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

Theorem cosval 16156
Description: Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
cosval (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))

Proof of Theorem cosval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . 5 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6911 . . . 4 (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3 oveq2 7439 . . . . 5 (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
43fveq2d 6911 . . . 4 (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
52, 4oveq12d 7449 . . 3 (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))
65oveq1d 7446 . 2 (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
7 df-cos 16103 . 2 cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
8 ovex 7464 . 2 (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ V
96, 7, 8fvmpt 7016 1 (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  cc 11151  ici 11155   + caddc 11156   · cmul 11158  -cneg 11491   / cdiv 11918  2c2 12319  expce 16094  cosccos 16097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-cos 16103
This theorem is referenced by:  tanval2  16166  tanval3  16167  recosval  16169  cosneg  16180  efival  16185  coshval  16188  cosadd  16198  cosper  26539  pige3ALT  26577  cosargd  26665  asinsin  26950  cosasin  26962  cosatan  26979
  Copyright terms: Public domain W3C validator