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Theorem atanbnd 24698

Description: The arctangent function is bounded by π / 2 on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)

**Assertion**
Ref	Expression
atanbnd	⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

**Proof of Theorem atanbnd**
Step	Hyp	Ref	Expression
1		atanre 24657	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
2	1	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ dom arctan)
3		atanneg 24679	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘-𝐴) = -(arctan‘𝐴))
5		renegcl 10382	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
7		lt0neg1 10572	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴))
8	7	biimpa 500	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴)
9	6, 8	elrpd 11907	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ⁺)
10		atanbndlem 24697	. . . . . 6 ⊢ (-𝐴 ∈ ℝ⁺ → (arctan‘-𝐴) ∈ (-(π / 2)(,)(π / 2)))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘-𝐴) ∈ (-(π / 2)(,)(π / 2)))
12	4, 11	eqeltrrd 2731	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
13		halfpire 24261	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14	13	recni 10090	. . . . . 6 ⊢ (π / 2) ∈ ℂ
15	14	negnegi 10389	. . . . 5 ⊢ --(π / 2) = (π / 2)
16	15	oveq2i 6701	. . . 4 ⊢ (-(π / 2)(,)--(π / 2)) = (-(π / 2)(,)(π / 2))
17	12, 16	syl6eleqr 2741	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2)))
18		neghalfpire 24262	. . . . 5 ⊢ -(π / 2) ∈ ℝ
19	18	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(π / 2) ∈ ℝ)
20	13	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (π / 2) ∈ ℝ)
21		atanrecl 24683	. . . . 5 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
22	21	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ ℝ)
23		iooneg 12330	. . . 4 ⊢ ((-(π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
24	19, 20, 22, 23	syl3anc 1366	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
25	17, 24	mpbird 247	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
26		simpr 476	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → 𝐴 = 0)
27	26	fveq2d 6233	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = (arctan‘0))
28		atan0 24680	. . . 4 ⊢ (arctan‘0) = 0
29	27, 28	syl6eq 2701	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = 0)
30		0re 10078	. . . 4 ⊢ 0 ∈ ℝ
31		pirp 24258	. . . . . 6 ⊢ π ∈ ℝ⁺
32		rphalfcl 11896	. . . . . 6 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
33		rpgt0 11882	. . . . . 6 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
34	31, 32, 33	mp2b 10	. . . . 5 ⊢ 0 < (π / 2)
35		lt0neg2 10573	. . . . . 6 ⊢ ((π / 2) ∈ ℝ → (0 < (π / 2) ↔ -(π / 2) < 0))
36	13, 35	ax-mp 5	. . . . 5 ⊢ (0 < (π / 2) ↔ -(π / 2) < 0)
37	34, 36	mpbi 220	. . . 4 ⊢ -(π / 2) < 0
38	18	rexri 10135	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
39	13	rexri 10135	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
40		elioo2 12254	. . . . 5 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (0 ∈ (-(π / 2)(,)(π / 2)) ↔ (0 ∈ ℝ ∧ -(π / 2) < 0 ∧ 0 < (π / 2))))
41	38, 39, 40	mp2an 708	. . . 4 ⊢ (0 ∈ (-(π / 2)(,)(π / 2)) ↔ (0 ∈ ℝ ∧ -(π / 2) < 0 ∧ 0 < (π / 2)))
42	30, 37, 34, 41	mpbir3an 1263	. . 3 ⊢ 0 ∈ (-(π / 2)(,)(π / 2))
43	29, 42	syl6eqel 2738	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
44		elrp 11872	. . 3 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45		atanbndlem 24697	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
46	44, 45	sylbir 225	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
47		lttri4 10160	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
48	30, 47	mpan2 707	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
49	25, 43, 46, 48	mpjao3dan 1435	1 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∨ w3o 1053 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 ℝ^*cxr 10111 < clt 10112 -cneg 10305 / cdiv 10722 2c2 11108 ℝ⁺crp 11870 (,)cioo 12213 πcpi 14841 arctancatan 24636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-tan 14846 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348 df-atan 24639

This theorem is referenced by: atanord 24699

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