Nearby theorems
|
|
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 25206 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
| 2 | 1 | fveq2i 6661 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
| 3 | pire 25150 | . . . . 5 ⊢ π ∈ ℝ | |
| 4 | 4nn 11757 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | nndivre 11715 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
| 6 | 3, 4, 5 | mp2an 691 | . . . 4 ⊢ (π / 4) ∈ ℝ |
| 7 | 6 | recni 10693 | . . 3 ⊢ (π / 4) ∈ ℂ |
| 8 | rere 14529 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
| 9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
| 10 | pirp 25153 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
| 11 | rphalfcl 12457 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
| 13 | rpgt0 12442 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
| 15 | halfpire 25156 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
| 16 | lt0neg2 11185 | . . . . . . . 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 233 | . . . . . 6 ⊢ -(π / 2) < 0 |
| 19 | nnrp 12441 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
| 20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
| 21 | rpdivcl 12455 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
| 22 | 10, 20, 21 | mp2an 691 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
| 23 | rpgt0 12442 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
| 24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
| 25 | neghalfpire 25157 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
| 26 | 0re 10681 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 27 | 25, 26, 6 | lttri 10804 | . . . . . 6 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 4)) → -(π / 2) < (π / 4)) |
| 28 | 18, 24, 27 | mp2an 691 | . . . . 5 ⊢ -(π / 2) < (π / 4) |
| 29 | 3 | recni 10693 | . . . . . . . 8 ⊢ π ∈ ℂ |
| 30 | 2cnne0 11884 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 31 | divdiv1 11389 | . . . . . . . 8 ⊢ ((π ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π / 2) / 2) = (π / (2 · 2))) | |
| 32 | 29, 30, 30, 31 | mp3an 1458 | . . . . . . 7 ⊢ ((π / 2) / 2) = (π / (2 · 2)) |
| 33 | 2t2e4 11838 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 34 | 33 | oveq2i 7161 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
| 35 | 32, 34 | eqtri 2781 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
| 36 | rphalflt 12459 | . . . . . . 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 5055 | . . . . 5 ⊢ (π / 4) < (π / 2) |
| 39 | 25 | rexri 10737 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
| 40 | 15 | rexri 10737 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
| 41 | elioo2 12820 | . . . . . 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 691 | . . . . 5 ⊢ ((π / 4) ∈ (-(π / 2)(,)(π / 2)) ↔ ((π / 4) ∈ ℝ ∧ -(π / 2) < (π / 4) ∧ (π / 4) < (π / 2))) |
| 43 | 6, 28, 38, 42 | mpbir3an 1338 | . . . 4 ⊢ (π / 4) ∈ (-(π / 2)(,)(π / 2)) |
| 44 | 9, 43 | eqeltri 2848 | . . 3 ⊢ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2)) |
| 45 | atantan 25608 | . . 3 ⊢ (((π / 4) ∈ ℂ ∧ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2))) → (arctan‘(tan‘(π / 4))) = (π / 4)) | |
| 46 | 7, 44, 45 | mp2an 691 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (π / 4) |
| 47 | 2, 46 | eqtr3i 2783 | 1 ⊢ (arctan‘1) = (π / 4) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 class class class wbr 5032 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 · cmul 10580 ℝ*cxr 10712 < clt 10713 -cneg 10909 / cdiv 11335 ℕcn 11674 2c2 11729 4c4 11731 ℝ+crp 12430 (,)cioo 12779 ℜcre 14504 tanctan 15467 πcpi 15468 arctancatan 25549 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655 |
| This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ioo 12783 df-ioc 12784 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-mod 13287 df-seq 13419 df-exp 13480 df-fac 13684 df-bc 13713 df-hash 13741 df-shft 14474 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-limsup 14876 df-clim 14893 df-rlim 14894 df-sum 15091 df-ef 15469 df-sin 15471 df-cos 15472 df-tan 15473 df-pi 15474 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-limc 24565 df-dv 24566 df-log 25247 df-atan 25552 |
| This theorem is referenced by: leibpi 25627 |
| Copyright terms: Public domain | W3C validator |