Theorem cosval 16048

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 7366	. . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6838	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3		oveq2 7366	. . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
4	3	fveq2d 6838	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
5	2, 4	oveq12d 7376	. . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))
6	5	oveq1d 7373	. 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
7		df-cos 15993	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
8		ovex 7391	. 2 ⊢ (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ V
9	6, 7, 8	fvmpt 6941	1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  ℂcc 11024  ici 11028   + caddc 11029   · cmul 11031  -cneg 11365   / cdiv 11794  2c2 12200  expce 15984  cosccos 15987

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-cos 15993

This theorem is referenced by:  tanval2  16058  tanval3  16059  recosval  16061  cosneg  16072  efival  16077  coshval  16080  cosadd  16090  cosper  26447  pige3ALT  26485  cosargd  26573  asinsin  26858  cosasin  26870  cosatan  26887