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| 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 25869 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
| 2 | 1 | fveq2i 6845 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
| 3 | pire 25813 | . . . . 5 ⊢ π ∈ ℝ | |
| 4 | 4nn 12235 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | nndivre 12193 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
| 6 | 3, 4, 5 | mp2an 690 | . . . 4 ⊢ (π / 4) ∈ ℝ |
| 7 | 6 | recni 11168 | . . 3 ⊢ (π / 4) ∈ ℂ |
| 8 | rere 15006 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
| 9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
| 10 | pirp 25816 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
| 11 | rphalfcl 12941 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
| 13 | rpgt0 12926 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
| 15 | halfpire 25819 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
| 16 | lt0neg2 11661 | . . . . . . . 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 229 | . . . . . 6 ⊢ -(π / 2) < 0 |
| 19 | nnrp 12925 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
| 20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
| 21 | rpdivcl 12939 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
| 22 | 10, 20, 21 | mp2an 690 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
| 23 | rpgt0 12926 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
| 24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
| 25 | neghalfpire 25820 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
| 26 | 0re 11156 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 27 | 25, 26, 6 | lttri 11280 | . . . . . 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 11168 | . . . . . . . 8 ⊢ π ∈ ℂ |
| 30 | 2cnne0 12362 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 31 | divdiv1 11865 | . . . . . . . 8 ⊢ ((π ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π / 2) / 2) = (π / (2 · 2))) | |
| 32 | 29, 30, 30, 31 | mp3an 1461 | . . . . . . 7 ⊢ ((π / 2) / 2) = (π / (2 · 2)) |
| 33 | 2t2e4 12316 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 34 | 33 | oveq2i 7367 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
| 35 | 32, 34 | eqtri 2764 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
| 36 | rphalflt 12943 | . . . . . . 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 5128 | . . . . 5 ⊢ (π / 4) < (π / 2) |
| 39 | 25 | rexri 11212 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
| 40 | 15 | rexri 11212 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
| 41 | elioo2 13304 | . . . . . 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 1341 | . . . 4 ⊢ (π / 4) ∈ (-(π / 2)(,)(π / 2)) |
| 44 | 9, 43 | eqeltri 2834 | . . 3 ⊢ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2)) |
| 45 | atantan 26271 | . . 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 2766 | 1 ⊢ (arctan‘1) = (π / 4) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7356 ℂcc 11048 ℝcr 11049 0cc0 11050 1c1 11051 · cmul 11055 ℝ*cxr 11187 < clt 11188 -cneg 11385 / cdiv 11811 ℕcn 12152 2c2 12207 4c4 12209 ℝ+crp 12914 (,)cioo 13263 ℜcre 14981 tanctan 15947 πcpi 15948 arctancatan 26212 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-mod 13774 df-seq 13906 df-exp 13967 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 df-sin 15951 df-cos 15952 df-tan 15953 df-pi 15954 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-rest 17303 df-topn 17304 df-0g 17322 df-gsum 17323 df-topgen 17324 df-pt 17325 df-prds 17328 df-xrs 17383 df-qtop 17388 df-imas 17389 df-xps 17391 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-mulg 18871 df-cntz 19095 df-cmn 19562 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-lp 22485 df-perf 22486 df-cn 22576 df-cnp 22577 df-haus 22664 df-tx 22911 df-hmeo 23104 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-xms 23671 df-ms 23672 df-tms 23673 df-cncf 24239 df-limc 25228 df-dv 25229 df-log 25910 df-atan 26215 |
| This theorem is referenced by: leibpi 26290 |
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