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Mirrors > Home > MPE Home > Th. List > resinval | Structured version Visualization version GIF version |
Description: The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
resinval | ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10394 | . . . . . . . 8 ⊢ i ∈ ℂ | |
2 | recn 10425 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | cjmul 14362 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) | |
4 | 1, 2, 3 | sylancr 578 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) |
5 | cji 14379 | . . . . . . . . 9 ⊢ (∗‘i) = -i | |
6 | 5 | oveq1i 6986 | . . . . . . . 8 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴)) |
7 | cjre 14359 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
8 | 7 | oveq2d 6992 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (-i · (∗‘𝐴)) = (-i · 𝐴)) |
9 | 6, 8 | syl5eq 2827 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((∗‘i) · (∗‘𝐴)) = (-i · 𝐴)) |
10 | 4, 9 | eqtrd 2815 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = (-i · 𝐴)) |
11 | 10 | fveq2d 6503 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (exp‘(∗‘(i · 𝐴))) = (exp‘(-i · 𝐴))) |
12 | mulcl 10419 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
13 | 1, 2, 12 | sylancr 578 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
14 | efcj 15305 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(∗‘(i · 𝐴))) = (∗‘(exp‘(i · 𝐴)))) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (exp‘(∗‘(i · 𝐴))) = (∗‘(exp‘(i · 𝐴)))) |
16 | 11, 15 | eqtr3d 2817 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(-i · 𝐴)) = (∗‘(exp‘(i · 𝐴)))) |
17 | 16 | oveq2d 6992 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴))))) |
18 | 17 | oveq1d 6991 | . 2 ⊢ (𝐴 ∈ ℝ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴)))) / (2 · i))) |
19 | sinval 15335 | . . 3 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) | |
20 | 2, 19 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
21 | efcl 15296 | . . 3 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
22 | imval2 14371 | . . 3 ⊢ ((exp‘(i · 𝐴)) ∈ ℂ → (ℑ‘(exp‘(i · 𝐴))) = (((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴)))) / (2 · i))) | |
23 | 13, 21, 22 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘(exp‘(i · 𝐴))) = (((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴)))) / (2 · i))) |
24 | 18, 20, 23 | 3eqtr4d 2825 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 ℝcr 10334 ici 10337 · cmul 10340 − cmin 10670 -cneg 10671 / cdiv 11098 2c2 11495 ∗ccj 14316 ℑcim 14318 expce 15275 sincsin 15277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-ico 12560 df-fz 12709 df-fzo 12850 df-fl 12977 df-seq 13185 df-exp 13245 df-fac 13449 df-hash 13506 df-shft 14287 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-limsup 14689 df-clim 14706 df-rlim 14707 df-sum 14904 df-ef 15281 df-sin 15283 |
This theorem is referenced by: resin4p 15351 resincl 15353 argimgt0 24896 |
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