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Theorem acosval 26927
Description: Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
acosval (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))

Proof of Theorem acosval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6905 . . 3 (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴))
21oveq2d 7448 . 2 (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴)))
3 df-acos 26910 . 2 arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
4 ovex 7465 . 2 ((π / 2) − (arcsin‘𝐴)) ∈ V
52, 3, 4fvmpt 7015 1 (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  cc 11154  cmin 11493   / cdiv 11921  2c2 12322  πcpi 16103  arcsincasin 26906  arccoscacos 26907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-acos 26910
This theorem is referenced by:  acosneg  26931  cosacos  26934  acoscos  26937  acos1  26939  acosbnd  26944  acosrecl  26947  sinacos  26949
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