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Mirrors > Home > MPE Home > Th. List > coscl | Structured version Visualization version GIF version |
Description: Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
coscl | ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosf 15468 | . 2 ⊢ cos:ℂ⟶ℂ | |
2 | 1 | ffvelrni 6843 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6349 ℂcc 10524 cosccos 15408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-ico 12734 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-fac 13624 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-cos 15414 |
This theorem is referenced by: tanval 15471 tancl 15472 coscld 15474 tanneg 15491 efmival 15496 sinadd 15507 cosadd 15508 tanaddlem 15509 sinsub 15511 cossub 15512 subsin 15514 sinmul 15515 cosmul 15516 addcos 15517 subcos 15518 sincossq 15519 sin2t 15520 cos2t 15521 cos2tsin 15522 demoivreALT 15544 sinhalfpilem 24978 sinmpi 25002 cosmpi 25003 sinppi 25004 cosppi 25005 efimpi 25006 sinhalfpip 25007 sinhalfpim 25008 coshalfpip 25009 coshalfpim 25010 asinsin 25397 acoscos 25398 atandmtan 25425 atantan 25428 sin2h 34764 cos2h 34765 tan2h 34766 dvtan 34824 itgsinexplem1 42119 itgsinexp 42120 dirkertrigeqlem1 42264 dirkertrigeqlem3 42266 seccl 44747 cotcl 44749 recsec 44753 reccot 44755 rectan 44756 onetansqsecsq 44758 cotsqcscsq 44759 |
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