Metamath Proof Explorer


Theorem cxplt2d

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

Ref Expression
Hypotheses recxpcld.1 ( 𝜑𝐴 ∈ ℝ )
recxpcld.2 ( 𝜑 → 0 ≤ 𝐴 )
recxpcld.3 ( 𝜑𝐵 ∈ ℝ )
mulcxpd.4 ( 𝜑 → 0 ≤ 𝐵 )
cxple2d.5 ( 𝜑𝐶 ∈ ℝ+ )
Assertion cxplt2d ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 recxpcld.1 ( 𝜑𝐴 ∈ ℝ )
2 recxpcld.2 ( 𝜑 → 0 ≤ 𝐴 )
3 recxpcld.3 ( 𝜑𝐵 ∈ ℝ )
4 mulcxpd.4 ( 𝜑 → 0 ≤ 𝐵 )
5 cxple2d.5 ( 𝜑𝐶 ∈ ℝ+ )
6 cxplt2 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ) )
7 1 2 3 4 5 6 syl221anc ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ) )