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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 15842 | . 2 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) | |
2 | ax-icn 10931 | . . . . 5 ⊢ i ∈ ℂ | |
3 | recn 10962 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 10956 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | efcl 15790 | . . . 4 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) ∈ ℂ) |
8 | 7 | imcld 14904 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) ∈ ℝ) |
9 | 1, 8 | eqeltrd 2841 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 ici 10874 · cmul 10877 ℑcim 14807 expce 15769 sincsin 15771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-ico 13084 df-fz 13239 df-fzo 13382 df-fl 13510 df-seq 13720 df-exp 13781 df-fac 13986 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 |
This theorem is referenced by: retancl 15849 resincld 15850 efieq 15870 sinbnd 15887 cosbnd 15888 sinbnd2 15889 sin02gt0 15899 sin4lt0 15902 absefi 15903 sinhalfpilem 25618 sinq12gt0 25662 sincos4thpi 25668 abssinbd 42805 resincncf 43387 fourierdlem39 43658 fourierdlem66 43684 fourierdlem73 43691 recsccl 46425 recotcl 46426 |
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