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Theorem acosval 26249
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 6847 . . 3 (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴))
21oveq2d 7378 . 2 (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴)))
3 df-acos 26232 . 2 arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
4 ovex 7395 . 2 ((π / 2) − (arcsin‘𝐴)) ∈ V
52, 3, 4fvmpt 6953 1 (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6501  (class class class)co 7362  cc 11056  cmin 11392   / cdiv 11819  2c2 12215  πcpi 15956  arcsincasin 26228  arccoscacos 26229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-acos 26232
This theorem is referenced by:  acosneg  26253  cosacos  26256  acoscos  26259  acos1  26261  acosbnd  26266  acosrecl  26269  sinacos  26271
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