Theorem cxple2d

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

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	recxpcld.2	$$⊢\phi \to 0\le A$$
	recxpcld.3	$$⊢\phi \to B\in ℝ$$
	mulcxpd.4	$$⊢\phi \to 0\le B$$
	cxple2d.5	$$⊢\phi \to C\in {ℝ}^{+}$$
Assertion	cxple2d	$$⊢\phi \to \left(A\le B\leftrightarrow A^C\le B^C\right)$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		recxpcld.2	$$⊢\phi \to 0\le A$$
3		recxpcld.3	$$⊢\phi \to B\in ℝ$$
4		mulcxpd.4	$$⊢\phi \to 0\le B$$
5		cxple2d.5	$$⊢\phi \to C\in {ℝ}^{+}$$
6		cxple2	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\wedge C\in {ℝ}^{+}\right)\to \left(A\le B\leftrightarrow A^C\le B^C\right)$$
7	1 2 3 4 5 6	syl221anc	$$⊢\phi \to \left(A\le B\leftrightarrow A^C\le B^C\right)$$