Step |
Hyp |
Ref |
Expression |
1 |
|
rexr |
|- ( A e. RR -> A e. RR* ) |
2 |
|
sgnval |
|- ( A e. RR* -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) |
3 |
1 2
|
syl |
|- ( A e. RR -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) |
4 |
3
|
oveq1d |
|- ( A e. RR -> ( ( sgn ` A ) x. A ) = ( if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) x. A ) ) |
5 |
|
ovif |
|- ( if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) x. A ) = if ( A = 0 , ( 0 x. A ) , ( if ( A < 0 , -u 1 , 1 ) x. A ) ) |
6 |
|
ovif |
|- ( if ( A < 0 , -u 1 , 1 ) x. A ) = if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) |
7 |
|
ifeq2 |
|- ( ( if ( A < 0 , -u 1 , 1 ) x. A ) = if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) -> if ( A = 0 , ( 0 x. A ) , ( if ( A < 0 , -u 1 , 1 ) x. A ) ) = if ( A = 0 , ( 0 x. A ) , if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) ) ) |
8 |
6 7
|
ax-mp |
|- if ( A = 0 , ( 0 x. A ) , ( if ( A < 0 , -u 1 , 1 ) x. A ) ) = if ( A = 0 , ( 0 x. A ) , if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) ) |
9 |
5 8
|
eqtri |
|- ( if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) x. A ) = if ( A = 0 , ( 0 x. A ) , if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) ) |
10 |
|
mul02lem2 |
|- ( A e. RR -> ( 0 x. A ) = 0 ) |
11 |
10
|
adantr |
|- ( ( A e. RR /\ A = 0 ) -> ( 0 x. A ) = 0 ) |
12 |
|
simpr |
|- ( ( A e. RR /\ A = 0 ) -> A = 0 ) |
13 |
12
|
abs00bd |
|- ( ( A e. RR /\ A = 0 ) -> ( abs ` A ) = 0 ) |
14 |
11 13
|
eqtr4d |
|- ( ( A e. RR /\ A = 0 ) -> ( 0 x. A ) = ( abs ` A ) ) |
15 |
|
recn |
|- ( A e. RR -> A e. CC ) |
16 |
15
|
mulm1d |
|- ( A e. RR -> ( -u 1 x. A ) = -u A ) |
17 |
15
|
mulid2d |
|- ( A e. RR -> ( 1 x. A ) = A ) |
18 |
16 17
|
ifeq12d |
|- ( A e. RR -> if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) = if ( A < 0 , -u A , A ) ) |
19 |
|
reabsifneg |
|- ( A e. RR -> ( abs ` A ) = if ( A < 0 , -u A , A ) ) |
20 |
18 19
|
eqtr4d |
|- ( A e. RR -> if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) = ( abs ` A ) ) |
21 |
20
|
adantr |
|- ( ( A e. RR /\ -. A = 0 ) -> if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) = ( abs ` A ) ) |
22 |
14 21
|
ifeqda |
|- ( A e. RR -> if ( A = 0 , ( 0 x. A ) , if ( A < 0 , ( -u 1 x. A ) , ( 1 x. A ) ) ) = ( abs ` A ) ) |
23 |
9 22
|
syl5eq |
|- ( A e. RR -> ( if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) x. A ) = ( abs ` A ) ) |
24 |
4 23
|
eqtr2d |
|- ( A e. RR -> ( abs ` A ) = ( ( sgn ` A ) x. A ) ) |