![]() |
Mathbox for David A. Wheeler |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sinhval-named | Structured version Visualization version GIF version |
Description: Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 48087. See sinhval 16122 for a theorem to convert this further. See sinh-conventional 48093 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
Ref | Expression |
---|---|
sinhval-named | β’ (π΄ β β β (sinhβπ΄) = ((sinβ(i Β· π΄)) / i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7422 | . . . 4 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | 1 | fveq2d 6895 | . . 3 β’ (π₯ = π΄ β (sinβ(i Β· π₯)) = (sinβ(i Β· π΄))) |
3 | 2 | oveq1d 7429 | . 2 β’ (π₯ = π΄ β ((sinβ(i Β· π₯)) / i) = ((sinβ(i Β· π΄)) / i)) |
4 | df-sinh 48087 | . 2 β’ sinh = (π₯ β β β¦ ((sinβ(i Β· π₯)) / i)) | |
5 | ovex 7447 | . 2 β’ ((sinβ(i Β· π΄)) / i) β V | |
6 | 3, 4, 5 | fvmpt 6999 | 1 β’ (π΄ β β β (sinhβπ΄) = ((sinβ(i Β· π΄)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 βcc 11128 ici 11132 Β· cmul 11135 / cdiv 11893 sincsin 16031 sinhcsinh 48084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-sinh 48087 |
This theorem is referenced by: sinh-conventional 48093 sinhpcosh 48094 |
Copyright terms: Public domain | W3C validator |