Theorem cxple2ad

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 )
	cxple2ad.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	cxple2ad.5	⊢ ( 𝜑 → 0 ≤ 𝐶 )
	cxple2ad.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	cxple2ad	⊢ ( 𝜑 → ( 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		cxple2ad.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5		cxple2ad.5	⊢ ( 𝜑 → 0 ≤ 𝐶 )
6		cxple2ad.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
7		cxple2a	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) )
8	1 3 4 2 5 6 7	syl321anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) )