![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7412 | . . . . 5 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | oveq1 7411 | . . . . . . 7 β’ (π₯ = π΄ β (π₯β2) = (π΄β2)) | |
3 | 2 | oveq2d 7420 | . . . . . 6 β’ (π₯ = π΄ β (1 β (π₯β2)) = (1 β (π΄β2))) |
4 | 3 | fveq2d 6888 | . . . . 5 β’ (π₯ = π΄ β (ββ(1 β (π₯β2))) = (ββ(1 β (π΄β2)))) |
5 | 1, 4 | oveq12d 7422 | . . . 4 β’ (π₯ = π΄ β ((i Β· π₯) + (ββ(1 β (π₯β2)))) = ((i Β· π΄) + (ββ(1 β (π΄β2))))) |
6 | 5 | fveq2d 6888 | . . 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 7420 | . 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 26747 | . 2 β’ arcsin = (π₯ β β β¦ (-i Β· (logβ((i Β· π₯) + (ββ(1 β (π₯β2))))))) | |
9 | ovex 7437 | . 2 β’ (-i Β· (logβ((i Β· π΄) + (ββ(1 β (π΄β2)))))) β V | |
10 | 7, 8, 9 | fvmpt 6991 | 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 6536 (class class class)co 7404 βcc 11107 1c1 11110 ici 11111 + caddc 11112 Β· cmul 11114 β cmin 11445 -cneg 11446 2c2 12268 βcexp 14029 βcsqrt 15183 logclog 26438 arcsincasin 26744 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-asin 26747 |
This theorem is referenced by: asinneg 26768 efiasin 26770 asinsin 26774 asin1 26776 asinbnd 26781 areacirclem4 37091 |
Copyright terms: Public domain | W3C validator |