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Metamath Proof Explorer

Theorem cxplt3d

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

Ref Expression
Hypotheses rpcxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
rpcxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
cxplt3d.3 ⊢ ( 𝜑 → 𝐴 < 1 )
cxplt3d.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion cxplt3d ( 𝜑 → ( 𝐵 < 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 rpcxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2 rpcxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 cxplt3d.3 ⊢ ( 𝜑 → 𝐴 < 1 )
4 cxplt3d.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5 cxplt3 ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 < 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐴 ↑𝑐 𝐵 ) ) )
6 1 3 2 4 5 syl22anc ⊢ ( 𝜑 → ( 𝐵 < 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐴 ↑𝑐 𝐵 ) ) )