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Mirrors > Home > ILE Home > Th. List > resincl | 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 11667 | . 2 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) | |
2 | ax-icn 7858 | . . . . 5 ⊢ i ∈ ℂ | |
3 | recn 7896 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 7890 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 412 | . . . 4 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | efcl 11616 | . . . 4 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → (exp‘(i · 𝐴)) ∈ ℂ) |
8 | 7 | imcld 10892 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) ∈ ℝ) |
9 | 1, 8 | eqeltrd 2247 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5851 ℂcc 7761 ℝcr 7762 ici 7765 · cmul 7768 ℑcim 10794 expce 11594 sincsin 11596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6510 df-en 6716 df-dom 6717 df-fin 6718 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-n0 9125 df-z 9202 df-uz 9477 df-q 9568 df-rp 9600 df-ico 9840 df-fz 9955 df-fzo 10088 df-seqfrec 10391 df-exp 10465 df-fac 10649 df-ihash 10699 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-clim 11231 df-sumdc 11306 df-ef 11600 df-sin 11602 |
This theorem is referenced by: retanclap 11674 resincld 11675 efieq 11687 sinbnd 11704 cosbnd 11705 sinbnd2 11706 sin02gt0 11715 sin4lt0 11718 absefi 11720 sinhalfpilem 13467 sinq12gt0 13506 sincos4thpi 13516 |
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