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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 11198 | . . . . . . . 8 ⊢ i ∈ ℂ | |
2 | recn 11229 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | cjmul 15122 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) | |
4 | 1, 2, 3 | sylancr 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴))) |
5 | cji 15139 | . . . . . . . . 9 ⊢ (∗‘i) = -i | |
6 | 5 | oveq1i 7430 | . . . . . . . 8 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴)) |
7 | cjre 15119 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
8 | 7 | oveq2d 7436 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (-i · (∗‘𝐴)) = (-i · 𝐴)) |
9 | 6, 8 | eqtrid 2780 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((∗‘i) · (∗‘𝐴)) = (-i · 𝐴)) |
10 | 4, 9 | eqtrd 2768 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (∗‘(i · 𝐴)) = (-i · 𝐴)) |
11 | 10 | fveq2d 6901 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (exp‘(∗‘(i · 𝐴))) = (exp‘(-i · 𝐴))) |
12 | mulcl 11223 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
13 | 1, 2, 12 | sylancr 586 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
14 | efcj 16069 | . . . . . 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 2770 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘(-i · 𝐴)) = (∗‘(exp‘(i · 𝐴)))) |
17 | 16 | oveq2d 7436 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = ((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴))))) |
18 | 17 | oveq1d 7435 | . 2 ⊢ (𝐴 ∈ ℝ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = (((exp‘(i · 𝐴)) − (∗‘(exp‘(i · 𝐴)))) / (2 · i))) |
19 | sinval 16099 | . . 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 16059 | . . 3 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
22 | imval2 15131 | . . 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 2778 | 1 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (ℑ‘(exp‘(i · 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ℝcr 11138 ici 11141 · cmul 11144 − cmin 11475 -cneg 11476 / cdiv 11902 2c2 12298 ∗ccj 15076 ℑcim 15078 expce 16038 sincsin 16040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-ico 13363 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-fac 14266 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 |
This theorem is referenced by: resin4p 16115 resincl 16117 argimgt0 26559 |
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