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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 6907 | . . 3 ⊢ (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴)) | |
2 | 1 | oveq2d 7447 | . 2 ⊢ (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴))) |
3 | df-acos 26924 | . 2 ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) | |
4 | ovex 7464 | . 2 ⊢ ((π / 2) − (arcsin‘𝐴)) ∈ V | |
5 | 2, 3, 4 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 − cmin 11490 / cdiv 11918 2c2 12319 πcpi 16099 arcsincasin 26920 arccoscacos 26921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-acos 26924 |
This theorem is referenced by: acosneg 26945 cosacos 26948 acoscos 26951 acos1 26953 acosbnd 26958 acosrecl 26961 sinacos 26963 |
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