reasinsin - Metamath Proof Explorer

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Theorem reasinsin 26961

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 26530	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 11240	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 26529	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 11240	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 26526	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 13022	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 13006	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11694	. . . . . . . . 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 232	. . . . . . 7 ⊢ -(π / 2) < 0
13		0re 11183	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11309	. . . . . . 7 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2))
15	12, 9, 14	mp2an 702	. . . . . 6 ⊢ -(π / 2) < (π / 2)
16	1, 3, 15	ltleii 11306	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13485	. . . . 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 1482	. . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2))
19	18	eleq2i 2854	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ 𝐴 ∈ (-(π / 2)[,](π / 2)))
20		elun 4106	. . 3 ⊢ (𝐴 ∈ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
21	19, 20	bitr3i 279	. 2 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) ↔ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}))
22		elioore 13379	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 11210	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 15251	. . . . 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 2862	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
27		asinsin 26957	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
28	23, 26, 27	syl2anc 593	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)
29		elpri 4606	. . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 11131	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 26951	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1)
33		asin1 26959	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11423	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2785	. . . . . 6 ⊢ (arcsin‘-1) = -(π / 2)
36		fveq2 6867	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2)))
37	3	recni 11196	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 16178	. . . . . . . . . 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 26533	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11423	. . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1
42	39, 41	eqtri 2785	. . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1
43	36, 42	eqtrdi 2813	. . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1)
44	43	fveq2d 6871	. . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘-1))
45		id 22	. . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2))
46	35, 44, 45	3eqtr4a 2823	. . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
47		fveq2 6867	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2813	. . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1)
49	48	fveq2d 6871	. . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1))
50		id 22	. . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2))
51	33, 49, 50	3eqtr4a 2823	. . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
52	46, 51	jaoi 868	. . . 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 868	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ {-(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴)
55	21, 54	sylbi 219	1 ⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∨ wo 858 = wceq 1560 ∈ wcel 2142 ∪ cun 3902 {cpr 4584 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 -cneg 11415 / cdiv 11844 2c2 12272 ℝ⁺crp 12993 (,)cioo 13349 [,]cicc 13352 ℜcre 15124 sincsin 16093 πcpi 16096 arcsincasin 26927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-pi 16102 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-limc 25928 df-dv 25929 df-log 26621 df-asin 26930

This theorem is referenced by: asinrebnd 26966 asin1half 42966

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