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| Description: Value of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| acosval | ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . 3 ⊢ (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴)) | |
| 2 | 1 | oveq2d 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 | |
| 5 | 2, 3, 4 | fvmpt 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 ℂcc 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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