reasinsin - Metamath Proof Explorer

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Theorem reasinsin 27281

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 26851	. . . . . 6 ⊢ -(π / 2) ∈ ℝ
2	1	rexri 11406	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
3		halfpire 26850	. . . . . 6 ⊢ (π / 2) ∈ ℝ
4	3	rexri 11406	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
5		pirp 26847	. . . . . . . . . 10 ⊢ π ∈ ℝ⁺
6		rphalfcl 13189	. . . . . . . . . 10 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (π / 2) ∈ ℝ⁺
8		rpgt0 13173	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ 0 < (π / 2)
10		lt0neg2 11863	. . . . . . . . 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 11349	. . . . . . . 8 ⊢ 0 ∈ ℝ
14	1, 13, 3	lttri 11475	. . . . . . 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 11472	. . . . 5 ⊢ -(π / 2) ≤ (π / 2)
17		prunioo 13652	. . . . 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 1491	. . . 4 ⊢ (( -(π / 2)(,)(π / 2)) ∪ { -(π / 2), (π / 2)}) = ( -(π / 2)[,](π / 2))
19	18	eleq2i 2856	. . 3 ⊢ (𝐴 ∈ (( -(π / 2)(,)(π / 2)) ∪ { -(π / 2), (π / 2)}) ↔ 𝐴 ∈ ( -(π / 2)[,](π / 2)))
20		elun 4155	. . 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 13546	. . . . 5 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
23	22	recnd 11376	. . . 4 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
24	22	rered 15434	. . . . 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 2864	. . . 4 ⊢ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ ( -(π / 2)(,)(π / 2)))
27		asinsin 27277	. . . 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 4608	. . . 4 ⊢ (𝐴 ∈ { -(π / 2), (π / 2)} → (𝐴 = -(π / 2) ∨ 𝐴 = (π / 2)))
30		ax-1cn 11297	. . . . . . . 8 ⊢ 1 ∈ ℂ
31		asinneg 27271	. . . . . . . 8 ⊢ (1 ∈ ℂ → (arcsin‘ -1) = -(arcsin‘1))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (arcsin‘ -1) = -(arcsin‘1)
33		asin1 27279	. . . . . . . 8 ⊢ (arcsin‘1) = (π / 2)
34	33	negeqi 11589	. . . . . . 7 ⊢ -(arcsin‘1) = -(π / 2)
35	32, 34	eqtri 2786	. . . . . 6 ⊢ (arcsin‘ -1) = -(π / 2)
36		fveq2 6916	. . . . . . . 8 ⊢ (𝐴 = -(π / 2) → (sin‘𝐴) = (sin‘ -(π / 2)))
37	3	recni 11362	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
38		sinneg 16358	. . . . . . . . . 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 26854	. . . . . . . . . 10 ⊢ (sin‘(π / 2)) = 1
41	40	negeqi 11589	. . . . . . . . 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 6920	. . . . . 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 6916	. . . . . . . 8 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = (sin‘(π / 2)))
48	47, 40	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = (π / 2) → (sin‘𝐴) = 1)
49	48	fveq2d 6920	. . . . . 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 1570 ∈ wcel 2147 ∪ cun 4152 {cpr 4590 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 -cneg 11581 / cdiv 12013 2c2 12437 ℝ⁺crp 13160 (,)cioo 13516 [,]cicc 13519 ℜcre 15307 sincsin 16273 πcpi 16276 arcsincasin 27247

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-asin 27250

This theorem is used by: asinrebnd 27286 asin1half 43674

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