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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 24337 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ | |
2 | 1 | rexri 10210 | . . . . 5 ⊢ -(π / 2) ∈ ℝ* |
3 | halfpire 24336 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
4 | 3 | rexri 10210 | . . . . 5 ⊢ (π / 2) ∈ ℝ* |
5 | pirp 24333 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
6 | rphalfcl 11972 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
8 | rpgt0 11958 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
10 | lt0neg2 10648 | . . . . . . . . 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 220 | . . . . . . 7 ⊢ -(π / 2) < 0 |
13 | 0re 10153 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
14 | 1, 13, 3 | lttri 10276 | . . . . . . 7 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2)) |
15 | 12, 9, 14 | mp2an 710 | . . . . . 6 ⊢ -(π / 2) < (π / 2) |
16 | 1, 3, 15 | ltleii 10273 | . . . . 5 ⊢ -(π / 2) ≤ (π / 2) |
17 | prunioo 12415 | . . . . 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 1537 | . . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2)) |
19 | 18 | eleq2i 2795 | . . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ 𝐴 ∈ (-(π / 2)[,](π / 2))) |
20 | elun 3861 | . . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)})) | |
21 | 19, 20 | bitr3i 266 | . 2 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)})) |
22 | elioore 12319 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ) | |
23 | 22 | recnd 10181 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ) |
24 | 22 | rered 14084 | . . . . 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 2803 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
27 | asinsin 24739 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴) | |
28 | 23, 26, 27 | syl2anc 696 | . . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
29 | elpri 4305 | . . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2))) | |
30 | ax-1cn 10107 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
31 | asinneg 24733 | . . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1)) | |
32 | 30, 31 | ax-mp 5 | . . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1) |
33 | asin1 24741 | . . . . . . . 8 ⊢ (arcsin‘1) = (π / 2) | |
34 | 33 | negeqi 10387 | . . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2) |
35 | 32, 34 | eqtri 2746 | . . . . . 6 ⊢ (arcsin‘-1) = -(π / 2) |
36 | fveq2 6304 | . . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2))) | |
37 | 3 | recni 10165 | . . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ |
38 | sinneg 14996 | . . . . . . . . . 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 24340 | . . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1 | |
41 | 40 | negeqi 10387 | . . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1 |
42 | 39, 41 | eqtri 2746 | . . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1 |
43 | 36, 42 | syl6eq 2774 | . . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1) |
44 | 43 | fveq2d 6308 | . . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘-1)) |
45 | id 22 | . . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2)) | |
46 | 35, 44, 45 | 3eqtr4a 2784 | . . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴) |
47 | fveq2 6304 | . . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2))) | |
48 | 47, 40 | syl6eq 2774 | . . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1) |
49 | 48 | fveq2d 6308 | . . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1)) |
50 | id 22 | . . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2)) | |
51 | 33, 49, 50 | 3eqtr4a 2784 | . . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴) |
52 | 46, 51 | jaoi 393 | . . . 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 393 | . 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴) |
55 | 21, 54 | sylbi 207 | 1 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1596 ∈ wcel 2103 ∪ cun 3678 {cpr 4287 class class class wbr 4760 ‘cfv 6001 (class class class)co 6765 ℂcc 10047 ℝcr 10048 0cc0 10049 1c1 10050 ℝ*cxr 10186 < clt 10187 ≤ cle 10188 -cneg 10380 / cdiv 10797 2c2 11183 ℝ+crp 11946 (,)cioo 12289 [,]cicc 12292 ℜcre 13957 sincsin 14914 πcpi 14917 arcsincasin 24709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 ax-addf 10128 ax-mulf 10129 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-iin 4631 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-of 7014 df-om 7183 df-1st 7285 df-2nd 7286 df-supp 7416 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-2o 7681 df-oadd 7684 df-er 7862 df-map 7976 df-pm 7977 df-ixp 8026 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-fsupp 8392 df-fi 8433 df-sup 8464 df-inf 8465 df-oi 8531 df-card 8878 df-cda 9103 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-z 11491 df-dec 11607 df-uz 11801 df-q 11903 df-rp 11947 df-xneg 12060 df-xadd 12061 df-xmul 12062 df-ioo 12293 df-ioc 12294 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-fl 12708 df-mod 12784 df-seq 12917 df-exp 12976 df-fac 13176 df-bc 13205 df-hash 13233 df-shft 13927 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-limsup 14322 df-clim 14339 df-rlim 14340 df-sum 14537 df-ef 14918 df-sin 14920 df-cos 14921 df-pi 14923 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-starv 16079 df-sca 16080 df-vsca 16081 df-ip 16082 df-tset 16083 df-ple 16084 df-ds 16087 df-unif 16088 df-hom 16089 df-cco 16090 df-rest 16206 df-topn 16207 df-0g 16225 df-gsum 16226 df-topgen 16227 df-pt 16228 df-prds 16231 df-xrs 16285 df-qtop 16290 df-imas 16291 df-xps 16293 df-mre 16369 df-mrc 16370 df-acs 16372 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-submnd 17458 df-mulg 17663 df-cntz 17871 df-cmn 18316 df-psmet 19861 df-xmet 19862 df-met 19863 df-bl 19864 df-mopn 19865 df-fbas 19866 df-fg 19867 df-cnfld 19870 df-top 20822 df-topon 20839 df-topsp 20860 df-bases 20873 df-cld 20946 df-ntr 20947 df-cls 20948 df-nei 21025 df-lp 21063 df-perf 21064 df-cn 21154 df-cnp 21155 df-haus 21242 df-tx 21488 df-hmeo 21681 df-fil 21772 df-fm 21864 df-flim 21865 df-flf 21866 df-xms 22247 df-ms 22248 df-tms 22249 df-cncf 22803 df-limc 23750 df-dv 23751 df-log 24423 df-asin 24712 |
This theorem is referenced by: asinrebnd 24748 |
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