![]() |
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 25974 | . . . . . 6 β’ -(Ο / 2) β β | |
2 | 1 | rexri 11271 | . . . . 5 β’ -(Ο / 2) β β* |
3 | halfpire 25973 | . . . . . 6 β’ (Ο / 2) β β | |
4 | 3 | rexri 11271 | . . . . 5 β’ (Ο / 2) β β* |
5 | pirp 25970 | . . . . . . . . . 10 β’ Ο β β+ | |
6 | rphalfcl 13000 | . . . . . . . . . 10 β’ (Ο β β+ β (Ο / 2) β β+) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 β’ (Ο / 2) β β+ |
8 | rpgt0 12985 | . . . . . . . . 9 β’ ((Ο / 2) β β+ β 0 < (Ο / 2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 β’ 0 < (Ο / 2) |
10 | lt0neg2 11720 | . . . . . . . . 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 11215 | . . . . . . . 8 β’ 0 β β | |
14 | 1, 13, 3 | lttri 11339 | . . . . . . 7 β’ ((-(Ο / 2) < 0 β§ 0 < (Ο / 2)) β -(Ο / 2) < (Ο / 2)) |
15 | 12, 9, 14 | mp2an 690 | . . . . . 6 β’ -(Ο / 2) < (Ο / 2) |
16 | 1, 3, 15 | ltleii 11336 | . . . . 5 β’ -(Ο / 2) β€ (Ο / 2) |
17 | prunioo 13457 | . . . . 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 1461 | . . . 4 β’ ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) = (-(Ο / 2)[,](Ο / 2)) |
19 | 18 | eleq2i 2825 | . . 3 β’ (π΄ β ((-(Ο / 2)(,)(Ο / 2)) βͺ {-(Ο / 2), (Ο / 2)}) β π΄ β (-(Ο / 2)[,](Ο / 2))) |
20 | elun 4148 | . . 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 276 | . 2 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (π΄ β (-(Ο / 2)(,)(Ο / 2)) β¨ π΄ β {-(Ο / 2), (Ο / 2)})) |
22 | elioore 13353 | . . . . 5 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β π΄ β β) | |
23 | 22 | recnd 11241 | . . . 4 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β π΄ β β) |
24 | 22 | rered 15170 | . . . . 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 2833 | . . . 4 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β (ββπ΄) β (-(Ο / 2)(,)(Ο / 2))) |
27 | asinsin 26394 | . . . 4 β’ ((π΄ β β β§ (ββπ΄) β (-(Ο / 2)(,)(Ο / 2))) β (arcsinβ(sinβπ΄)) = π΄) | |
28 | 23, 26, 27 | syl2anc 584 | . . 3 β’ (π΄ β (-(Ο / 2)(,)(Ο / 2)) β (arcsinβ(sinβπ΄)) = π΄) |
29 | elpri 4650 | . . . 4 β’ (π΄ β {-(Ο / 2), (Ο / 2)} β (π΄ = -(Ο / 2) β¨ π΄ = (Ο / 2))) | |
30 | ax-1cn 11167 | . . . . . . . 8 β’ 1 β β | |
31 | asinneg 26388 | . . . . . . . 8 β’ (1 β β β (arcsinβ-1) = -(arcsinβ1)) | |
32 | 30, 31 | ax-mp 5 | . . . . . . 7 β’ (arcsinβ-1) = -(arcsinβ1) |
33 | asin1 26396 | . . . . . . . 8 β’ (arcsinβ1) = (Ο / 2) | |
34 | 33 | negeqi 11452 | . . . . . . 7 β’ -(arcsinβ1) = -(Ο / 2) |
35 | 32, 34 | eqtri 2760 | . . . . . 6 β’ (arcsinβ-1) = -(Ο / 2) |
36 | fveq2 6891 | . . . . . . . 8 β’ (π΄ = -(Ο / 2) β (sinβπ΄) = (sinβ-(Ο / 2))) | |
37 | 3 | recni 11227 | . . . . . . . . . 10 β’ (Ο / 2) β β |
38 | sinneg 16088 | . . . . . . . . . 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 25977 | . . . . . . . . . 10 β’ (sinβ(Ο / 2)) = 1 | |
41 | 40 | negeqi 11452 | . . . . . . . . 9 β’ -(sinβ(Ο / 2)) = -1 |
42 | 39, 41 | eqtri 2760 | . . . . . . . 8 β’ (sinβ-(Ο / 2)) = -1 |
43 | 36, 42 | eqtrdi 2788 | . . . . . . 7 β’ (π΄ = -(Ο / 2) β (sinβπ΄) = -1) |
44 | 43 | fveq2d 6895 | . . . . . 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 2798 | . . . . 5 β’ (π΄ = -(Ο / 2) β (arcsinβ(sinβπ΄)) = π΄) |
47 | fveq2 6891 | . . . . . . . 8 β’ (π΄ = (Ο / 2) β (sinβπ΄) = (sinβ(Ο / 2))) | |
48 | 47, 40 | eqtrdi 2788 | . . . . . . 7 β’ (π΄ = (Ο / 2) β (sinβπ΄) = 1) |
49 | 48 | fveq2d 6895 | . . . . . 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 2798 | . . . . 5 β’ (π΄ = (Ο / 2) β (arcsinβ(sinβπ΄)) = π΄) |
52 | 46, 51 | jaoi 855 | . . . 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 855 | . 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 845 = wceq 1541 β wcel 2106 βͺ cun 3946 {cpr 4630 class class class wbr 5148 βcfv 6543 (class class class)co 7408 βcc 11107 βcr 11108 0cc0 11109 1c1 11110 β*cxr 11246 < clt 11247 β€ cle 11248 -cneg 11444 / cdiv 11870 2c2 12266 β+crp 12973 (,)cioo 13323 [,]cicc 13326 βcre 15043 sincsin 16006 Οcpi 16009 arcsincasin 26364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064 df-asin 26367 |
This theorem is referenced by: asinrebnd 26403 |
Copyright terms: Public domain | W3C validator |