Theorem cxple2d

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

	Ref	Expression
Hypotheses	recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	mulcxpd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
	cxple2d.5	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	cxple2d	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4		mulcxpd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		cxple2d.5	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
6		cxple2	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ) )
7	1 2 3 4 5 6	syl221anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ) )