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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 6902 | . . 3 ⊢ (𝑥 = 𝐴 → (arcsin‘𝑥) = (arcsin‘𝐴)) | |
2 | 1 | oveq2d 7442 | . 2 ⊢ (𝑥 = 𝐴 → ((π / 2) − (arcsin‘𝑥)) = ((π / 2) − (arcsin‘𝐴))) |
3 | df-acos 26818 | . 2 ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥))) | |
4 | ovex 7459 | . 2 ⊢ ((π / 2) − (arcsin‘𝐴)) ∈ V | |
5 | 2, 3, 4 | fvmpt 7010 | 1 ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 − cmin 11482 / cdiv 11909 2c2 12305 πcpi 16050 arcsincasin 26814 arccoscacos 26815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-acos 26818 |
This theorem is referenced by: acosneg 26839 cosacos 26842 acoscos 26845 acos1 26847 acosbnd 26852 acosrecl 26855 sinacos 26857 |
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