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Mirrors > Home > MPE Home > Th. List > asinval | Structured version Visualization version GIF version |
Description: Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
asinval | ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7158 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | oveq1 7157 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
3 | 2 | oveq2d 7166 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (1 − (𝑥↑2)) = (1 − (𝐴↑2))) |
4 | 3 | fveq2d 6668 | . . . . 5 ⊢ (𝑥 = 𝐴 → (√‘(1 − (𝑥↑2))) = (√‘(1 − (𝐴↑2)))) |
5 | 1, 4 | oveq12d 7168 | . . . 4 ⊢ (𝑥 = 𝐴 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
6 | 5 | fveq2d 6668 | . . 3 ⊢ (𝑥 = 𝐴 → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) |
7 | 6 | oveq2d 7166 | . 2 ⊢ (𝑥 = 𝐴 → (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) |
8 | df-asin 25437 | . 2 ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) | |
9 | ovex 7183 | . 2 ⊢ (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6762 | 1 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 ici 10533 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 2c2 11686 ↑cexp 13423 √csqrt 14586 logclog 25132 arcsincasin 25434 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 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 7153 df-asin 25437 |
This theorem is referenced by: asinneg 25458 efiasin 25460 asinsin 25464 asin1 25466 asinbnd 25471 areacirclem4 34979 |
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