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Mirrors > Home > MPE Home > Th. List > sinval | Structured version Visualization version GIF version |
Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
Ref | Expression |
---|---|
sinval | ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7420 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6895 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴))) |
3 | oveq2 7420 | . . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴)) | |
4 | 3 | fveq2d 6895 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴))) |
5 | 2, 4 | oveq12d 7430 | . . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
6 | 5 | oveq1d 7427 | . 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
7 | df-sin 16020 | . 2 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) | |
8 | ovex 7445 | . 2 ⊢ (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6998 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ici 11118 · cmul 11121 − cmin 11451 -cneg 11452 / cdiv 11878 2c2 12274 expce 16012 sincsin 16014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-sin 16020 |
This theorem is referenced by: tanval2 16083 resinval 16085 sinneg 16096 efival 16102 sinhval 16104 sinadd 16114 dvsincos 25833 sinper 26331 sineq0 26373 efeq1 26377 sinasin 26735 sineq0ALT 44163 |
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