Theorem cxplead

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

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	cxplead.2	$$⊢\phi \to 1\le A$$
	cxplead.3	$$⊢\phi \to B\in ℝ$$
	cxplead.4	$$⊢\phi \to C\in ℝ$$
	cxplead.5	$$⊢\phi \to B\le C$$
Assertion	cxplead	$$⊢\phi \to A^B\le A^C$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		cxplead.2	$$⊢\phi \to 1\le A$$
3		cxplead.3	$$⊢\phi \to B\in ℝ$$
4		cxplead.4	$$⊢\phi \to C\in ℝ$$
5		cxplead.5	$$⊢\phi \to B\le C$$
6		cxplea	$$⊢\left(\left(A\in ℝ\wedge 1\le A\right)\wedge \left(B\in ℝ\wedge C\in ℝ\right)\wedge B\le C\right)\to A^B\le A^C$$
7	1 2 3 4 5 6	syl221anc	$$⊢\phi \to A^B\le A^C$$