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 26576 | . . 3 ⊢ (tan‘(π / 4)) = 1 | |
| 2 | 1 | fveq2i 6870 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (arctan‘1) |
| 3 | pire 26516 | . . . . 5 ⊢ π ∈ ℝ | |
| 4 | 4nn 12301 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 5 | nndivre 12254 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
| 6 | 3, 4, 5 | mp2an 702 | . . . 4 ⊢ (π / 4) ∈ ℝ |
| 7 | 6 | recni 11196 | . . 3 ⊢ (π / 4) ∈ ℂ |
| 8 | rere 15149 | . . . . 5 ⊢ ((π / 4) ∈ ℝ → (ℜ‘(π / 4)) = (π / 4)) | |
| 9 | 6, 8 | ax-mp 5 | . . . 4 ⊢ (ℜ‘(π / 4)) = (π / 4) |
| 10 | pirp 26523 | . . . . . . . . 9 ⊢ π ∈ ℝ+ | |
| 11 | rphalfcl 13022 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ+ |
| 13 | rpgt0 13006 | . . . . . . . 8 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 0 < (π / 2) |
| 15 | halfpire 26526 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
| 16 | lt0neg2 11694 | . . . . . . . 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 13005 | . . . . . . . . 9 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
| 20 | 4, 19 | ax-mp 5 | . . . . . . . 8 ⊢ 4 ∈ ℝ+ |
| 21 | rpdivcl 13020 | . . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 4 ∈ ℝ+) → (π / 4) ∈ ℝ+) | |
| 22 | 10, 20, 21 | mp2an 702 | . . . . . . 7 ⊢ (π / 4) ∈ ℝ+ |
| 23 | rpgt0 13006 | . . . . . . 7 ⊢ ((π / 4) ∈ ℝ+ → 0 < (π / 4)) | |
| 24 | 22, 23 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 4) |
| 25 | neghalfpire 26527 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
| 26 | 0re 11183 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 27 | 25, 26, 6 | lttri 11309 | . . . . . 6 ⊢ ((-(π / 2) < 0 ∧ 0 < (π / 4)) → -(π / 2) < (π / 4)) |
| 28 | 18, 24, 27 | mp2an 702 | . . . . 5 ⊢ -(π / 2) < (π / 4) |
| 29 | 3 | recni 11196 | . . . . . . . 8 ⊢ π ∈ ℂ |
| 30 | 2cnne0 12430 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 31 | divdiv1 11902 | . . . . . . . 8 ⊢ ((π ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((π / 2) / 2) = (π / (2 · 2))) | |
| 32 | 29, 30, 30, 31 | mp3an 1482 | . . . . . . 7 ⊢ ((π / 2) / 2) = (π / (2 · 2)) |
| 33 | 2t2e4 12381 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 34 | 33 | oveq2i 7407 | . . . . . . 7 ⊢ (π / (2 · 2)) = (π / 4) |
| 35 | 32, 34 | eqtri 2785 | . . . . . 6 ⊢ ((π / 2) / 2) = (π / 4) |
| 36 | rphalflt 13024 | . . . . . . 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 5123 | . . . . 5 ⊢ (π / 4) < (π / 2) |
| 39 | 25 | rexri 11240 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* |
| 40 | 15 | rexri 11240 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
| 41 | elioo2 13390 | . . . . . 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 702 | . . . . 5 ⊢ ((π / 4) ∈ (-(π / 2)(,)(π / 2)) ↔ ((π / 4) ∈ ℝ ∧ -(π / 2) < (π / 4) ∧ (π / 4) < (π / 2))) |
| 43 | 6, 28, 38, 42 | mpbir3an 1355 | . . . 4 ⊢ (π / 4) ∈ (-(π / 2)(,)(π / 2)) |
| 44 | 9, 43 | eqeltri 2858 | . . 3 ⊢ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2)) |
| 45 | atantan 26985 | . . 3 ⊢ (((π / 4) ∈ ℂ ∧ (ℜ‘(π / 4)) ∈ (-(π / 2)(,)(π / 2))) → (arctan‘(tan‘(π / 4))) = (π / 4)) | |
| 46 | 7, 44, 45 | mp2an 702 | . 2 ⊢ (arctan‘(tan‘(π / 4))) = (π / 4) |
| 47 | 2, 46 | eqtr3i 2787 | 1 ⊢ (arctan‘1) = (π / 4) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 · cmul 11078 ℝ*cxr 11215 < clt 11216 -cneg 11415 / cdiv 11844 ℕcn 12210 2c2 12272 4c4 12274 ℝ+crp 12993 (,)cioo 13349 ℜcre 15124 tanctan 16095 πcpi 16096 arctancatan 26926 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-tan 16101 df-pi 16102 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-fbas 21418 df-fg 21419 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-nei 23155 df-lp 23193 df-perf 23194 df-cn 23284 df-cnp 23285 df-haus 23372 df-tx 23619 df-hmeo 23812 df-fil 23903 df-fm 23995 df-flim 23996 df-flf 23997 df-xms 24377 df-ms 24378 df-tms 24379 df-cncf 24937 df-limc 25925 df-dv 25926 df-log 26618 df-atan 26929 |
| This theorem is referenced by: leibpi 27004 |
| Copyright terms: Public domain | W3C validator |