Step |
Hyp |
Ref |
Expression |
1 |
|
absnid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 ) |
2 |
1
|
eqcomd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → - 𝐴 = ( abs ‘ 𝐴 ) ) |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
|
ltnle |
⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 ↔ ¬ 𝐴 ≤ 0 ) ) |
5 |
|
ltle |
⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 → 0 ≤ 𝐴 ) ) |
6 |
4 5
|
sylbird |
⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ¬ 𝐴 ≤ 0 → 0 ≤ 𝐴 ) ) |
7 |
3 6
|
mpan |
⊢ ( 𝐴 ∈ ℝ → ( ¬ 𝐴 ≤ 0 → 0 ≤ 𝐴 ) ) |
8 |
7
|
imdistani |
⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ≤ 0 ) → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
9 |
|
absid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( abs ‘ 𝐴 ) = 𝐴 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = 𝐴 ) |
11 |
10
|
eqcomd |
⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ≤ 0 ) → 𝐴 = ( abs ‘ 𝐴 ) ) |
12 |
2 11
|
ifeqda |
⊢ ( 𝐴 ∈ ℝ → if ( 𝐴 ≤ 0 , - 𝐴 , 𝐴 ) = ( abs ‘ 𝐴 ) ) |
13 |
12
|
eqcomd |
⊢ ( 𝐴 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 𝐴 ≤ 0 , - 𝐴 , 𝐴 ) ) |