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Metamath Proof Explorer

Theorem cxple2ad

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

Ref Expression
Hypotheses recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
recxpcld.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
cxple2ad.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
cxple2ad.5 ⊢ ( 𝜑 → 0 ≤ 𝐶 )
cxple2ad.6 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion cxple2ad ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) )

Proof

Step Hyp Ref Expression
1 recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
3 recxpcld.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4 cxple2ad.4 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5 cxple2ad.5 ⊢ ( 𝜑 → 0 ≤ 𝐶 )
6 cxple2ad.6 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
7 cxple2a ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) )
8 1 3 4 2 5 6 7 syl321anc ⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) )