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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 6557 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴))) |
3 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴)) | |
4 | 3 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴))) |
5 | 2, 4 | oveq12d 6567 | . . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴)))) |
6 | 5 | oveq1d 6564 | . 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
7 | df-sin 14639 | . 2 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))) | |
8 | ovex 6577 | . 2 ⊢ (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ici 9817 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 expce 14631 sincsin 14633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-sin 14639 |
This theorem is referenced by: tanval2 14702 resinval 14704 sinneg 14715 efival 14721 sinhval 14723 sinadd 14733 dvsincos 23548 sinper 24037 sineq0 24077 efeq1 24079 sinasin 24416 sineq0ALT 38195 |
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