Theorem cxpled

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

	Ref	Expression
Hypotheses	recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	cxpltd.2	⊢ ( 𝜑 → 1 < 𝐴 )
	cxpltd.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	cxpltd.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	cxpled	⊢ ( 𝜑 → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐵 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )

Proof

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