![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (arcsinβ(sinβπ΄)) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neghalfpire 25975 | . . . . . 6 β’ -(Ο / 2) β β | |
2 | 1 | rexri 11272 | . . . . 5 β’ -(Ο / 2) β β* |
3 | halfpire 25974 | . . . . . 6 β’ (Ο / 2) β β | |
4 | 3 | rexri 11272 | . . . . 5 β’ (Ο / 2) β β* |
5 | pirp 25971 | . . . . . . . . . 10 β’ Ο β β+ | |
6 | rphalfcl 13001 | . . . . . . . . . 10 β’ (Ο β β+ β (Ο / 2) β β+) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 β’ (Ο / 2) β β+ |
8 | rpgt0 12986 | . . . . . . . . 9 β’ ((Ο / 2) β β+ β 0 < (Ο / 2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 β’ 0 < (Ο / 2) |
10 | lt0neg2 11721 | . . . . . . . . 9 β’ ((Ο / 2) β β β (0 < (Ο / 2) β -(Ο / 2) < 0)) | |
11 | 3, 10 | ax-mp 5 | . . . . . . . 8 β’ (0 < (Ο / 2) β -(Ο / 2) < 0) |
12 | 9, 11 | mpbi 229 | . . . . . . 7 β’ -(Ο / 2) < 0 |
13 | 0re 11216 | . . . . . . . 8 β’ 0 β β | |
14 | 1, 13, 3 | lttri 11340 | . . . . . . 7 β’ ((-(Ο / 2) < 0 β§ 0 < (Ο / 2)) β -(Ο / 2) < (Ο / 2)) |
15 | 12, 9, 14 | mp2an 691 | . . . . . 6 β’ -(Ο / 2) < (Ο / 2) |
16 | 1, 3, 15 | ltleii 11337 | . . . . 5 β’ -(Ο / 2) β€ (Ο / 2) |
17 | prunioo 13458 | . . . . 5 β’ ((-(Ο / 2) β β* β§ (Ο / 2) β β* β§ -(Ο / 2) β€ (Ο / 2)) β ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) = (-(Ο / 2)[,](Ο / 2))) | |
18 | 2, 4, 16, 17 | mp3an 1462 | . . . 4 β’ ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) = (-(Ο / 2)[,](Ο / 2)) |
19 | 18 | eleq2i 2826 | . . 3 β’ (π΄ β ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) β π΄ β (-(Ο / 2)[,](Ο / 2))) |
20 | elun 4149 | . . 3 β’ (π΄ β ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) β (π΄ β (-(Ο / 2)(,)(Ο / 2)) β¨ π΄ β {-(Ο / 2), (Ο / 2)})) | |
21 | 19, 20 | bitr3i 277 | . 2 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (π΄ β (-(Ο / 2)(,)(Ο / 2)) β¨ π΄ β {-(Ο / 2), (Ο / 2)})) |
22 | elioore 13354 | . . . . 5 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β π΄ β β) | |
23 | 22 | recnd 11242 | . . . 4 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β π΄ β β) |
24 | 22 | rered 15171 | . . . . 5 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β (ββπ΄) = π΄) |
25 | id 22 | . . . . 5 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β π΄ β (-(Ο / 2)(,)(Ο / 2))) | |
26 | 24, 25 | eqeltrd 2834 | . . . 4 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β (ββπ΄) β (-(Ο / 2)(,)(Ο / 2))) |
27 | asinsin 26397 | . . . 4 β’ ((π΄ β β β§ (ββπ΄) β (-(Ο / 2)(,)(Ο / 2))) β (arcsinβ(sinβπ΄)) = π΄) | |
28 | 23, 26, 27 | syl2anc 585 | . . 3 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β (arcsinβ(sinβπ΄)) = π΄) |
29 | elpri 4651 | . . . 4 β’ (π΄ β {-(Ο / 2), (Ο / 2)} β (π΄ = -(Ο / 2) β¨ π΄ = (Ο / 2))) | |
30 | ax-1cn 11168 | . . . . . . . 8 β’ 1 β β | |
31 | asinneg 26391 | . . . . . . . 8 β’ (1 β β β (arcsinβ-1) = -(arcsinβ1)) | |
32 | 30, 31 | ax-mp 5 | . . . . . . 7 β’ (arcsinβ-1) = -(arcsinβ1) |
33 | asin1 26399 | . . . . . . . 8 β’ (arcsinβ1) = (Ο / 2) | |
34 | 33 | negeqi 11453 | . . . . . . 7 β’ -(arcsinβ1) = -(Ο / 2) |
35 | 32, 34 | eqtri 2761 | . . . . . 6 β’ (arcsinβ-1) = -(Ο / 2) |
36 | fveq2 6892 | . . . . . . . 8 β’ (π΄ = -(Ο / 2) β (sinβπ΄) = (sinβ-(Ο / 2))) | |
37 | 3 | recni 11228 | . . . . . . . . . 10 β’ (Ο / 2) β β |
38 | sinneg 16089 | . . . . . . . . . 10 β’ ((Ο / 2) β β β (sinβ-(Ο / 2)) = -(sinβ(Ο / 2))) | |
39 | 37, 38 | ax-mp 5 | . . . . . . . . 9 β’ (sinβ-(Ο / 2)) = -(sinβ(Ο / 2)) |
40 | sinhalfpi 25978 | . . . . . . . . . 10 β’ (sinβ(Ο / 2)) = 1 | |
41 | 40 | negeqi 11453 | . . . . . . . . 9 β’ -(sinβ(Ο / 2)) = -1 |
42 | 39, 41 | eqtri 2761 | . . . . . . . 8 β’ (sinβ-(Ο / 2)) = -1 |
43 | 36, 42 | eqtrdi 2789 | . . . . . . 7 β’ (π΄ = -(Ο / 2) β (sinβπ΄) = -1) |
44 | 43 | fveq2d 6896 | . . . . . 6 β’ (π΄ = -(Ο / 2) β (arcsinβ(sinβπ΄)) = (arcsinβ-1)) |
45 | id 22 | . . . . . 6 β’ (π΄ = -(Ο / 2) β π΄ = -(Ο / 2)) | |
46 | 35, 44, 45 | 3eqtr4a 2799 | . . . . 5 β’ (π΄ = -(Ο / 2) β (arcsinβ(sinβπ΄)) = π΄) |
47 | fveq2 6892 | . . . . . . . 8 β’ (π΄ = (Ο / 2) β (sinβπ΄) = (sinβ(Ο / 2))) | |
48 | 47, 40 | eqtrdi 2789 | . . . . . . 7 β’ (π΄ = (Ο / 2) β (sinβπ΄) = 1) |
49 | 48 | fveq2d 6896 | . . . . . 6 β’ (π΄ = (Ο / 2) β (arcsinβ(sinβπ΄)) = (arcsinβ1)) |
50 | id 22 | . . . . . 6 β’ (π΄ = (Ο / 2) β π΄ = (Ο / 2)) | |
51 | 33, 49, 50 | 3eqtr4a 2799 | . . . . 5 β’ (π΄ = (Ο / 2) β (arcsinβ(sinβπ΄)) = π΄) |
52 | 46, 51 | jaoi 856 | . . . 4 β’ ((π΄ = -(Ο / 2) β¨ π΄ = (Ο / 2)) β (arcsinβ(sinβπ΄)) = π΄) |
53 | 29, 52 | syl 17 | . . 3 β’ (π΄ β {-(Ο / 2), (Ο / 2)} β (arcsinβ(sinβπ΄)) = π΄) |
54 | 28, 53 | jaoi 856 | . 2 β’ ((π΄ β (-(Ο / 2)(,)(Ο / 2)) β¨ π΄ β {-(Ο / 2), (Ο / 2)}) β (arcsinβ(sinβπ΄)) = π΄) |
55 | 21, 54 | sylbi 216 | 1 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (arcsinβ(sinβπ΄)) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β¨ wo 846 = wceq 1542 β wcel 2107 βͺ cun 3947 {cpr 4631 class class class wbr 5149 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 0cc0 11110 1c1 11111 β*cxr 11247 < clt 11248 β€ cle 11249 -cneg 11445 / cdiv 11871 2c2 12267 β+crp 12974 (,)cioo 13324 [,]cicc 13327 βcre 15044 sincsin 16007 Οcpi 16010 arcsincasin 26367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-asin 26370 |
This theorem is referenced by: asinrebnd 26406 |
Copyright terms: Public domain | W3C validator |