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 . See sinhval for a theorem to convert this further. See sinh-conventional 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 | ||
|---|---|---|---|
| Assertion | sinhval-named | |- ( A e. CC -> ( sinh ` A ) = ( ( sin ` ( _i x. A ) ) / _i ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 | |- ( x = A -> ( _i x. x ) = ( _i x. A ) ) |
|
| 2 | 1 | fveq2d | |- ( x = A -> ( sin ` ( _i x. x ) ) = ( sin ` ( _i x. A ) ) ) |
| 3 | 2 | oveq1d | |- ( x = A -> ( ( sin ` ( _i x. x ) ) / _i ) = ( ( sin ` ( _i x. A ) ) / _i ) ) |
| 4 | df-sinh | |- sinh = ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) ) |
|
| 5 | ovex | |- ( ( sin ` ( _i x. A ) ) / _i ) e. _V |
|
| 6 | 3 4 5 | fvmpt | |- ( A e. CC -> ( sinh ` A ) = ( ( sin ` ( _i x. A ) ) / _i ) ) |