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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 | β’ (π΄ β β β (arcsinβπ΄) = (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7434 | . . . . 5 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | oveq1 7433 | . . . . . . 7 β’ (π₯ = π΄ β (π₯β2) = (π΄β2)) | |
3 | 2 | oveq2d 7442 | . . . . . 6 β’ (π₯ = π΄ β (1 β (π₯β2)) = (1 β (π΄β2))) |
4 | 3 | fveq2d 6906 | . . . . 5 β’ (π₯ = π΄ β (ββ(1 β (π₯β2))) = (ββ(1 β (π΄β2)))) |
5 | 1, 4 | oveq12d 7444 | . . . 4 β’ (π₯ = π΄ β ((i Β· π₯) + (ββ(1 β (π₯β2)))) = ((i Β· π΄) + (ββ(1 β (π΄β2))))) |
6 | 5 | fveq2d 6906 | . . 3 β’ (π₯ = π΄ β (logβ((i Β· π₯) + (ββ(1 β (π₯β2))))) = (logβ((i Β· π΄) + (ββ(1 β (π΄β2)))))) |
7 | 6 | oveq2d 7442 | . 2 β’ (π₯ = π΄ β (-i Β· (logβ((i Β· π₯) + (ββ(1 β (π₯β2)))))) = (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2))))))) |
8 | df-asin 26817 | . 2 β’ arcsin = (π₯ β β β¦ (-i Β· (logβ((i Β· π₯) + (ββ(1 β (π₯β2))))))) | |
9 | ovex 7459 | . 2 β’ (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2)))))) β V | |
10 | 7, 8, 9 | fvmpt 7010 | 1 β’ (π΄ β β β (arcsinβπ΄) = (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2))))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11144 1c1 11147 ici 11148 + caddc 11149 Β· cmul 11151 β cmin 11482 -cneg 11483 2c2 12305 βcexp 14066 βcsqrt 15220 logclog 26508 arcsincasin 26814 |
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-asin 26817 |
This theorem is referenced by: asinneg 26838 efiasin 26840 asinsin 26844 asin1 26846 asinbnd 26851 areacirclem4 37217 |
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