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 ∈ ℂ → ( sinh ‘ 𝐴 ) = ( ( sin ‘ ( i · 𝐴 ) ) / i ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 | ⊢ ( 𝑥 = 𝐴 → ( i · 𝑥 ) = ( i · 𝐴 ) ) | |
| 2 | 1 | fveq2d | ⊢ ( 𝑥 = 𝐴 → ( sin ‘ ( i · 𝑥 ) ) = ( sin ‘ ( i · 𝐴 ) ) ) |
| 3 | 2 | oveq1d | ⊢ ( 𝑥 = 𝐴 → ( ( sin ‘ ( i · 𝑥 ) ) / i ) = ( ( sin ‘ ( i · 𝐴 ) ) / i ) ) |
| 4 | df-sinh | ⊢ sinh = ( 𝑥 ∈ ℂ ↦ ( ( sin ‘ ( i · 𝑥 ) ) / i ) ) | |
| 5 | ovex | ⊢ ( ( sin ‘ ( i · 𝐴 ) ) / i ) ∈ V | |
| 6 | 3 4 5 | fvmpt | ⊢ ( 𝐴 ∈ ℂ → ( sinh ‘ 𝐴 ) = ( ( sin ‘ ( i · 𝐴 ) ) / i ) ) |