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Mirrors > Home > MPE Home > Th. List > sincld | Structured version Visualization version GIF version |
Description: Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
sincld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
sincld | ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sincld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | sincl 15473 | . 2 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (sin‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6350 ℂcc 10529 sincsin 15411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 |
This theorem is referenced by: retanhcl 15506 tanhlt1 15507 tanadd 15514 addsin 15517 sincossq 15523 sinkpi 25101 coseq1 25104 efif1olem4 25123 heron 25410 sin2h 34876 dvtan 34936 sineq0ALT 41264 sinmulcos 42138 dvcosre 42188 dvasinbx 42197 dvcosax 42203 itgsin0pilem1 42227 ibliccsinexp 42228 iblioosinexp 42230 itgsinexplem1 42231 itgsinexp 42232 itgcoscmulx 42246 itgsincmulx 42251 wallispilem2 42344 dirker2re 42370 dirkerdenne0 42371 dirkerper 42374 dirkertrigeqlem2 42377 dirkertrigeqlem3 42378 dirkeritg 42380 dirkercncflem2 42382 dirkercncflem4 42384 fourierdlem39 42424 fourierdlem43 42428 fourierdlem44 42429 fourierdlem56 42440 fourierdlem57 42441 fourierdlem58 42442 fourierdlem62 42446 fourierdlem68 42452 fourierdlem72 42456 fourierdlem73 42457 fourierdlem76 42460 fourierdlem80 42464 fourierdlem103 42487 fourierdlem104 42488 sqwvfoura 42506 sqwvfourb 42507 fouriersw 42509 sinh-conventional 44831 |
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