reasinsin - Metamath Proof Explorer

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Theorem reasinsin 26936

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 26505	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 11235	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 26504	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 11235	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 26501	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 13017	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 13001	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11689	. . . . . . . . 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 11178	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11304	. . . . . . 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 11301	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13480	. . . . 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 1481	. . . 4 ⊢ ((-(π / 2)(,)(π / 2)) ∪ {-(π / 2), (π / 2)}) = (-(π / 2)[,](π / 2))
19	18	eleq2i 2853	. . 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 13374	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 11205	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 15232	. . . . 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 2861	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
27		asinsin 26932	. . . 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 4605	. . . 4 ⊢ (𝐴 ∈ {-(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 11126	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 26926	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘-1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘-1) = -(arcsin‘1)
33		asin1 26934	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11418	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2784	. . . . . 6 ⊢ (arcsin‘-1) = -(π / 2)
36		fveq2 6861	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘-(π / 2)))
37	3	recni 11191	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 16159	. . . . . . . . . 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 26508	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11418	. . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1
42	39, 41	eqtri 2784	. . . . . . . 8 ⊢ (sin‘-(π / 2)) = -1
43	36, 42	eqtrdi 2812	. . . . . . 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 2822	. . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
47		fveq2 6861	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2812	. . . . . . 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 2822	. . . . 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 1559 ∈ wcel 2141 ∪ cun 3902 {cpr 4583 class class class wbr 5099 ‘cfv 6515 (class class class)co 7390 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 ℝ^*cxr 11210 < clt 11211 ≤ cle 11212 -cneg 11410 / cdiv 11839 2c2 12267 ℝ⁺crp 12988 (,)cioo 13344 [,]cicc 13347 ℜcre 15105 sincsin 16074 πcpi 16077 arcsincasin 26902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-inf2 9591 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 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 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9303 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9453 df-card 9892 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-sub 11411 df-neg 11412 df-div 11840 df-nn 12206 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12477 df-z 12564 df-dec 12684 df-uz 12835 df-q 12945 df-rp 12989 df-xneg 13109 df-xadd 13110 df-xmul 13111 df-ioo 13348 df-ioc 13349 df-ico 13350 df-icc 13351 df-fz 13508 df-fzo 13655 df-fl 13797 df-mod 13875 df-seq 14010 df-exp 14070 df-fac 14282 df-bc 14311 df-hash 14339 df-shft 15075 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-limsup 15479 df-clim 15496 df-rlim 15497 df-sum 15695 df-ef 16078 df-sin 16080 df-cos 16081 df-pi 16083 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17248 df-plusg 17280 df-mulr 17281 df-starv 17282 df-sca 17283 df-vsca 17284 df-ip 17285 df-tset 17286 df-ple 17287 df-ds 17289 df-unif 17290 df-hom 17291 df-cco 17292 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17513 df-qtop 17518 df-imas 17519 df-xps 17521 df-mre 17595 df-mrc 17596 df-acs 17598 df-mgm 18655 df-sgrp 18734 df-mnd 18750 df-submnd 18799 df-mulg 19091 df-cntz 19338 df-cmn 19803 df-psmet 21394 df-xmet 21395 df-met 21396 df-bl 21397 df-mopn 21398 df-fbas 21399 df-fg 21400 df-cnfld 21403 df-top 22932 df-topon 22949 df-topsp 22971 df-bases 22984 df-cld 23057 df-ntr 23058 df-cls 23059 df-nei 23136 df-lp 23174 df-perf 23175 df-cn 23265 df-cnp 23266 df-haus 23353 df-tx 23600 df-hmeo 23793 df-fil 23884 df-fm 23976 df-flim 23977 df-flf 23978 df-xms 24358 df-ms 24359 df-tms 24360 df-cncf 24918 df-limc 25906 df-dv 25907 df-log 26596 df-asin 26905

This theorem is referenced by: asinrebnd 26941 asin1half 42919

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