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Mirrors > Home > MPE Home > Th. List > coscld | Structured version Visualization version GIF version |
Description: Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
sincld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
coscld | ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sincld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | coscl 16124 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ‘cfv 6553 ℂcc 11152 cosccos 16061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-pm 8857 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-ico 13379 df-fz 13534 df-fzo 13677 df-fl 13807 df-seq 14017 df-exp 14077 df-fac 14286 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-cos 16067 |
This theorem is referenced by: tanadd 16164 addsin 16167 sincossq 16173 pilem2 26474 ptolemy 26516 efif1olem4 26564 ssscongptld 26842 chordthmlem 26852 heron 26858 cos2h 37272 tan2h 37273 dvtan 37331 sinmulcos 45435 dvsinax 45483 dvasinbx 45490 itgsin0pilem1 45520 itgsinexplem1 45524 itgcoscmulx 45539 itgsincmulx 45544 dirkertrigeqlem1 45668 dirkertrigeqlem2 45669 dirkertrigeqlem3 45670 dirkeritg 45672 dirkercncflem2 45674 fourierdlem39 45716 fourierdlem56 45732 fourierdlem57 45733 fourierdlem58 45734 fourierdlem62 45738 fourierdlem68 45744 fourierdlem73 45749 fouriersw 45801 |
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