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 ) ) |