Theorem avgle

Description: The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005) (Revised by Mario Carneiro, 28-May-2014)

	Ref	Expression
Assertion	avgle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐴 ∨ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		letric	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) )
2	1	orcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ∨ 𝐴 ≤ 𝐵 ) )
3		avgle2	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) ≤ 𝐴 ) )
4	3	ancoms	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) ≤ 𝐴 ) )
5		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
6		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
7		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
8	5 6 7	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
9	8	oveq1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) / 2 ) = ( ( 𝐵 + 𝐴 ) / 2 ) )
10	9	breq1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐴 ↔ ( ( 𝐵 + 𝐴 ) / 2 ) ≤ 𝐴 ) )
11	4 10	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐴 ) )
12		avgle2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) )
13	11 12	orbi12d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐵 ≤ 𝐴 ∨ 𝐴 ≤ 𝐵 ) ↔ ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐴 ∨ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) ) )
14	2 13	mpbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐴 ∨ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) )