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Mirrors > Home > MPE Home > Th. List > atanf | Structured version Visualization version GIF version |
Description: Domain and range of the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
atanf | ⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-atan 24815 | . 2 ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) | |
2 | ovex 6823 | . . . . 5 ⊢ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ V | |
3 | 2, 1 | dmmpti 6163 | . . . 4 ⊢ dom arctan = (ℂ ∖ {-i, i}) |
4 | 3 | eleq2i 2842 | . . 3 ⊢ (𝑥 ∈ dom arctan ↔ 𝑥 ∈ (ℂ ∖ {-i, i})) |
5 | ax-icn 10197 | . . . . 5 ⊢ i ∈ ℂ | |
6 | halfcl 11459 | . . . . 5 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (i / 2) ∈ ℂ |
8 | ax-1cn 10196 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | atandm2 24825 | . . . . . . . . 9 ⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 − (i · 𝑥)) ≠ 0 ∧ (1 + (i · 𝑥)) ≠ 0)) | |
10 | 9 | simp1bi 1139 | . . . . . . . 8 ⊢ (𝑥 ∈ dom arctan → 𝑥 ∈ ℂ) |
11 | mulcl 10222 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ) | |
12 | 5, 10, 11 | sylancr 575 | . . . . . . 7 ⊢ (𝑥 ∈ dom arctan → (i · 𝑥) ∈ ℂ) |
13 | subcl 10482 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 − (i · 𝑥)) ∈ ℂ) | |
14 | 8, 12, 13 | sylancr 575 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 − (i · 𝑥)) ∈ ℂ) |
15 | 9 | simp2bi 1140 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 − (i · 𝑥)) ≠ 0) |
16 | 14, 15 | logcld 24538 | . . . . 5 ⊢ (𝑥 ∈ dom arctan → (log‘(1 − (i · 𝑥))) ∈ ℂ) |
17 | addcl 10220 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 + (i · 𝑥)) ∈ ℂ) | |
18 | 8, 12, 17 | sylancr 575 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 + (i · 𝑥)) ∈ ℂ) |
19 | 9 | simp3bi 1141 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 + (i · 𝑥)) ≠ 0) |
20 | 18, 19 | logcld 24538 | . . . . 5 ⊢ (𝑥 ∈ dom arctan → (log‘(1 + (i · 𝑥))) ∈ ℂ) |
21 | 16, 20 | subcld 10594 | . . . 4 ⊢ (𝑥 ∈ dom arctan → ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) ∈ ℂ) |
22 | mulcl 10222 | . . . 4 ⊢ (((i / 2) ∈ ℂ ∧ ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) ∈ ℂ) → ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ ℂ) | |
23 | 7, 21, 22 | sylancr 575 | . . 3 ⊢ (𝑥 ∈ dom arctan → ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ ℂ) |
24 | 4, 23 | sylbir 225 | . 2 ⊢ (𝑥 ∈ (ℂ ∖ {-i, i}) → ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ ℂ) |
25 | 1, 24 | fmpti 6525 | 1 ⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 {cpr 4318 dom cdm 5249 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 0cc0 10138 1c1 10139 ici 10140 + caddc 10141 · cmul 10143 − cmin 10468 -cneg 10469 / cdiv 10886 2c2 11272 logclog 24522 arctancatan 24812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851 df-log 24524 df-atan 24815 |
This theorem is referenced by: atancl 24829 atanval 24832 dvatan 24883 atancn 24884 |
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