:: EXTREAL1 semantic presentation
theorem Th1: :: EXTREAL1:1
theorem Th2: :: EXTREAL1:2
canceled;
theorem Th3: :: EXTREAL1:3
canceled;
theorem Th4: :: EXTREAL1:4
theorem Th5: :: EXTREAL1:5
for b
1, b
2 being
R_eal holds b
1 - (- b2) = b
1 + b
2
theorem Th6: :: EXTREAL1:6
canceled;
theorem Th7: :: EXTREAL1:7
Lemma4:
for b1 being R_eal holds
( b1 in REAL implies ( b1 + -infty = -infty & b1 + +infty = +infty ) )
by SUPINF_1:1, SUPINF_1:6, SUPINF_2:def 2;
Lemma5:
for b1, b2 being R_eal holds
( b1 in REAL & b2 in REAL implies b1 + b2 in REAL )
theorem Th8: :: EXTREAL1:8
theorem Th9: :: EXTREAL1:9
theorem Th10: :: EXTREAL1:10
canceled;
theorem Th11: :: EXTREAL1:11
:: deftheorem Def1 defines * EXTREAL1:def 1 :
theorem Th12: :: EXTREAL1:12
canceled;
theorem Th13: :: EXTREAL1:13
for b
1, b
2 being
R_ealfor b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies b
1 * b
2 = b
3 * b
4 )
Lemma10:
for b1 being R_eal
for b2 being Real holds
not ( b1 = b2 & 0 < b2 & not 0. < b1 )
by SUPINF_2:def 1;
Lemma11:
for b1 being R_eal
for b2 being Real holds
not ( b1 = b2 & b2 < 0 & not b1 < 0. )
by SUPINF_2:def 1;
theorem Th14: :: EXTREAL1:14
theorem Th15: :: EXTREAL1:15
theorem Th16: :: EXTREAL1:16
theorem Th17: :: EXTREAL1:17
for b
1, b
2 being
R_eal holds b
1 * b
2 = b
2 * b
1
theorem Th18: :: EXTREAL1:18
theorem Th19: :: EXTREAL1:19
theorem Th20: :: EXTREAL1:20
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 )
theorem Th21: :: EXTREAL1:21
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. )
theorem Th22: :: EXTREAL1:22
theorem Th23: :: EXTREAL1:23
for b
1, b
2, b
3 being
R_eal holds
(b1 * b2) * b
3 = b
1 * (b2 * b3)
theorem Th24: :: EXTREAL1:24
theorem Th25: :: EXTREAL1:25
theorem Th26: :: EXTREAL1:26
for b
1, b
2 being
R_eal holds
(
- (b1 * b2) = b
1 * (- b2) &
- (b1 * b2) = (- b1) * b
2 )
theorem Th27: :: EXTREAL1:27
theorem Th28: :: EXTREAL1:28
Lemma20:
for b1, b2, b3 being R_eal holds
( b1 <> +infty & b1 <> -infty implies b1 * (b2 + b3) = (b1 * b2) + (b1 * b3) )
theorem Th29: :: EXTREAL1:29
theorem Th30: :: EXTREAL1:30
:: deftheorem Def2 defines / EXTREAL1:def 2 :
theorem Th31: :: EXTREAL1:31
canceled;
theorem Th32: :: EXTREAL1:32
for b
1, b
2 being
R_eal holds
( b
2 <> 0. implies for b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies b
1 / b
2 = b
3 / b
4 ) )
theorem Th33: :: EXTREAL1:33
theorem Th34: :: EXTREAL1:34
:: deftheorem Def3 defines |. EXTREAL1:def 3 :
theorem Th35: :: EXTREAL1:35
canceled;
theorem Th36: :: EXTREAL1:36
theorem Th37: :: EXTREAL1:37
theorem Th38: :: EXTREAL1:38
theorem Th39: :: EXTREAL1:39
theorem Th40: :: EXTREAL1:40
theorem Th41: :: EXTREAL1:41
theorem Th42: :: EXTREAL1:42
for b
1, b
2, b
3 being
R_eal holds
( b
1 <= b
2 &
0. <= b
3 implies b
1 * b
3 <= b
2 * b
3 )
theorem Th43: :: EXTREAL1:43
for b
1, b
2, b
3 being
R_eal holds
( b
1 <= b
2 & b
3 <= 0. implies b
2 * b
3 <= b
1 * b
3 )
theorem Th44: :: EXTREAL1:44
theorem Th45: :: EXTREAL1:45
theorem Th46: :: EXTREAL1:46
for b
1, b
2 being
R_eal holds
( b
1 is
Real & b
2 is
Real implies ( not ( b
1 < b
2 & ( for b
3, b
4 being
Real holds
not ( b
3 = b
1 & b
4 = b
2 & b
3 < b
4 ) ) ) & not ( ex b
3, b
4 being
Real st
( b
3 = b
1 & b
4 = b
2 & b
3 < b
4 ) & not b
1 < b
2 ) ) ) ;
theorem Th47: :: EXTREAL1:47
theorem Th48: :: EXTREAL1:48
theorem Th49: :: EXTREAL1:49
theorem Th50: :: EXTREAL1:50
theorem Th51: :: EXTREAL1:51
theorem Th52: :: EXTREAL1:52