Theorem efadd

Description: Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006) (Proof shortened by Mario Carneiro, 29-Apr-2014)

	Ref	Expression
Assertion	efadd	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to {\mathrm{e}}^{A+B}={\mathrm{e}}^A{\mathrm{e}}^B$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\left(n\in {\mathbb{N}}_0⟼\frac{A^n}{n!}\right)=\left(n\in {\mathbb{N}}_0⟼\frac{A^n}{n!}\right)$$
2		eqid	$$⊢\left(n\in {\mathbb{N}}_0⟼\frac{B^n}{n!}\right)=\left(n\in {\mathbb{N}}_0⟼\frac{B^n}{n!}\right)$$
3		eqid	$$⊢\left(n\in {\mathbb{N}}_0⟼\frac{{\left(A+B\right)}^n}{n!}\right)=\left(n\in {\mathbb{N}}_0⟼\frac{{\left(A+B\right)}^n}{n!}\right)$$
4		simpl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A\in ℂ$$
5		simpr	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to B\in ℂ$$
6	1 2 3 4 5	efaddlem	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to {\mathrm{e}}^{A+B}={\mathrm{e}}^A{\mathrm{e}}^B$$