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Metamath Proof Explorer

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 ) )