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Mirrors > Home > ILE Home > Th. List > recosval | GIF version |
Description: The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
recosval | ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7862 | . . . . . . . 8 ⊢ i ∈ ℂ | |
2 | recn 7900 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | cjmul 10842 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) | |
4 | 1, 2, 3 | sylancr 412 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) |
5 | cji 10859 | . . . . . . . . 9 ⊢ (∗‘i) = -i | |
6 | 5 | oveq1i 5861 | . . . . . . . 8 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴)) |
7 | cjre 10839 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
8 | 7 | oveq2d 5867 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (-i · (∗‘𝐴)) = (-i · 𝐴)) |
9 | 6, 8 | eqtrid 2215 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((∗‘i) · (∗‘𝐴)) = (-i · 𝐴)) |
10 | 4, 9 | eqtrd 2203 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = (-i · 𝐴)) |
11 | 10 | fveq2d 5498 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (exp‘(∗‘(i · 𝐴))) = (exp‘(-i · 𝐴))) |
12 | mulcl 7894 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
13 | 1, 2, 12 | sylancr 412 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
14 | efcj 11629 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(∗‘(i · 𝐴))) = (∗‘(exp‘(i · 𝐴)))) | |
15 | 13, 14 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (exp‘(∗‘(i · 𝐴))) = (∗‘(exp‘(i · 𝐴)))) |
16 | 11, 15 | eqtr3d 2205 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(-i · 𝐴)) = (∗‘(exp‘(i · 𝐴)))) |
17 | 16 | oveq2d 5867 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) = ((exp‘(i · 𝐴)) + (∗‘(exp‘(i · 𝐴))))) |
18 | 17 | oveq1d 5866 | . 2 ⊢ (𝐴 ∈ ℝ → (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) = (((exp‘(i · 𝐴)) + (∗‘(exp‘(i · 𝐴)))) / 2)) |
19 | cosval 11659 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | |
20 | 2, 19 | syl 14 | . 2 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
21 | efcl 11620 | . . 3 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
22 | reval 10806 | . . 3 ⊢ ((exp‘(i · 𝐴)) ∈ ℂ → (ℜ‘(exp‘(i · 𝐴))) = (((exp‘(i · 𝐴)) + (∗‘(exp‘(i · 𝐴)))) / 2)) | |
23 | 13, 21, 22 | 3syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (ℜ‘(exp‘(i · 𝐴))) = (((exp‘(i · 𝐴)) + (∗‘(exp‘(i · 𝐴)))) / 2)) |
24 | 18, 20, 23 | 3eqtr4d 2213 | 1 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = (ℜ‘(exp‘(i · 𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5851 ℂcc 7765 ℝcr 7766 ici 7769 + caddc 7770 · cmul 7772 -cneg 8084 / cdiv 8582 2c2 8922 ∗ccj 10796 ℜcre 10797 expce 11598 cosccos 11601 |
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 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
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 6511 df-en 6717 df-dom 6718 df-fin 6719 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-ico 9844 df-fz 9959 df-fzo 10092 df-seqfrec 10395 df-exp 10469 df-fac 10653 df-ihash 10703 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-clim 11235 df-sumdc 11310 df-ef 11604 df-cos 11607 |
This theorem is referenced by: recos4p 11675 recoscl 11677 cos0 11686 |
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