Theorem cosval 15468

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.

Step	Hyp	Ref	Expression
1		oveq2 7143	. . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6649	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3		oveq2 7143	. . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
4	3	fveq2d 6649	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
5	2, 4	oveq12d 7153	. . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))
6	5	oveq1d 7150	. 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
7		df-cos 15416	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
8		ovex 7168	. 2 ⊢ (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ V
9	6, 7, 8	fvmpt 6745	1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  (class class class)co 7135  ℂcc 10524  ici 10528   + caddc 10529   · cmul 10531  -cneg 10860   / cdiv 11286  2c2 11680  expce 15407  cosccos 15410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-cos 15416

This theorem is referenced by:  tanval2  15478  tanval3  15479  recosval  15481  cosneg  15492  efival  15497  coshval  15500  cosadd  15510  cosper  25075  pige3ALT  25112  cosargd  25199  asinsin  25478  cosasin  25490  cosatan  25507