Theorem expsubd

Description: Exponent subtraction law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	$$⊢\phi \to A\in ℂ$$
	sqrecd.1	$$⊢\phi \to A\ne 0$$
	expclzd.3	$$⊢\phi \to N\in \mathbb{Z}$$
	expsubd.3	$$⊢\phi \to M\in \mathbb{Z}$$
Assertion	expsubd	$$⊢\phi \to A^{M-N}=\frac{A^M}{A^N}$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		sqrecd.1	$$⊢\phi \to A\ne 0$$
3		expclzd.3	$$⊢\phi \to N\in \mathbb{Z}$$
4		expsubd.3	$$⊢\phi \to M\in \mathbb{Z}$$
5		expsub	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(M\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\right)\to A^{M-N}=\frac{A^M}{A^N}$$
6	1 2 4 3 5	syl22anc	$$⊢\phi \to A^{M-N}=\frac{A^M}{A^N}$$