Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  acosval Structured version   Visualization version   GIF version

Theorem acosval 24591
 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 6178 . . 3 (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴))
21oveq2d 6651 . 2 (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴)))
3 df-acos 24574 . 2 arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
4 ovex 6663 . 2 ((π / 2) − (arcsin‘𝐴)) ∈ V
52, 3, 4fvmpt 6269 1 (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  ‘cfv 5876  (class class class)co 6635  ℂcc 9919   − cmin 10251   / cdiv 10669  2c2 11055  πcpi 14778  arcsincasin 24570  arccoscacos 24571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-acos 24574 This theorem is referenced by:  acosneg  24595  cosacos  24598  acoscos  24601  acos1  24603  acosbnd  24608  acosrecl  24611  sinacos  24613
 Copyright terms: Public domain W3C validator