reasinsin - Metamath Proof Explorer

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Theorem reasinsin 27236

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 26806	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 11377	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 26805	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 11377	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 26802	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 13159	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 13143	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11833	. . . . . . . . 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 233	. . . . . . 7 ⊢ -(π / 2) < 0
13		0re 11320	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11446	. . . . . . 7 ⊢ (( -(π / 2) < 0 ∧ 0 < (π / 2)) → -(π / 2) < (π / 2))
15	12, 9, 14	mp2an 705	. . . . . 6 ⊢ -(π / 2) < (π / 2)
16	1, 3, 15	ltleii 11443	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13622	. . . . 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 1490	. . . 4 ⊢ (( -(π / 2)(,)(π / 2)) ∪ { -(π / 2), (π / 2)}) = ( -(π / 2)[,](π / 2))
19	18	eleq2i 2855	. . 3 ⊢ (𝐴 ∈ (( -(π / 2)(,)(π / 2)) ∪ { -(π / 2), (π / 2)}) ↔ 𝐴 ∈ ( -(π / 2)[,](π / 2)))
20		elun 4154	. . 3 ⊢ (𝐴 ∈ (( -(π / 2)(,)(π / 2)) ∪ { -(π / 2), (π / 2)}) ↔ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ { -(π / 2), (π / 2)}))
21	19, 20	bitr3i 280	. 2 ⊢ (𝐴 ∈ ( -(π / 2)[,](π / 2)) ↔ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ { -(π / 2), (π / 2)}))
22		elioore 13516	. . . . 5 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 11347	. . . 4 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 15403	. . . . 5 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (ℜ‘𝐴) = 𝐴)
25		id 23	. . . . 5 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ( -(π / 2)(,)(π / 2)))
26	24, 25	eqeltrd 2863	. . . 4 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2)))
27		asinsin 27232	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)
28	23, 26, 27	syl2anc 596	. . 3 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)
29		elpri 4607	. . . 4 ⊢ (𝐴 ∈ { -(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 11268	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 27226	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘ -1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘ -1) = -(arcsin‘1)
33		asin1 27234	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11560	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2786	. . . . . 6 ⊢ (arcsin‘ -1) = -(π / 2)
36		fveq2 6893	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘ -(π / 2)))
37	3	recni 11333	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 16326	. . . . . . . . . 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 26809	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11560	. . . . . . . . 9 ⊢ -(sin‘(π / 2)) = -1
42	39, 41	eqtri 2786	. . . . . . . 8 ⊢ (sin‘ -(π / 2)) = -1
43	36, 42	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = -1)
44	43	fveq2d 6897	. . . . . 6 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘ -1))
45		id 23	. . . . . 6 ⊢ (𝐴 = -(π / 2) → 𝐴 = -(π / 2))
46	35, 44, 45	3eqtr4a 2824	. . . . 5 ⊢ (𝐴 = -(π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
47		fveq2 6893	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1)
49	48	fveq2d 6897	. . . . . 6 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = (arcsin‘1))
50		id 23	. . . . . 6 ⊢ (𝐴 = (π / 2) → 𝐴 = (π / 2))
51	33, 49, 50	3eqtr4a 2824	. . . . 5 ⊢ (𝐴 = (π / 2) → (arcsin‘(sin‘𝐴)) = 𝐴)
52	46, 51	jaoi 871	. . . 4 ⊢ ((𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)
53	29, 52	syl 18	. . 3 ⊢ (𝐴 ∈ { -(π / 2), (π / 2)} → (arcsin‘(sin‘𝐴)) = 𝐴)
54	28, 53	jaoi 871	. 2 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∨ 𝐴 ∈ { -(π / 2), (π / 2)}) → (arcsin‘(sin‘𝐴)) = 𝐴)
55	21, 54	sylbi 220	1 ⊢ (𝐴 ∈ ( -(π / 2)[,](π / 2)) → (arcsin‘(sin‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∨ wo 861 = wceq 1569 ∈ wcel 2146 ∪ cun 4151 {cpr 4589 class class class wbr 5105 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 ℝcr 11209 0cc0 11210 1c1 11211 ℝ^*cxr 11352 < clt 11353 ≤ cle 11354 -cneg 11552 / cdiv 11983 2c2 12407 ℝ⁺crp 13130 (,)cioo 13486 [,]cicc 13489 ℜcre 15276 sincsin 16241 πcpi 16244 arcsincasin 27202

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-pm 8861 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-fi 9413 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-ioo 13490 df-ioc 13491 df-ico 13492 df-icc 13493 df-fz 13650 df-fzo 13799 df-fl 13942 df-mod 14020 df-seq 14155 df-exp 14215 df-fac 14428 df-bc 14457 df-hash 14485 df-shft 15232 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-limsup 15650 df-clim 15667 df-rlim 15668 df-sum 15866 df-ef 16245 df-sin 16247 df-cos 16248 df-pi 16250 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-hom 17464 df-cco 17465 df-rest 17605 df-topn 17606 df-0g 17624 df-gsum 17625 df-topgen 17626 df-pt 17627 df-prds 17630 df-xrs 17686 df-qtop 17691 df-imas 17692 df-xps 17694 df-mre 17768 df-mrc 17769 df-acs 17771 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-mulg 19290 df-cntz 19543 df-cmn 20008 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-fbas 21687 df-fg 21688 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cld 23349 df-ntr 23350 df-cls 23351 df-nei 23428 df-lp 23466 df-perf 23467 df-cn 23557 df-cnp 23558 df-haus 23645 df-tx 23893 df-hmeo 24086 df-fil 24177 df-fm 24269 df-flim 24270 df-flf 24271 df-xms 24651 df-ms 24652 df-tms 24653 df-cncf 25211 df-limc 26198 df-dv 26199 df-log 26896 df-asin 27205

This theorem is used by: asinrebnd 27241 asin1half 43548

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