Theorem cxpled

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	cxpltd.2	$$⊢\phi \to 1<A$$
	cxpltd.3	$$⊢\phi \to B\in ℝ$$
	cxpltd.4	$$⊢\phi \to C\in ℝ$$
Assertion	cxpled	$$⊢\phi \to \left(B\le C\leftrightarrow A^B\le A^C\right)$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		cxpltd.2	$$⊢\phi \to 1<A$$
3		cxpltd.3	$$⊢\phi \to B\in ℝ$$
4		cxpltd.4	$$⊢\phi \to C\in ℝ$$
5		cxple	$$⊢\left(\left(A\in ℝ\wedge 1<A\right)\wedge \left(B\in ℝ\wedge C\in ℝ\right)\right)\to \left(B\le C\leftrightarrow A^B\le A^C\right)$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to \left(B\le C\leftrightarrow A^B\le A^C\right)$$