Theorem tanhval-named

Description: Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh . (Contributed by David A. Wheeler, 10-May-2015)

	Ref	Expression
Assertion	tanhval-named	⊢ ( 𝐴 ∈ ( ◡ cosh “ ( ℂ ∖ { 0 } ) ) → ( tanh ‘ 𝐴 ) = ( ( tan ‘ ( i · 𝐴 ) ) / i ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑥 = 𝐴 → ( i · 𝑥 ) = ( i · 𝐴 ) )
2	1	fveq2d	⊢ ( 𝑥 = 𝐴 → ( tan ‘ ( i · 𝑥 ) ) = ( tan ‘ ( i · 𝐴 ) ) )
3	2	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( tan ‘ ( i · 𝑥 ) ) / i ) = ( ( tan ‘ ( i · 𝐴 ) ) / i ) )
4		df-tanh	⊢ tanh = ( 𝑥 ∈ ( ◡ cosh “ ( ℂ ∖ { 0 } ) ) ↦ ( ( tan ‘ ( i · 𝑥 ) ) / i ) )
5		ovex	⊢ ( ( tan ‘ ( i · 𝐴 ) ) / i ) ∈ V
6	3 4 5	fvmpt	⊢ ( 𝐴 ∈ ( ◡ cosh “ ( ℂ ∖ { 0 } ) ) → ( tanh ‘ 𝐴 ) = ( ( tan ‘ ( i · 𝐴 ) ) / i ) )