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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 26421 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
| 2 | 1 | fveq2i 6825 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
| 3 | pire 26364 | . . . . 5 ⊢ π ∈ ℝ | |
| 4 | 4nn 12211 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | nndivre 12169 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
| 6 | 3, 4, 5 | mp2an 692 | . . . 4 ⊢ (π / 4) ∈ ℝ |
| 7 | 6 | recni 11129 | . . 3 ⊢ (π / 4) ∈ ℂ |
| 8 | rere 15029 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
| 9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
| 10 | pirp 26368 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
| 11 | rphalfcl 12922 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
| 13 | rpgt0 12906 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
| 15 | halfpire 26371 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
| 16 | lt0neg2 11627 | . . . . . . . 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 230 | . . . . . 6 ⊢ -(π / 2) < 0 |
| 19 | nnrp 12905 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
| 20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
| 21 | rpdivcl 12920 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
| 22 | 10, 20, 21 | mp2an 692 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
| 23 | rpgt0 12906 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
| 24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
| 25 | neghalfpire 26372 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
| 26 | 0re 11117 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 27 | 25, 26, 6 | lttri 11242 | . . . . . 6 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 4)) → -(π / 2) < (π / 4)) |
| 28 | 18, 24, 27 | mp2an 692 | . . . . 5 ⊢ -(π / 2) < (π / 4) |
| 29 | 3 | recni 11129 | . . . . . . . 8 ⊢ π ∈ ℂ |
| 30 | 2cnne0 12333 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 31 | divdiv1 11835 | . . . . . . . 8 ⊢ ((π ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π / 2) / 2) = (π / (2 · 2))) | |
| 32 | 29, 30, 30, 31 | mp3an 1463 | . . . . . . 7 ⊢ ((π / 2) / 2) = (π / (2 · 2)) |
| 33 | 2t2e4 12287 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 34 | 33 | oveq2i 7360 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
| 35 | 32, 34 | eqtri 2752 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
| 36 | rphalflt 12924 | . . . . . . 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 5115 | . . . . 5 ⊢ (π / 4) < (π / 2) |
| 39 | 25 | rexri 11173 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
| 40 | 15 | rexri 11173 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
| 41 | elioo2 13289 | . . . . . 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 692 | . . . . 5 ⊢ ((π / 4) ∈ (-(π / 2)(,)(π / 2)) ↔ ((π / 4) ∈ ℝ ∧ -(π / 2) < (π / 4) ∧ (π / 4) < (π / 2))) |
| 43 | 6, 28, 38, 42 | mpbir3an 1342 | . . . 4 ⊢ (π / 4) ∈ (-(π / 2)(,)(π / 2)) |
| 44 | 9, 43 | eqeltri 2824 | . . 3 ⊢ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2)) |
| 45 | atantan 26831 | . . 3 ⊢ (((π / 4) ∈ ℂ ∧ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2))) → (arctan‘(tan‘(π / 4))) = (π / 4)) | |
| 46 | 7, 44, 45 | mp2an 692 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (π / 4) |
| 47 | 2, 46 | eqtr3i 2754 | 1 ⊢ (arctan‘1) = (π / 4) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 ℝ*cxr 11148 < clt 11149 -cneg 11348 / cdiv 11777 ℕcn 12128 2c2 12183 4c4 12185 ℝ+crp 12893 (,)cioo 13248 ℜcre 15004 tanctan 15972 πcpi 15973 arctancatan 26772 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-atan 26775 |
| This theorem is referenced by: leibpi 26850 |
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