Theorem efzval

Description: Value of the exponential function for integers. Special case of efval . Equation 30 of Rudin p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006) (Revised by Mario Carneiro, 5-Jun-2014)

	Ref	Expression
Assertion	efzval	⊢ ( 𝑁 ∈ ℤ → ( exp ‘ 𝑁 ) = ( e ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		zcn	⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ )
2	1	mulridd	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 · 1 ) = 𝑁 )
3	2	fveq2d	⊢ ( 𝑁 ∈ ℤ → ( exp ‘ ( 𝑁 · 1 ) ) = ( exp ‘ 𝑁 ) )
4		ax-1cn	⊢ 1 ∈ ℂ
5		efexp	⊢ ( ( 1 ∈ ℂ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · 1 ) ) = ( ( exp ‘ 1 ) ↑ 𝑁 ) )
6	4 5	mpan	⊢ ( 𝑁 ∈ ℤ → ( exp ‘ ( 𝑁 · 1 ) ) = ( ( exp ‘ 1 ) ↑ 𝑁 ) )
7	3 6	eqtr3d	⊢ ( 𝑁 ∈ ℤ → ( exp ‘ 𝑁 ) = ( ( exp ‘ 1 ) ↑ 𝑁 ) )
8		df-e	⊢ e = ( exp ‘ 1 )
9	8	oveq1i	⊢ ( e ↑ 𝑁 ) = ( ( exp ‘ 1 ) ↑ 𝑁 )
10	7 9	eqtr4di	⊢ ( 𝑁 ∈ ℤ → ( exp ‘ 𝑁 ) = ( e ↑ 𝑁 ) )