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Mirrors > Home > MPE Home > Th. List > atanf | Structured version Visualization version GIF version |
Description: Domain and codoamin 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 26924 | . 2 ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) | |
2 | ovex 7463 | . . . . 5 ⊢ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ V | |
3 | 2, 1 | dmmpti 6712 | . . . 4 ⊢ dom arctan = (ℂ ∖ {-i, i}) |
4 | 3 | eleq2i 2830 | . . 3 ⊢ (𝑥 ∈ dom arctan ↔ 𝑥 ∈ (ℂ ∖ {-i, i})) |
5 | ax-icn 11211 | . . . . 5 ⊢ i ∈ ℂ | |
6 | halfcl 12488 | . . . . 5 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (i / 2) ∈ ℂ |
8 | ax-1cn 11210 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | atandm2 26934 | . . . . . . . . 9 ⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 − (i · 𝑥)) ≠ 0 ∧ (1 + (i · 𝑥)) ≠ 0)) | |
10 | 9 | simp1bi 1144 | . . . . . . . 8 ⊢ (𝑥 ∈ dom arctan → 𝑥 ∈ ℂ) |
11 | mulcl 11236 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ) | |
12 | 5, 10, 11 | sylancr 587 | . . . . . . 7 ⊢ (𝑥 ∈ dom arctan → (i · 𝑥) ∈ ℂ) |
13 | subcl 11504 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 − (i · 𝑥)) ∈ ℂ) | |
14 | 8, 12, 13 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 − (i · 𝑥)) ∈ ℂ) |
15 | 9 | simp2bi 1145 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 − (i · 𝑥)) ≠ 0) |
16 | 14, 15 | logcld 26626 | . . . . 5 ⊢ (𝑥 ∈ dom arctan → (log‘(1 − (i · 𝑥))) ∈ ℂ) |
17 | addcl 11234 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 + (i · 𝑥)) ∈ ℂ) | |
18 | 8, 12, 17 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 + (i · 𝑥)) ∈ ℂ) |
19 | 9 | simp3bi 1146 | . . . . . 6 ⊢ (𝑥 ∈ dom arctan → (1 + (i · 𝑥)) ≠ 0) |
20 | 18, 19 | logcld 26626 | . . . . 5 ⊢ (𝑥 ∈ dom arctan → (log‘(1 + (i · 𝑥))) ∈ ℂ) |
21 | 16, 20 | subcld 11617 | . . . 4 ⊢ (𝑥 ∈ dom arctan → ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) ∈ ℂ) |
22 | mulcl 11236 | . . . 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 587 | . . 3 ⊢ (𝑥 ∈ dom arctan → ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ ℂ) |
24 | 4, 23 | sylbir 235 | . 2 ⊢ (𝑥 ∈ (ℂ ∖ {-i, i}) → ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ ℂ) |
25 | 1, 24 | fmpti 7131 | 1 ⊢ arctan:(ℂ ∖ {-i, i})⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {cpr 4632 dom cdm 5688 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 ici 11154 + caddc 11155 · cmul 11157 − cmin 11489 -cneg 11490 / cdiv 11917 2c2 12318 logclog 26610 arctancatan 26921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-atan 26924 |
This theorem is referenced by: atancl 26938 atanval 26941 dvatan 26992 atancn 26993 |
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