reasinsin - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > reasinsin		Structured version Visualization version GIF version

Theorem reasinsin 25989

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 25565	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 10980	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 25564	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 10980	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 25561	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 12702	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 12687	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11428	. . . . . . . . 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 229	. . . . . . 7 ⊢ -(π / 2) < 0
13		0re 10924	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11047	. . . . . . 7 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2))
15	12, 9, 14	mp2an 688	. . . . . 6 ⊢ -(π / 2) < (π / 2)
16	1, 3, 15	ltleii 11044	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13158	. . . . 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 1459	. . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2))
19	18	eleq2i 2828	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ 𝐴 ∈ (-(π / 2)[,](π / 2)))
20		elun 4084	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
21	19, 20	bitr3i 276	. 2 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
22		elioore 13054	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 10950	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 14879	. . . . 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 2837	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
27		asinsin 25985	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
28	23, 26, 27	syl2anc 583	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)
29		elpri 4585	. . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 10876	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 25979	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1)
33		asin1 25987	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11160	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2765	. . . . . 6 ⊢ (arcsin‘-1) = -(π / 2)
36		fveq2 6761	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2)))
37	3	recni 10936	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 15799	. . . . . . . . . 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 25568	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11160	. . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1
42	39, 41	eqtri 2765	. . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1
43	36, 42	eqtrdi 2793	. . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1)
44	43	fveq2d 6765	. . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘-1))
45		id 22	. . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2))
46	35, 44, 45	3eqtr4a 2803	. . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
47		fveq2 6761	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2793	. . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1)
49	48	fveq2d 6765	. . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1))
50		id 22	. . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2))
51	33, 49, 50	3eqtr4a 2803	. . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
52	46, 51	jaoi 853	. . . 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 853	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴)
55	21, 54	sylbi 216	1 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ wo 843 = wceq 1539 ∈ wcel 2107 ∪ cun 3886 {cpr 4565 class class class wbr 5075 ‘cfv 6423 (class class class)co 7260 ℂcc 10816 ℝcr 10817 0cc0 10818 1c1 10819 ℝ^*cxr 10955 < clt 10956 ≤ cle 10957 -cneg 11152 / cdiv 11578 2c2 11974 ℝ⁺crp 12675 (,)cioo 13024 [,]cicc 13027 ℜcre 14752 sincsin 15717 πcpi 15720 arcsincasin 25955

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 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 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-inf2 9345 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 ax-pre-sup 10896 ax-addf 10897 ax-mulf 10898

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-isom 6432 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-of 7516 df-om 7693 df-1st 7809 df-2nd 7810 df-supp 7954 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-2o 8273 df-er 8461 df-map 8580 df-pm 8581 df-ixp 8649 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-fsupp 9075 df-fi 9116 df-sup 9147 df-inf 9148 df-oi 9215 df-card 9644 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-3 11983 df-4 11984 df-5 11985 df-6 11986 df-7 11987 df-8 11988 df-9 11989 df-n0 12180 df-z 12266 df-dec 12383 df-uz 12528 df-q 12634 df-rp 12676 df-xneg 12793 df-xadd 12794 df-xmul 12795 df-ioo 13028 df-ioc 13029 df-ico 13030 df-icc 13031 df-fz 13185 df-fzo 13328 df-fl 13456 df-mod 13534 df-seq 13666 df-exp 13727 df-fac 13932 df-bc 13961 df-hash 13989 df-shft 14722 df-cj 14754 df-re 14755 df-im 14756 df-sqrt 14890 df-abs 14891 df-limsup 15124 df-clim 15141 df-rlim 15142 df-sum 15342 df-ef 15721 df-sin 15723 df-cos 15724 df-pi 15726 df-struct 16792 df-sets 16809 df-slot 16827 df-ndx 16839 df-base 16857 df-ress 16886 df-plusg 16919 df-mulr 16920 df-starv 16921 df-sca 16922 df-vsca 16923 df-ip 16924 df-tset 16925 df-ple 16926 df-ds 16928 df-unif 16929 df-hom 16930 df-cco 16931 df-rest 17077 df-topn 17078 df-0g 17096 df-gsum 17097 df-topgen 17098 df-pt 17099 df-prds 17102 df-xrs 17157 df-qtop 17162 df-imas 17163 df-xps 17165 df-mre 17239 df-mrc 17240 df-acs 17242 df-mgm 18270 df-sgrp 18319 df-mnd 18330 df-submnd 18375 df-mulg 18645 df-cntz 18867 df-cmn 19332 df-psmet 20533 df-xmet 20534 df-met 20535 df-bl 20536 df-mopn 20537 df-fbas 20538 df-fg 20539 df-cnfld 20542 df-top 21987 df-topon 22004 df-topsp 22026 df-bases 22040 df-cld 22114 df-ntr 22115 df-cls 22116 df-nei 22193 df-lp 22231 df-perf 22232 df-cn 22322 df-cnp 22323 df-haus 22410 df-tx 22657 df-hmeo 22850 df-fil 22941 df-fm 23033 df-flim 23034 df-flf 23035 df-xms 23417 df-ms 23418 df-tms 23419 df-cncf 23985 df-limc 24973 df-dv 24974 df-log 25655 df-asin 25958

This theorem is referenced by: asinrebnd 25994

W3C validator