:: ABSVALUE semantic presentation
:: deftheorem Def1 defines |. ABSVALUE:def 1 :
theorem :: ABSVALUE:1
Lm1:
for x being real number holds 0 <= abs x
by COMPLEX1:132;
Lm2:
for x being real number st x <> 0 holds
0 < abs x
by COMPLEX1:133;
theorem :: ABSVALUE:2
canceled;
theorem :: ABSVALUE:3
canceled;
theorem :: ABSVALUE:4
canceled;
theorem :: ABSVALUE:5
canceled;
theorem :: ABSVALUE:6
canceled;
theorem :: ABSVALUE:7
theorem :: ABSVALUE:8
canceled;
theorem :: ABSVALUE:9
Lm3:
for x, y being real number holds abs (x * y) = (abs x) * (abs y)
by COMPLEX1:151;
theorem :: ABSVALUE:10
canceled;
theorem :: ABSVALUE:11
theorem :: ABSVALUE:12
Lm4:
for x, y being real number holds abs (x + y) <= (abs x) + (abs y)
by COMPLEX1:142;
theorem :: ABSVALUE:13
canceled;
theorem Th14: :: ABSVALUE:14
theorem :: ABSVALUE:15
Lm5:
for x, y being real number holds abs (x / y) = (abs x) / (abs y)
by COMPLEX1:153;
theorem :: ABSVALUE:16
canceled;
theorem :: ABSVALUE:17
canceled;
theorem :: ABSVALUE:18
canceled;
theorem :: ABSVALUE:19
canceled;
theorem :: ABSVALUE:20
theorem :: ABSVALUE:21
theorem :: ABSVALUE:22
canceled;
theorem :: ABSVALUE:23
theorem :: ABSVALUE:24
theorem :: ABSVALUE:25
theorem :: ABSVALUE:26
:: deftheorem Def2 defines sgn ABSVALUE:def 2 :
for
x being
real number holds
( ( 0
< x implies
sgn x = 1 ) & (
x < 0 implies
sgn x = - 1 ) & ( not 0
< x & not
x < 0 implies
sgn x = 0 ) );
theorem :: ABSVALUE:27
canceled;
theorem :: ABSVALUE:28
canceled;
theorem :: ABSVALUE:29
canceled;
theorem :: ABSVALUE:30
canceled;
theorem :: ABSVALUE:31
theorem :: ABSVALUE:32
theorem Th33: :: ABSVALUE:33
theorem :: ABSVALUE:34
theorem Th35: :: ABSVALUE:35
theorem :: ABSVALUE:36
theorem :: ABSVALUE:37
theorem Th38: :: ABSVALUE:38
theorem Th39: :: ABSVALUE:39
theorem :: ABSVALUE:40
theorem :: ABSVALUE:41
theorem :: ABSVALUE:42