:: 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
for b
1, b
2 being
R_eal holds
( 0
<= b
1 & 0
<= b
2 implies 0
<= b
1 + b
2 )
theorem Th8: :: EXTREAL2:8
canceled;
theorem Th9: :: EXTREAL2:9
for b
1, b
2 being
R_eal holds
( b
1 <= 0 & b
2 <= 0 implies b
1 + b
2 <= 0 )
theorem Th10: :: EXTREAL2:10
canceled;
theorem Th11: :: EXTREAL2:11
theorem Th12: :: EXTREAL2:12
not +infty in REAL
by XXREAL_0:8;
then Lemma4:
+infty + -infty = 0.
by SUPINF_2:def 2;
ex b1 being Real st
( 0. = b1 & - 0. = - b1 )
by SUPINF_2:def 3;
then Lemma5:
- 0. = 0
;
Lemma6:
- +infty = -infty
by SUPINF_2:def 3;
theorem Th13: :: EXTREAL2:13
for b
1 being
R_eal holds b
1 - b
1 = 0
E7: - (+infty + -infty ) =
- (+infty + -infty )
.=
- 0.
by Lemma4
.=
0
by Lemma5
.=
+infty + -infty
by Lemma4
.=
+infty - +infty
by SUPINF_2:def 3
.=
(- -infty ) - +infty
by SUPINF_2:def 3, XXREAL_0:11
;
Lemma8:
for b1 being R_eal holds
( b1 in REAL implies - (b1 + +infty ) = (- +infty ) + (- b1) )
Lemma9:
for b1 being R_eal holds
( b1 in REAL implies - (b1 + -infty ) = (- -infty ) + (- b1) )
theorem Th14: :: EXTREAL2:14
for b
1, b
2 being
R_eal holds
- (b1 + b2) = (- b2) - b
1
theorem Th15: :: EXTREAL2:15
for b
1, b
2 being
R_eal holds
(
- (b1 - b2) = (- b1) + b
2 &
- (b1 - b2) = b
2 - b
1 )
theorem Th16: :: EXTREAL2:16
for b
1, b
2 being
R_eal holds
(
- ((- b1) + b2) = b
1 - b
2 &
- ((- b1) + b2) = b
1 + (- b2) )
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 b
1, b
2 being
R_eal holds
( not ( ( ( 0
< b
1 & 0
< b
2 ) or ( b
1 < 0 & b
2 < 0 ) ) & not 0
< b
1 * b
2 ) & not ( 0
< b
1 * b
2 & not ( 0
< b
1 & 0
< b
2 ) & not ( b
1 < 0 & b
2 < 0 ) ) )
theorem Th44: :: EXTREAL2:44
for b
1, b
2 being
R_eal holds
( not ( ( ( 0
< b
1 & b
2 < 0 ) or ( b
1 < 0 & 0
< b
2 ) ) & not b
1 * b
2 < 0 ) & not ( b
1 * b
2 < 0 & not ( 0
< b
1 & b
2 < 0 ) & not ( b
1 < 0 & 0
< b
2 ) ) )
theorem Th45: :: EXTREAL2:45
for b
1, b
2 being
R_eal holds
( ( ( not ( not 0
<= b
1 & not 0
< b
1 ) & not ( not 0
<= b
2 & not 0
< b
2 ) ) or ( not ( not b
1 <= 0 & not b
1 < 0 ) & not ( not b
2 <= 0 & not b
2 < 0 ) ) ) iff 0
<= b
1 * b
2 )
by Th44;
theorem Th46: :: EXTREAL2:46
for b
1, b
2 being
R_eal holds
( ( ( not ( not b
1 <= 0 & not b
1 < 0 ) & not ( not 0
<= b
2 & not 0
< b
2 ) ) or ( not ( not 0
<= b
1 & not 0
< b
1 ) & not ( not b
2 <= 0 & not b
2 < 0 ) ) ) iff b
1 * b
2 <= 0 )
by Th43;
theorem Th47: :: EXTREAL2:47
for b
1, b
2 being
R_eal holds
( ( b
1 <= - b
2 implies b
2 <= - b
1 ) & (
- b
1 <= b
2 implies
- b
2 <= b
1 ) )
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
for b
1, b
2 being
R_eal holds
(
|.b1.| < b
2 implies (
- b
2 < b
1 & b
1 < b
2 ) )
theorem Th59: :: EXTREAL2:59
for b
1, b
2 being
R_eal holds
(
- b
1 < b
2 & b
2 < b
1 implies ( 0
< b
1 &
|.b2.| < b
1 ) )
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 b
1, b
2 being
R_ealfor b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies ( not ( b
4 < b
3 & not b
2 < b
1 ) & not ( b
2 < b
1 & not b
4 < b
3 ) & ( b
4 <= b
3 implies b
2 <= b
1 ) & ( b
2 <= b
1 implies b
4 <= b
3 ) ) ) ;
theorem Th75: :: EXTREAL2:75
theorem Th76: :: EXTREAL2:76
theorem Th77: :: EXTREAL2:77
for b
1, b
2 being
R_ealfor b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies (
min b
1,b
2 = min b
3,b
4 &
max b
1,b
2 = max b
3,b
4 ) ) ;
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
for b
1, b
2 being
R_eal holds
(
max b
1,b
2 = b
1 or
max b
1,b
2 = b
2 )
theorem Th96: :: EXTREAL2:96
for b
1, b
2, b
3 being
R_eal holds
( ( b
1 <= b
2 & b
3 <= b
2 ) iff
max b
1,b
3 <= b
2 )
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 b
1, b
2, b
3 being
R_eal holds
(
min b
1,
(max b2,b3) = max (min b1,b2),
(min b1,b3) &
min (max b2,b3),b
1 = max (min b2,b1),
(min b3,b1) )
by XXREAL_0:38;
theorem Th104: :: EXTREAL2:104
for b
1, b
2, b
3 being
R_eal holds
(
max b
1,
(min b2,b3) = min (max b1,b2),
(max b1,b3) &
max (min b2,b3),b
1 = min (max b2,b1),
(max b3,b1) )
by XXREAL_0:39;