Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > atan1 | Structured version Visualization version GIF version |
Description: The arctangent of 1 is π / 4. (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
atan1 | ⊢ (arctan‘1) = (π / 4) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tan4thpi 25099 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
2 | 1 | fveq2i 6672 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
3 | pire 25043 | . . . . 5 ⊢ π ∈ ℝ | |
4 | 4nn 11719 | . . . . 5 ⊢ 4 ∈ ℕ | |
5 | nndivre 11677 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 690 | . . . 4 ⊢ (π / 4) ∈ ℝ |
7 | 6 | recni 10654 | . . 3 ⊢ (π / 4) ∈ ℂ |
8 | rere 14480 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
10 | pirp 25046 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
11 | rphalfcl 12415 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
13 | rpgt0 12400 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
15 | halfpire 25049 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
16 | lt0neg2 11146 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ → (0 < (π / 2) ↔ -(π / 2) < 0)) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ (0 < (π / 2) ↔ -(π / 2) < 0) |
18 | 14, 17 | mpbi 232 | . . . . . 6 ⊢ -(π / 2) < 0 |
19 | nnrp 12399 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
21 | rpdivcl 12413 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
22 | 10, 20, 21 | mp2an 690 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
23 | rpgt0 12400 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
25 | neghalfpire 25050 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
26 | 0re 10642 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
27 | 25, 26, 6 | lttri 10765 | . . . . . 6 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 4)) → -(π / 2) < (π / 4)) |
28 | 18, 24, 27 | mp2an 690 | . . . . 5 ⊢ -(π / 2) < (π / 4) |
29 | 3 | recni 10654 | . . . . . . . 8 ⊢ π ∈ ℂ |
30 | 2cnne0 11846 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
31 | divdiv1 11350 | . . . . . . . 8 ⊢ ((π ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π / 2) / 2) = (π / (2 · 2))) | |
32 | 29, 30, 30, 31 | mp3an 1457 | . . . . . . 7 ⊢ ((π / 2) / 2) = (π / (2 · 2)) |
33 | 2t2e4 11800 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
34 | 33 | oveq2i 7166 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
35 | 32, 34 | eqtri 2844 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
36 | rphalflt 12417 | . . . . . . 7 ⊢ ((π / 2) ∈ ℝ+ → ((π / 2) / 2) < (π / 2)) | |
37 | 12, 36 | ax-mp 5 | . . . . . 6 ⊢ ((π / 2) / 2) < (π / 2) |
38 | 35, 37 | eqbrtrri 5088 | . . . . 5 ⊢ (π / 4) < (π / 2) |
39 | 25 | rexri 10698 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
40 | 15 | rexri 10698 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
41 | elioo2 12778 | . . . . . 6 ⊢ ((-(π / 2) ∈ ℝ* ∧ (π / 2) ∈ ℝ*) → ((π / 4) ∈ (-(π / 2)(,)(π / 2)) ↔ ((π / 4) ∈ ℝ ∧ -(π / 2) < (π / 4) ∧ (π / 4) < (π / 2)))) | |
42 | 39, 40, 41 | mp2an 690 | . . . . 5 ⊢ ((π / 4) ∈ (-(π / 2)(,)(π / 2)) ↔ ((π / 4) ∈ ℝ ∧ -(π / 2) < (π / 4) ∧ (π / 4) < (π / 2))) |
43 | 6, 28, 38, 42 | mpbir3an 1337 | . . . 4 ⊢ (π / 4) ∈ (-(π / 2)(,)(π / 2)) |
44 | 9, 43 | eqeltri 2909 | . . 3 ⊢ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2)) |
45 | atantan 25500 | . . 3 ⊢ (((π / 4) ∈ ℂ ∧ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2))) → (arctan‘(tan‘(π / 4))) = (π / 4)) | |
46 | 7, 44, 45 | mp2an 690 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (π / 4) |
47 | 2, 46 | eqtr3i 2846 | 1 ⊢ (arctan‘1) = (π / 4) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 · cmul 10541 ℝ*cxr 10673 < clt 10674 -cneg 10870 / cdiv 11296 ℕcn 11637 2c2 11691 4c4 11693 ℝ+crp 12388 (,)cioo 12737 ℜcre 14455 tanctan 15418 πcpi 15419 arctancatan 25441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-tan 15424 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464 df-log 25139 df-atan 25444 |
This theorem is referenced by: leibpi 25519 |
Copyright terms: Public domain | W3C validator |