MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sinval Structured version   Visualization version   GIF version

Theorem sinval 16099
Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
sinval (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))

Proof of Theorem sinval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7428 . . . . 5 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6901 . . . 4 (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3 oveq2 7428 . . . . 5 (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
43fveq2d 6901 . . . 4 (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
52, 4oveq12d 7438 . . 3 (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))))
65oveq1d 7435 . 2 (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
7 df-sin 16046 . 2 sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
8 ovex 7453 . 2 (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ V
96, 7, 8fvmpt 7005 1 (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  cc 11137  ici 11141   · cmul 11144  cmin 11475  -cneg 11476   / cdiv 11902  2c2 12298  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-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-sin 16046
This theorem is referenced by:  tanval2  16110  resinval  16112  sinneg  16123  efival  16129  sinhval  16131  sinadd  16141  dvsincos  25926  sinper  26429  sineq0  26471  efeq1  26475  sinasin  26834  sineq0ALT  44376
  Copyright terms: Public domain W3C validator