Theorem cxple3d

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

	Ref	Expression
Hypotheses	rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	cxplt3d.3	⊢ ( 𝜑 → 𝐴 < 1 )
	cxplt3d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	cxple3d	⊢ ( 𝜑 → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		cxplt3d.3	⊢ ( 𝜑 → 𝐴 < 1 )
4		cxplt3d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5		cxple3	⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ) )