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Mirrors > Home > MPE Home > Th. List > atanbnd | Structured version Visualization version GIF version |
Description: The arctangent function is bounded by π / 2 on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.) |
Ref | Expression |
---|---|
atanbnd | ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atanre 24412 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ∈ dom arctan) |
3 | atanneg 24434 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (arctan‘-𝐴) = -(arctan‘𝐴)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘-𝐴) = -(arctan‘𝐴)) |
5 | renegcl 10223 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
7 | lt0neg1 10413 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
8 | 7 | biimpa 500 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴) |
9 | 6, 8 | elrpd 11745 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ+) |
10 | atanbndlem 24452 | . . . . . 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 2689 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
13 | halfpire 24020 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ | |
14 | 13 | recni 9931 | . . . . . 6 ⊢ (π / 2) ∈ ℂ |
15 | 14 | negnegi 10230 | . . . . 5 ⊢ --(π / 2) = (π / 2) |
16 | 15 | oveq2i 6560 | . . . 4 ⊢ (-(π / 2)(,)--(π / 2)) = (-(π / 2)(,)(π / 2)) |
17 | 12, 16 | syl6eleqr 2699 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2))) |
18 | neghalfpire 24021 | . . . . 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 24438 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ) | |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ ℝ) |
23 | iooneg 12163 | . . . 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 1318 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(arctan‘𝐴) ∈ (-(π / 2)(,)--(π / 2)))) |
25 | 17, 24 | mpbird 246 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
26 | simpr 476 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → 𝐴 = 0) | |
27 | 26 | fveq2d 6107 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = (arctan‘0)) |
28 | atan0 24435 | . . . 4 ⊢ (arctan‘0) = 0 | |
29 | 27, 28 | syl6eq 2660 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) = 0) |
30 | 0re 9919 | . . . 4 ⊢ 0 ∈ ℝ | |
31 | pirp 24017 | . . . . . 6 ⊢ π ∈ ℝ+ | |
32 | rphalfcl 11734 | . . . . . 6 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
33 | rpgt0 11720 | . . . . . 6 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
34 | 31, 32, 33 | mp2b 10 | . . . . 5 ⊢ 0 < (π / 2) |
35 | lt0neg2 10414 | . . . . . 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 219 | . . . 4 ⊢ -(π / 2) < 0 |
38 | 18 | rexri 9976 | . . . . 5 ⊢ -(π / 2) ∈ ℝ* |
39 | 13 | rexri 9976 | . . . . 5 ⊢ (π / 2) ∈ ℝ* |
40 | elioo2 12087 | . . . . 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 704 | . . . 4 ⊢ (0 ∈ (-(π / 2)(,)(π / 2)) ↔ (0 ∈ ℝ ∧ -(π / 2) < 0 ∧ 0 < (π / 2))) |
42 | 30, 37, 34, 41 | mpbir3an 1237 | . . 3 ⊢ 0 ∈ (-(π / 2)(,)(π / 2)) |
43 | 29, 42 | syl6eqel 2696 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
44 | elrp 11710 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
45 | atanbndlem 24452 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) | |
46 | 44, 45 | sylbir 224 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
47 | lttri4 10001 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴)) | |
48 | 30, 47 | mpan2 703 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴)) |
49 | 25, 43, 46, 48 | mpjao3dan 1387 | 1 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ℝ*cxr 9952 < clt 9953 -cneg 10146 / cdiv 10563 2c2 10947 ℝ+crp 11708 (,)cioo 12046 πcpi 14636 arctancatan 24391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-atan 24394 |
This theorem is referenced by: atanord 24454 |
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