Theorem coshval-named

Description: Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh . See coshval for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015)

	Ref	Expression
Assertion	coshval-named	⊢ ( 𝐴 ∈ ℂ → ( cosh ‘ 𝐴 ) = ( cos ‘ ( i · 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑥 = 𝐴 → ( i · 𝑥 ) = ( i · 𝐴 ) )
2	1	fveq2d	⊢ ( 𝑥 = 𝐴 → ( cos ‘ ( i · 𝑥 ) ) = ( cos ‘ ( i · 𝐴 ) ) )
3		df-cosh	⊢ cosh = ( 𝑥 ∈ ℂ ↦ ( cos ‘ ( i · 𝑥 ) ) )
4		fvex	⊢ ( cos ‘ ( i · 𝐴 ) ) ∈ V
5	2 3 4	fvmpt	⊢ ( 𝐴 ∈ ℂ → ( cosh ‘ 𝐴 ) = ( cos ‘ ( i · 𝐴 ) ) )