Theorem cxple3d

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

	Ref	Expression
Hypotheses	rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpcxpcld.2	$$⊢\phi \to B\in ℝ$$
	cxplt3d.3	$$⊢\phi \to A<1$$
	cxplt3d.4	$$⊢\phi \to C\in ℝ$$
Assertion	cxple3d	$$⊢\phi \to \left(B\le C\leftrightarrow A^C\le A^B\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpcxpcld.2	$$⊢\phi \to B\in ℝ$$
3		cxplt3d.3	$$⊢\phi \to A<1$$
4		cxplt3d.4	$$⊢\phi \to C\in ℝ$$
5		cxple3	$$⊢\left(\left(A\in {ℝ}^{+}\wedge A<1\right)\wedge \left(B\in ℝ\wedge C\in ℝ\right)\right)\to \left(B\le C\leftrightarrow A^C\le A^B\right)$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to \left(B\le C\leftrightarrow A^C\le A^B\right)$$