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Metamath Proof Explorer

Theorem cxpltd

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

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

Proof

Step Hyp Ref Expression
1 recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 cxpltd.2 ⊢ ( 𝜑 → 1 < 𝐴 )
3 cxpltd.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4 cxpltd.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5 cxplt ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 < 𝐶 ↔ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → ( 𝐵 < 𝐶 ↔ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )