:: EXTREAL2 semantic presentation
theorem :: EXTREAL2:1
canceled;
theorem :: EXTREAL2:2
theorem :: EXTREAL2:3
theorem Th4: :: EXTREAL2:4
theorem :: EXTREAL2:5
theorem :: EXTREAL2:6
canceled;
theorem Th7: :: EXTREAL2:7
theorem :: EXTREAL2:8
canceled;
theorem Th9: :: EXTREAL2:9
theorem :: EXTREAL2:10
canceled;
theorem :: EXTREAL2:11
theorem :: EXTREAL2:12
not +infty in REAL
by XXREAL_0:8;
then Lm1:
+infty + -infty = 0.
by SUPINF_2:def 2;
ex a being Real st
( 0. = a & - 0. = - a )
by SUPINF_2:def 3;
then Lm2:
- 0. = 0
;
Lm3:
- +infty = -infty
by SUPINF_2:def 3;
theorem :: EXTREAL2:13
Lm4: - (+infty + -infty ) =
+infty - +infty
by Lm1, Lm2, SUPINF_2:def 3
.=
(- -infty ) - +infty
by SUPINF_2:def 3, XXREAL_0:11
;
Lm5:
for x being R_eal st x in REAL holds
- (x + +infty ) = (- +infty ) + (- x)
Lm6:
for x being R_eal st x in REAL holds
- (x + -infty ) = (- -infty ) + (- x)
theorem Th14: :: EXTREAL2:14
theorem Th15: :: EXTREAL2:15
for
x,
y being
R_eal holds
(
- (x - y) = (- x) + y &
- (x - y) = y - x )
theorem :: EXTREAL2:16
for
x,
y being
R_eal holds
(
- ((- x) + y) = x - y &
- ((- x) + y) = x + (- y) )
theorem Th17: :: EXTREAL2:17
theorem Th18: :: EXTREAL2:18
theorem Th19: :: EXTREAL2:19
theorem Th20: :: EXTREAL2:20
theorem :: EXTREAL2:21
theorem :: EXTREAL2:22
theorem Th23: :: EXTREAL2:23
theorem Th24: :: EXTREAL2:24
theorem Th25: :: EXTREAL2:25
theorem Th26: :: EXTREAL2:26
theorem :: EXTREAL2:27
theorem :: EXTREAL2:28
theorem :: EXTREAL2:29
theorem :: EXTREAL2:30
theorem :: EXTREAL2:31
theorem :: EXTREAL2:32
theorem :: EXTREAL2:33
theorem Th34: :: EXTREAL2:34
theorem :: EXTREAL2:35
canceled;
theorem :: EXTREAL2:36
canceled;
theorem :: EXTREAL2:37
canceled;
theorem :: EXTREAL2:38
canceled;
theorem :: EXTREAL2:39
canceled;
theorem :: EXTREAL2:40
theorem :: EXTREAL2:41
theorem :: EXTREAL2:42
theorem Th43: :: EXTREAL2:43
for
x,
y being
R_eal holds
( ( ( 0
< x & 0
< y ) or (
x < 0 &
y < 0 ) ) iff 0
< x * y )
theorem Th44: :: EXTREAL2:44
for
x,
y being
R_eal holds
( ( ( 0
< x &
y < 0 ) or (
x < 0 & 0
< y ) ) iff
x * y < 0 )
theorem :: EXTREAL2:45
for
x,
y being
R_eal holds
( ( ( ( 0
<= x or 0
< x ) & ( 0
<= y or 0
< y ) ) or ( (
x <= 0 or
x < 0 ) & (
y <= 0 or
y < 0 ) ) ) iff 0
<= x * y )
by Th44;
theorem :: EXTREAL2:46
for
x,
y being
R_eal holds
( ( ( (
x <= 0 or
x < 0 ) & ( 0
<= y or 0
< y ) ) or ( ( 0
<= x or 0
< x ) & (
y <= 0 or
y < 0 ) ) ) iff
x * y <= 0 )
by Th43;
theorem Th47: :: EXTREAL2:47
for
x,
y being
R_eal holds
( (
x <= - y implies
y <= - x ) & (
- x <= y implies
- y <= x ) )
theorem :: EXTREAL2:48
canceled;
theorem Th49: :: EXTREAL2:49
theorem Th50: :: EXTREAL2:50
theorem Th51: :: EXTREAL2:51
theorem Th52: :: EXTREAL2:52
theorem :: EXTREAL2:53
theorem :: EXTREAL2:54
theorem Th55: :: EXTREAL2:55
theorem :: EXTREAL2:56
theorem Th57: :: EXTREAL2:57
theorem Th58: :: EXTREAL2:58
theorem Th59: :: EXTREAL2:59
theorem Th60: :: EXTREAL2:60
theorem Th61: :: EXTREAL2:61
theorem Th62: :: EXTREAL2:62
theorem :: EXTREAL2:63
theorem :: EXTREAL2:64
theorem :: EXTREAL2:65
theorem Th66: :: EXTREAL2:66
theorem Th67: :: EXTREAL2:67
theorem Th68: :: EXTREAL2:68
theorem :: EXTREAL2:69
theorem :: EXTREAL2:70
theorem :: EXTREAL2:71
theorem :: EXTREAL2:72
theorem :: EXTREAL2:73
theorem :: EXTREAL2:74
for
x,
y being
R_eal for
a,
b being
Real st
x = a &
y = b holds
( (
b < a implies
y < x ) & (
y < x implies
b < a ) & (
b <= a implies
y <= x ) & (
y <= x implies
b <= a ) ) ;
Lm7:
for x, y being R_eal st ( x = -infty or y = -infty ) holds
min x,y = -infty
by XXREAL_0:44;
Lm8:
for x, y being R_eal st ( x = +infty or y = +infty ) holds
max x,y = +infty
by XXREAL_0:41;
theorem :: EXTREAL2:75
canceled;
theorem :: EXTREAL2:76
canceled;
theorem :: EXTREAL2:77
theorem :: EXTREAL2:78
canceled;
theorem :: EXTREAL2:79
canceled;
theorem :: EXTREAL2:80
theorem :: EXTREAL2:81
canceled;
theorem :: EXTREAL2:82
canceled;
theorem :: EXTREAL2:83
theorem :: EXTREAL2:84
canceled;
theorem :: EXTREAL2:85
Lm9:
for x, y being R_eal st x <= y holds
max x,y = y
by XXREAL_0:def 9;
theorem :: EXTREAL2:86
canceled;
theorem :: EXTREAL2:87
canceled;
theorem :: EXTREAL2:88
canceled;
theorem :: EXTREAL2:89
canceled;
theorem :: EXTREAL2:90
canceled;
theorem :: EXTREAL2:91
theorem Th92: :: EXTREAL2:92
theorem :: EXTREAL2:93
canceled;
theorem :: EXTREAL2:94
theorem Th95: :: EXTREAL2:95
theorem :: EXTREAL2:96
for
y,
x,
z being
R_eal holds
( (
y <= x &
z <= x ) iff
max y,
z <= x )
theorem :: EXTREAL2:97
canceled;
theorem :: EXTREAL2:98
canceled;
theorem :: EXTREAL2:99
canceled;
theorem :: EXTREAL2:100