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Mirrors > Home > MPE Home > Th. List > sincl | Structured version Visualization version GIF version |
Description: Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
sincl | ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sinf 16120 | . 2 ⊢ sin:ℂ⟶ℂ | |
2 | 1 | ffvelcdmi 7088 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ‘cfv 6545 ℂcc 11146 sincsin 16059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-inf2 9676 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-isom 6554 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8849 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-sup 9477 df-inf 9478 df-oi 9545 df-card 9974 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-n0 12518 df-z 12604 df-uz 12868 df-rp 13022 df-ico 13377 df-fz 13532 df-fzo 13675 df-fl 13805 df-seq 14015 df-exp 14075 df-fac 14285 df-hash 14342 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15467 df-clim 15484 df-rlim 15485 df-sum 15685 df-ef 16063 df-sin 16065 |
This theorem is referenced by: tancl 16125 sincld 16126 tanneg 16144 sin0 16145 efmival 16149 sinadd 16160 cosadd 16161 tanaddlem 16162 sinsub 16164 cossub 16165 subsin 16167 sinmul 16168 cosmul 16169 addcos 16170 subcos 16171 sincossq 16172 sin2t 16173 cos2t 16174 cos2tsin 16175 demoivreALT 16197 sinhalfpilem 26487 sinmpi 26511 cosmpi 26512 sinppi 26513 cosppi 26514 efimpi 26515 sinhalfpip 26516 sinhalfpim 26517 coshalfpip 26518 coshalfpim 26519 sincos6thpi 26539 abssinper 26544 asinsin 26916 atandmtan 26944 atantan 26947 sin2h 37323 tan2h 37325 dvtan 37383 dvcosax 45582 itgsinexplem1 45610 itgsinexp 45611 csccl 48532 cotcl 48533 reccsc 48538 reccot 48539 rectan 48540 onetansqsecsq 48542 cotsqcscsq 48543 |
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