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Metamath Proof Explorer

Theorem cxplt2

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion cxplt2 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 cxple2 ⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )
2 1 3com12 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )
3 2 notbid ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ¬ 𝐵 ≤ 𝐴 ↔ ¬ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )
4 simp1l ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ℝ )
5 simp2l ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐵 ∈ ℝ )
6 4 5 ltnled ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
7 simp1r ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 0 ≤ 𝐴 )
8 rpre ⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℝ )
9 8 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ℝ )
10 recxpcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ )
11 4 7 9 10 syl3anc ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ )
12 simp2r ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 0 ≤ 𝐵 )
13 recxpcl ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℝ )
14 5 12 9 13 syl3anc ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℝ )
15 11 14 ltnled ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐵 ↑𝑐 𝐶 ) ↔ ¬ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )
16 3 6 15 3bitr4d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) < ( 𝐵 ↑𝑐 𝐶 ) ) )