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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 26399 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
| 2 | 1 | fveq2i 6843 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
| 3 | pire 26342 | . . . . 5 ⊢ π ∈ ℝ | |
| 4 | 4nn 12245 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | nndivre 12203 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
| 6 | 3, 4, 5 | mp2an 692 | . . . 4 ⊢ (π / 4) ∈ ℝ |
| 7 | 6 | recni 11164 | . . 3 ⊢ (π / 4) ∈ ℂ |
| 8 | rere 15064 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
| 9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
| 10 | pirp 26346 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
| 11 | rphalfcl 12956 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
| 13 | rpgt0 12940 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
| 15 | halfpire 26349 | . . . . . . . 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 230 | . . . . . 6 ⊢ -(π / 2) < 0 |
| 19 | nnrp 12939 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
| 20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
| 21 | rpdivcl 12954 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
| 22 | 10, 20, 21 | mp2an 692 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
| 23 | rpgt0 12940 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
| 24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
| 25 | neghalfpire 26350 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
| 26 | 0re 11152 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 27 | 25, 26, 6 | lttri 11276 | . . . . . 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 11164 | . . . . . . . 8 ⊢ π ∈ ℂ |
| 30 | 2cnne0 12367 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 31 | divdiv1 11869 | . . . . . . . 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 12321 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 34 | 33 | oveq2i 7380 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
| 35 | 32, 34 | eqtri 2752 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
| 36 | rphalflt 12958 | . . . . . . 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 5125 | . . . . 5 ⊢ (π / 4) < (π / 2) |
| 39 | 25 | rexri 11208 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
| 40 | 15 | rexri 11208 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
| 41 | elioo2 13323 | . . . . . 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 26809 | . . 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 5102 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 ℝ*cxr 11183 < clt 11184 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 4c4 12219 ℝ+crp 12927 (,)cioo 13282 ℜcre 15039 tanctan 16007 πcpi 16008 arctancatan 26750 |
| 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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 |
| 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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-tan 16013 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-log 26441 df-atan 26753 |
| This theorem is referenced by: leibpi 26828 |
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