Theorem efval

Description: Value of the exponential function. (Contributed by NM, 8-Jan-2006) (Revised by Mario Carneiro, 10-Nov-2013)

	Ref	Expression
Assertion	efval	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐴 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ↑ 𝑘 ) = ( 𝐴 ↑ 𝑘 ) )
2	1	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( ( 𝐴 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) )
3	2	sumeq2sdv	⊢ ( 𝑥 = 𝐴 → Σ 𝑘 ∈ ℕ0 ( ( 𝑥 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐴 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) )
4		df-ef	⊢ exp = ( 𝑥 ∈ ℂ ↦ Σ 𝑘 ∈ ℕ0 ( ( 𝑥 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) )
5		sumex	⊢ Σ 𝑘 ∈ ℕ0 ( ( 𝐴 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ V
6	3 4 5	fvmpt	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐴 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) )