Theorem cxplead

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

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

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		cxplead.2	⊢ ( 𝜑 → 1 ≤ 𝐴 )
3		cxplead.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4		cxplead.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5		cxplead.5	⊢ ( 𝜑 → 𝐵 ≤ 𝐶 )
6		cxplea	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐵 ≤ 𝐶 ) → ( 𝐴 ↑𝑐 𝐵 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) )
7	1 2 3 4 5 6	syl221anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) )