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 ↔ ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 cxple2 ( ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵𝐴 ↔ ( 𝐵𝑐 𝐶 ) ≤ ( 𝐴𝑐 𝐶 ) ) )
2 1 3com12 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵𝐴 ↔ ( 𝐵𝑐 𝐶 ) ≤ ( 𝐴𝑐 𝐶 ) ) )
3 2 notbid ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ¬ 𝐵𝐴 ↔ ¬ ( 𝐵𝑐 𝐶 ) ≤ ( 𝐴𝑐 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 ↔ ¬ 𝐵𝐴 ) )
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 ≤ 𝐴𝐶 ∈ ℝ ) → ( 𝐴𝑐 𝐶 ) ∈ ℝ )
11 4 7 9 10 syl3anc ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴𝑐 𝐶 ) ∈ ℝ )
12 simp2r ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → 0 ≤ 𝐵 )
13 recxpcl ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵𝐶 ∈ ℝ ) → ( 𝐵𝑐 𝐶 ) ∈ ℝ )
14 5 12 9 13 syl3anc ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵𝑐 𝐶 ) ∈ ℝ )
15 11 14 ltnled ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ↔ ¬ ( 𝐵𝑐 𝐶 ) ≤ ( 𝐴𝑐 𝐶 ) ) )
16 3 6 15 3bitr4d ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴𝑐 𝐶 ) < ( 𝐵𝑐 𝐶 ) ) )