reasinsin - Metamath Proof Explorer

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Theorem reasinsin 26813

Description: The arcsine function composed with sin is equal to the identity. (Contributed by Mario Carneiro, 2-Apr-2015.)

**Assertion**
Ref	Expression
reasinsin	⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)

**Proof of Theorem reasinsin**
Step	Hyp	Ref	Expression
1		neghalfpire 26381	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 11239	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 26380	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 11239	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 26377	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 12987	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 12971	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11692	. . . . . . . . 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 230	. . . . . . 7 ⊢ -(π / 2) < 0
13		0re 11183	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11307	. . . . . . 7 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2))
15	12, 9, 14	mp2an 692	. . . . . 6 ⊢ -(π / 2) < (π / 2)
16	1, 3, 15	ltleii 11304	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13449	. . . . 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 1463	. . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2))
19	18	eleq2i 2821	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ 𝐴 ∈ (-(π / 2)[,](π / 2)))
20		elun 4119	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
21	19, 20	bitr3i 277	. 2 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
22		elioore 13343	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 11209	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 15197	. . . . 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 2829	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
27		asinsin 26809	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
28	23, 26, 27	syl2anc 584	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)
29		elpri 4616	. . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 11133	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 26803	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1)
33		asin1 26811	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11421	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2753	. . . . . 6 ⊢ (arcsin‘-1) = -(π / 2)
36		fveq2 6861	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2)))
37	3	recni 11195	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 16121	. . . . . . . . . 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 26384	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11421	. . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1
42	39, 41	eqtri 2753	. . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1
43	36, 42	eqtrdi 2781	. . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1)
44	43	fveq2d 6865	. . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘-1))
45		id 22	. . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2))
46	35, 44, 45	3eqtr4a 2791	. . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
47		fveq2 6861	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2781	. . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1)
49	48	fveq2d 6865	. . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1))
50		id 22	. . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2))
51	33, 49, 50	3eqtr4a 2791	. . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
52	46, 51	jaoi 857	. . . 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 857	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴)
55	21, 54	sylbi 217	1 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∪ cun 3915 {cpr 4594 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 -cneg 11413 / cdiv 11842 2c2 12248 ℝ⁺crp 12958 (,)cioo 13313 [,]cicc 13316 ℜcre 15070 sincsin 16036 πcpi 16039 arcsincasin 26779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472 df-asin 26782

This theorem is referenced by: asinrebnd 26818 asin1half 42352

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