Theorem cosval 14897

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 6698	. . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6233	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3		oveq2 6698	. . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
4	3	fveq2d 6233	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
5	2, 4	oveq12d 6708	. . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))
6	5	oveq1d 6705	. 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
7		df-cos 14845	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
8		ovex 6718	. 2 ⊢ (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ V
9	6, 7, 8	fvmpt 6321	1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  ici 9976   + caddc 9977   · cmul 9979  -cneg 10305   / cdiv 10722  2c2 11108  expce 14836  cosccos 14839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-cos 14845

This theorem is referenced by:  tanval2  14907  tanval3  14908  recosval  14910  cosneg  14921  efival  14926  coshval  14929  cosadd  14939  cosper  24279  pige3  24314  cosargd  24399  asinsin  24664  cosasin  24676  cosatan  24693