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Theorem atanbnd 27049

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 27008	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
2	1	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ dom arctan)
3		atanneg 27030	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴))
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘-𝐴) = -(arctan‘𝐴))
5		renegcl 11509	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
6	5	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
7		lt0neg1 11708	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴))
8	7	biimpa 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴)
9	6, 8	elrpd 13048	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ⁺)
10		atanbndlem 27048	. . . . . 6 ⊢ (-𝐴 ∈ ℝ⁺ → (arctan‘-𝐴) ∈ (-(π / 2)(,)(π / 2)))
11	9, 10	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘-𝐴) ∈ (-(π / 2)(,)(π / 2)))
12	4, 11	eqeltrrd 2866	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
13		halfpire 26587	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14	13	recni 11211	. . . . . 6 ⊢ (π / 2) ∈ ℂ
15	14	negnegi 11516	. . . . 5 ⊢ --(π / 2) = (π / 2)
16	15	oveq2i 7411	. . . 4 ⊢ (-(π / 2)(,)--(π / 2)) = (-(π / 2)(,)(π / 2))
17	12, 16	eleqtrrdi 2876	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2)))
18		neghalfpire 26588	. . . 4 ⊢ -(π / 2) ∈ ℝ
19		atanrecl 27034	. . . . 5 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
20	19	adantr 485	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ ℝ)
21		iooneg 13489	. . . 4 ⊢ ((-(π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
22	18, 13, 20, 21	mp3an12i 1489	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
23	17, 22	mpbird 260	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
24		simpr 489	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → 𝐴 = 0)
25	24	fveq2d 6875	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = (arctan‘0))
26		atan0 27031	. . . 4 ⊢ (arctan‘0) = 0
27	25, 26	eqtrdi 2816	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = 0)
28		0re 11198	. . . 4 ⊢ 0 ∈ ℝ
29		pirp 26584	. . . . . 6 ⊢ π ∈ ℝ⁺
30		rphalfcl 13036	. . . . . 6 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
31		rpgt0 13020	. . . . . 6 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
32	29, 30, 31	mp2b 10	. . . . 5 ⊢ 0 < (π / 2)
33		lt0neg2 11709	. . . . . 6 ⊢ ((π / 2) ∈ ℝ → (0 < (π / 2) ↔ -(π / 2) < 0))
34	13, 33	ax-mp 5	. . . . 5 ⊢ (0 < (π / 2) ↔ -(π / 2) < 0)
35	32, 34	mpbi 233	. . . 4 ⊢ -(π / 2) < 0
36	18	rexri 11255	. . . . 5 ⊢ -(π / 2) ∈ ℝ^*
37	13	rexri 11255	. . . . 5 ⊢ (π / 2) ∈ ℝ^*
38		elioo2 13404	. . . . 5 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (0 ∈ (-(π / 2)(,)(π / 2)) ↔ (0 ∈ ℝ ∧ -(π / 2) < 0 ∧ 0 < (π / 2))))
39	36, 37, 38	mp2an 704	. . . 4 ⊢ (0 ∈ (-(π / 2)(,)(π / 2)) ↔ (0 ∈ ℝ ∧ -(π / 2) < 0 ∧ 0 < (π / 2)))
40	28, 35, 32, 39	mpbir3an 1358	. . 3 ⊢ 0 ∈ (-(π / 2)(,)(π / 2))
41	27, 40	eqeltrdi 2873	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
42		elrp 13009	. . 3 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
43		atanbndlem 27048	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
44	42, 43	sylbir 238	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
45		lttri4 11282	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
46	28, 45	mpan2 703	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
47	23, 41, 44, 46	mpjao3dan 1455	1 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ w3o 1100 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 class class class wbr 5105 dom cdm 5652 ‘cfv 6525 (class class class)co 7400 ℝcr 11087 0cc0 11088 ℝ^*cxr 11230 < clt 11231 -cneg 11430 / cdiv 11859 2c2 12286 ℝ⁺crp 13007 (,)cioo 13363 πcpi 16110 arctancatan 26987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-tan 16115 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-log 26679 df-atan 26990

This theorem is referenced by: atanord 27050

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