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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 7370 | . . . . 5 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | oveq1 7369 | . . . . . . 7 β’ (π₯ = π΄ β (π₯β2) = (π΄β2)) | |
3 | 2 | oveq2d 7378 | . . . . . 6 β’ (π₯ = π΄ β (1 β (π₯β2)) = (1 β (π΄β2))) |
4 | 3 | fveq2d 6851 | . . . . 5 β’ (π₯ = π΄ β (ββ(1 β (π₯β2))) = (ββ(1 β (π΄β2)))) |
5 | 1, 4 | oveq12d 7380 | . . . 4 β’ (π₯ = π΄ β ((i Β· π₯) + (ββ(1 β (π₯β2)))) = ((i Β· π΄) + (ββ(1 β (π΄β2))))) |
6 | 5 | fveq2d 6851 | . . 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 7378 | . 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 26231 | . 2 β’ arcsin = (π₯ β β β¦ (-i Β· (logβ((i Β· π₯) + (ββ(1 β (π₯β2))))))) | |
9 | ovex 7395 | . 2 β’ (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2)))))) β V | |
10 | 7, 8, 9 | fvmpt 6953 | 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 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 βcc 11056 1c1 11059 ici 11060 + caddc 11061 Β· cmul 11063 β cmin 11392 -cneg 11393 2c2 12215 βcexp 13974 βcsqrt 15125 logclog 25926 arcsincasin 26228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-asin 26231 |
This theorem is referenced by: asinneg 26252 efiasin 26254 asinsin 26258 asin1 26260 asinbnd 26265 areacirclem4 36198 |
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