Step |
Hyp |
Ref |
Expression |
1 |
|
absid |
|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A ) |
2 |
1
|
eqcomd |
|- ( ( A e. RR /\ 0 <_ A ) -> A = ( abs ` A ) ) |
3 |
|
0re |
|- 0 e. RR |
4 |
|
ltnle |
|- ( ( A e. RR /\ 0 e. RR ) -> ( A < 0 <-> -. 0 <_ A ) ) |
5 |
|
ltle |
|- ( ( A e. RR /\ 0 e. RR ) -> ( A < 0 -> A <_ 0 ) ) |
6 |
4 5
|
sylbird |
|- ( ( A e. RR /\ 0 e. RR ) -> ( -. 0 <_ A -> A <_ 0 ) ) |
7 |
3 6
|
mpan2 |
|- ( A e. RR -> ( -. 0 <_ A -> A <_ 0 ) ) |
8 |
7
|
imdistani |
|- ( ( A e. RR /\ -. 0 <_ A ) -> ( A e. RR /\ A <_ 0 ) ) |
9 |
|
absnid |
|- ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A ) |
10 |
8 9
|
syl |
|- ( ( A e. RR /\ -. 0 <_ A ) -> ( abs ` A ) = -u A ) |
11 |
10
|
eqcomd |
|- ( ( A e. RR /\ -. 0 <_ A ) -> -u A = ( abs ` A ) ) |
12 |
2 11
|
ifeqda |
|- ( A e. RR -> if ( 0 <_ A , A , -u A ) = ( abs ` A ) ) |
13 |
12
|
eqcomd |
|- ( A e. RR -> ( abs ` A ) = if ( 0 <_ A , A , -u A ) ) |