Theorem cxple2ad

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	$$⊢\phi \to B\in ℝ$$
	cxple2ad.4	$$⊢\phi \to C\in ℝ$$
	cxple2ad.5	$$⊢\phi \to 0\le C$$
	cxple2ad.6	$$⊢\phi \to A\le B$$
Assertion	cxple2ad	$$⊢\phi \to A^C\le B^C$$

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		cxple2ad.4	$$⊢\phi \to C\in ℝ$$
5		cxple2ad.5	$$⊢\phi \to 0\le C$$
6		cxple2ad.6	$$⊢\phi \to A\le B$$
7		cxple2a	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\wedge \left(0\le A\wedge 0\le C\right)\wedge A\le B\right)\to A^C\le B^C$$
8	1 3 4 2 5 6 7	syl321anc	$$⊢\phi \to A^C\le B^C$$