679 [,]cicc 12728 sincsin 15402 πcpi 15405 arcsincasin 25426 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
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 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ioc 12730 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14411 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-limsup 14813 df-clim 14830 df-rlim 14831 df-sum 15028 df-ef 15406 df-sin 15408 df-cos 15409 df-pi 15411 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 df-perf 21728 df-cn 21818 df-cnp 21819 df-haus 21906 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-xms 22913 df-ms 22914 df-tms 22915 df-cncf 23469 df-limc 24449 df-dv 24450 df-log 25126 df-asin 25429 |
This theorem is referenced by: asinrecl 25466 |
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