![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > resincl | Structured version Visualization version GIF version |
Description: The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
resincl | ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resinval 15237 | . 2 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) | |
2 | ax-icn 10311 | . . . . 5 ⊢ i ∈ ℂ | |
3 | recn 10342 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 10336 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 583 | . . . 4 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | efcl 15185 | . . . 4 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) ∈ ℂ) |
8 | 7 | imcld 14312 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) ∈ ℝ) |
9 | 1, 8 | eqeltrd 2906 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 ici 10254 · cmul 10257 ℑcim 14215 expce 15164 sincsin 15166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ico 12469 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-fac 13354 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-sum 14794 df-ef 15170 df-sin 15172 |
This theorem is referenced by: retancl 15244 resincld 15245 efieq 15265 sinbnd 15282 cosbnd 15283 sinbnd2 15284 sin02gt0 15294 sin4lt0 15297 absefi 15298 sinhalfpilem 24615 sinq12gt0 24659 sincos4thpi 24665 abssinbd 40307 resincncf 40883 fourierdlem39 41157 fourierdlem66 41183 fourierdlem73 41190 recsccl 43393 recotcl 43394 |
Copyright terms: Public domain | W3C validator |