Theorem avgle2

Description: Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013) (Revised by Mario Carneiro, 28-May-2014)

	Ref	Expression
Assertion	avgle2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		avglt1	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ 𝐵 < ( ( 𝐵 + 𝐴 ) / 2 ) ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ 𝐵 < ( ( 𝐵 + 𝐴 ) / 2 ) ) )
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	breq2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < ( ( 𝐴 + 𝐵 ) / 2 ) ↔ 𝐵 < ( ( 𝐵 + 𝐴 ) / 2 ) ) )
9	2 8	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ 𝐵 < ( ( 𝐴 + 𝐵 ) / 2 ) ) )
10	9	notbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ 𝐵 < ( ( 𝐴 + 𝐵 ) / 2 ) ) )
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 + 𝐵 ) / 2 ) ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( ( 𝐴 + 𝐵 ) / 2 ) ) )
16	14 15	sylancom	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( ( 𝐴 + 𝐵 ) / 2 ) ) )
17	10 11 16	3bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( ( 𝐴 + 𝐵 ) / 2 ) ≤ 𝐵 ) )