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Mirrors > Home > MPE Home > Th. List > acosval | Structured version Visualization version GIF version |
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 6664 | . . 3 ⊢ (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴)) | |
2 | 1 | oveq2d 7161 | . 2 ⊢ (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴))) |
3 | df-acos 25371 | . 2 ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) | |
4 | ovex 7178 | . 2 ⊢ ((π / 2) − (arcsin‘𝐴)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6762 | 1 ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 − cmin 10859 / cdiv 11286 2c2 11681 πcpi 15410 arcsincasin 25367 arccoscacos 25368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7148 df-acos 25371 |
This theorem is referenced by: acosneg 25392 cosacos 25395 acoscos 25398 acos1 25400 acosbnd 25405 acosrecl 25408 sinacos 25410 |
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