| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 16162 | . 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 2108 ‘cfv 6561 ℂcc 11153 sincsin 16099 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 |
| This theorem is referenced by: retanhcl 16195 tanhlt1 16196 tanadd 16203 addsin 16206 sincossq 16212 sinkpi 26564 coseq1 26567 efif1olem4 26587 heron 26881 sin2h 37617 dvtan 37677 sineq0ALT 44957 sinmulcos 45880 dvcosre 45927 dvasinbx 45935 dvcosax 45941 itgsin0pilem1 45965 ibliccsinexp 45966 iblioosinexp 45968 itgsinexplem1 45969 itgsinexp 45970 itgcoscmulx 45984 itgsincmulx 45989 wallispilem2 46081 dirker2re 46107 dirkerdenne0 46108 dirkerper 46111 dirkertrigeqlem2 46114 dirkertrigeqlem3 46115 dirkeritg 46117 dirkercncflem2 46119 dirkercncflem4 46121 fourierdlem39 46161 fourierdlem43 46165 fourierdlem44 46166 fourierdlem56 46177 fourierdlem57 46178 fourierdlem58 46179 fourierdlem62 46183 fourierdlem68 46189 fourierdlem72 46193 fourierdlem73 46194 fourierdlem76 46197 fourierdlem80 46201 fourierdlem103 46224 fourierdlem104 46225 sqwvfoura 46243 sqwvfourb 46244 fouriersw 46246 sinh-conventional 49258 |
| Copyright terms: Public domain | W3C validator |