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Metamath Proof Explorer

Theorem avgle1

Description: Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014)

Ref Expression
Assertion avgle1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ( ( 𝐴 + 𝐵 ) / 2 ) ) )

Proof

Step Hyp Ref Expression
1 avglt2 ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) < 𝐴 ) )
2 1 ancoms ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) < 𝐴 ) )
3 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
5 addcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6 3 4 5 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
7 6 oveq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) / 2 ) = ( ( 𝐵 + 𝐴 ) / 2 ) )
8 7 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) < 𝐴 ) )
9 2 8 bitr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐴 ) )
10 9 notbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐴 ) )
11 lenlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
12 readdcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
13 rehalfcl ⊢ ( ( 𝐴 + 𝐵 ) ∈ ℝ → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℝ )
14 12 13 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℝ )
15 lenlt ⊢ ( ( 𝐴 ∈ ℝ ∧ ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℝ ) → ( 𝐴 ≤ ( ( 𝐴 + 𝐵 ) / 2 ) ↔ ¬ ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐴 ) )
16 14 15 syldan ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ ( ( 𝐴 + 𝐵 ) / 2 ) ↔ ¬ ( ( 𝐴 + 𝐵 ) / 2 ) < 𝐴 ) )
17 10 11 16 3bitr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ 𝐴 ≤ ( ( 𝐴 + 𝐵 ) / 2 ) ) )