:: EXTREAL2 semantic presentation
theorem Th1: :: EXTREAL2:1
canceled;
theorem Th2: :: EXTREAL2:2
theorem Th3: :: EXTREAL2:3
theorem Th4: :: EXTREAL2:4
theorem Th5: :: EXTREAL2:5
theorem Th6: :: EXTREAL2:6
canceled;
theorem Th7: :: EXTREAL2:7
theorem Th8: :: EXTREAL2:8
canceled;
theorem Th9: :: EXTREAL2:9
theorem Th10: :: EXTREAL2:10
canceled;
theorem Th11: :: EXTREAL2:11
theorem Th12: :: EXTREAL2:12
not +infty in REAL
by XXREAL_0:8;
then Lemma27:
+infty + -infty = 0.
by SUPINF_2:def 2;
ex a being Real st
( 0. = a & - 0. = - a )
by SUPINF_2:def 3;
then Lemma28:
- 0. = 0
;
Lemma29:
- +infty = -infty
by SUPINF_2:def 3;
theorem Th13: :: EXTREAL2:13
E30: - (+infty + -infty ) =
+infty - +infty
by , , SUPINF_2:def 3
.=
(- -infty ) - +infty
by SUPINF_2:def 3, XXREAL_0:11
;
Lemma31:
for x being R_eal st x in REAL holds
- (x + +infty ) = (- +infty ) + (- x)
Lemma32:
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 Th16: :: 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 Th21: :: EXTREAL2:21
theorem Th22: :: EXTREAL2:22
theorem Th23: :: EXTREAL2:23
theorem Th24: :: EXTREAL2:24
theorem Th25: :: EXTREAL2:25
theorem Th26: :: EXTREAL2:26
theorem Th27: :: EXTREAL2:27
theorem Th28: :: EXTREAL2:28
theorem Th29: :: EXTREAL2:29
theorem Th30: :: EXTREAL2:30
theorem Th31: :: EXTREAL2:31
theorem Th32: :: EXTREAL2:32
theorem Th33: :: EXTREAL2:33
theorem Th34: :: EXTREAL2:34
theorem Th35: :: EXTREAL2:35
canceled;
theorem Th36: :: EXTREAL2:36
canceled;
theorem Th37: :: EXTREAL2:37
canceled;
theorem Th38: :: EXTREAL2:38
canceled;
theorem Th39: :: EXTREAL2:39
canceled;
theorem Th40: :: EXTREAL2:40
theorem Th41: :: EXTREAL2:41
theorem Th42: :: 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 Th45: :: 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 Th7;
theorem Th46: :: 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 Th4;
theorem Th47: :: EXTREAL2:47
for
x,
y being
R_eal holds
( (
x <= - y implies
y <= - x ) & (
- x <= y implies
- y <= x ) )
theorem Th48: :: EXTREAL2:48
canceled;
theorem Th49: :: EXTREAL2:49
theorem Th50: :: EXTREAL2:50
theorem Th51: :: EXTREAL2:51
theorem Th52: :: EXTREAL2:52
theorem Th53: :: EXTREAL2:53
theorem Th54: :: EXTREAL2:54
theorem Th55: :: EXTREAL2:55
theorem Th56: :: 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 Th63: :: EXTREAL2:63
theorem Th64: :: EXTREAL2:64
theorem Th65: :: EXTREAL2:65
theorem Th66: :: EXTREAL2:66
theorem Th67: :: EXTREAL2:67
theorem Th68: :: EXTREAL2:68
theorem Th69: :: EXTREAL2:69
theorem Th70: :: EXTREAL2:70
theorem Th71: :: EXTREAL2:71
theorem Th72: :: EXTREAL2:72
theorem Th73: :: EXTREAL2:73
theorem Th74: :: 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 ) ) ;
theorem Th75: :: EXTREAL2:75
theorem Th76: :: EXTREAL2:76
theorem Th77: :: EXTREAL2:77
theorem Th78: :: EXTREAL2:78
theorem Th79: :: EXTREAL2:79
theorem Th80: :: EXTREAL2:80
theorem Th81: :: EXTREAL2:81
theorem Th82: :: EXTREAL2:82
canceled;
theorem Th83: :: EXTREAL2:83
theorem Th84: :: EXTREAL2:84
theorem Th85: :: EXTREAL2:85
theorem Th86: :: EXTREAL2:86
canceled;
theorem Th87: :: EXTREAL2:87
theorem Th88: :: EXTREAL2:88
theorem Th89: :: EXTREAL2:89
theorem Th90: :: EXTREAL2:90
theorem Th91: :: EXTREAL2:91
theorem Th92: :: EXTREAL2:92
theorem Th93: :: EXTREAL2:93
canceled;
theorem Th94: :: EXTREAL2:94
theorem Th95: :: EXTREAL2:95
theorem Th96: :: EXTREAL2:96
for
y,
x,
z being
R_eal holds
( (
y <= x &
z <= x ) iff
max y,
z <= x )
theorem Th97: :: EXTREAL2:97
canceled;
theorem Th98: :: EXTREAL2:98
theorem Th99: :: EXTREAL2:99
theorem Th100: :: EXTREAL2:100
theorem Th101: :: EXTREAL2:101
theorem Th102: :: EXTREAL2:102
theorem Th103: :: EXTREAL2:103
for
x,
y,
z being
R_eal holds
(
min x,
(max y,z) = max (min x,y),
(min x,z) &
min (max y,z),
x = max (min y,x),
(min z,x) )
by XXREAL_0:38;
theorem Th104: :: EXTREAL2:104
for
x,
y,
z being
R_eal holds
(
max x,
(min y,z) = min (max x,y),
(max x,z) &
max (min y,z),
x = min (max y,x),
(max z,x) )
by XXREAL_0:39;