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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 6658 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | oveq1 6657 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
3 | 2 | oveq2d 6666 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (1 − (𝑥↑2)) = (1 − (𝐴↑2))) |
4 | 3 | fveq2d 6195 | . . . . 5 ⊢ (𝑥 = 𝐴 → (√‘(1 − (𝑥↑2))) = (√‘(1 − (𝐴↑2)))) |
5 | 1, 4 | oveq12d 6668 | . . . 4 ⊢ (𝑥 = 𝐴 → ((i · 𝑥) + (√‘(1 − (𝑥↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
6 | 5 | fveq2d 6195 | . . 3 ⊢ (𝑥 = 𝐴 → (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))) = (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) |
7 | 6 | oveq2d 6666 | . 2 ⊢ (𝑥 = 𝐴 → (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) |
8 | df-asin 24592 | . 2 ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2))))))) | |
9 | ovex 6678 | . 2 ⊢ (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2)))))) ∈ V | |
10 | 7, 8, 9 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) = (-i · (log‘((i · 𝐴) + (√‘(1 − (𝐴↑2))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 ici 9938 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 2c2 11070 ↑cexp 12860 √csqrt 13973 logclog 24301 arcsincasin 24589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-asin 24592 |
This theorem is referenced by: asinneg 24613 efiasin 24615 asinsin 24619 asin1 24621 asinbnd 24626 areacirclem4 33503 |
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