Theorem cosval 16320

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 7451	. . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6912	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3		oveq2 7451	. . . . 5 ⊢ (𝑥 = 𝐴 → ( -i · 𝑥) = ( -i · 𝐴))
4	3	fveq2d 6912	. . . 4 ⊢ (𝑥 = 𝐴 → (exp‘( -i · 𝑥)) = (exp‘( -i · 𝐴)))
5	2, 4	oveq12d 7461	. . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))))
6	5	oveq1d 7458	. 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))) / 2))
7		df-cos 16265	. 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))
8		ovex 7476	. 2 ⊢ (((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))) / 2) ∈ V
9	6, 7, 8	fvmpt 7016	1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘( -i · 𝐴))) / 2))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6558  (class class class)co 7443  ℂcc 11225  ici 11229   + caddc 11230   · cmul 11232   -cneg 11569   / cdiv 12000  2c2 12424  expce 16256  cosccos 16259

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-cos 16265

This theorem is used by:  tanval2  16330  tanval3  16331  recosval  16333  cosneg  16344  efival  16349  coshval  16352  cosadd  16362  cosper  26841  pige3ALT  26878  cosargd  26966  asinsin  27250  cosasin  27262  cosatan  27279