:: XXREAL_0 semantic presentation
:: deftheorem Def1 defines ext-real XXREAL_0:def 1 :
:: deftheorem Def2 defines +infty XXREAL_0:def 2 :
:: deftheorem Def3 defines -infty XXREAL_0:def 3 :
:: deftheorem Def4 defines ExtREAL XXREAL_0:def 4 :
definition
let c
1, c
2 be
ext-real number ;
pred c
1 <= c
2 means :
Def5:
:: XXREAL_0:def 5
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = b
1 & a
2 = b
2 & b
1 <=' b
2 )
if ( a
1 in REAL+ & a
2 in REAL+ )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = [0,b1] & a
2 = [0,b2] & b
2 <=' b
1 )
if ( a
1 in [:{0},REAL+ :] & a
2 in [:{0},REAL+ :] )
otherwise not ( not ( a
2 in REAL+ & a
1 in [:{0},REAL+ :] ) & not a
1 = -infty & not a
2 = +infty );
consistency
( c1 in REAL+ & c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] implies ( not ( ex b1, b2 being Element of REAL+ st
( c1 = b1 & c2 = b2 & b1 <=' b2 ) & ( for b1, b2 being Element of REAL+ holds
not ( c1 = [0,b1] & c2 = [0,b2] & b2 <=' b1 ) ) ) & not ( ex b1, b2 being Element of REAL+ st
( c1 = [0,b1] & c2 = [0,b2] & b2 <=' b1 ) & ( for b1, b2 being Element of REAL+ holds
not ( c1 = b1 & c2 = b2 & b1 <=' b2 ) ) ) ) )
by ARYTM_0:5, XBOOLE_0:3;
reflexivity
for b1 being ext-real number holds
( not ( b1 in REAL+ & b1 in REAL+ & ( for b2, b3 being Element of REAL+ holds
not ( b1 = b2 & b1 = b3 & b2 <=' b3 ) ) ) & not ( b1 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] & ( for b2, b3 being Element of REAL+ holds
not ( b1 = [0,b2] & b1 = [0,b3] & b3 <=' b2 ) ) ) & not ( not ( b1 in REAL+ & b1 in REAL+ ) & not ( b1 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] ) & not ( b1 in REAL+ & b1 in [:{0},REAL+ :] ) & not b1 = -infty & not b1 = +infty ) )
connectedness
for b1, b2 being ext-real number holds
( not ( not ( b1 in REAL+ & b2 in REAL+ & ( for b3, b4 being Element of REAL+ holds
not ( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ( for b3, b4 being Element of REAL+ holds
not ( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) ) & not ( not ( b1 in REAL+ & b2 in REAL+ ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] ) & not ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) & not b1 = -infty & not b2 = +infty ) ) implies ( not ( b2 in REAL+ & b1 in REAL+ & ( for b3, b4 being Element of REAL+ holds
not ( b2 = b3 & b1 = b4 & b3 <=' b4 ) ) ) & not ( b2 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] & ( for b3, b4 being Element of REAL+ holds
not ( b2 = [0,b3] & b1 = [0,b4] & b4 <=' b3 ) ) ) & not ( not ( b2 in REAL+ & b1 in REAL+ ) & not ( b2 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] ) & not ( b1 in REAL+ & b2 in [:{0},REAL+ :] ) & not b2 = -infty & not b1 = +infty ) ) )
end;
:: deftheorem Def5 defines <= XXREAL_0:def 5 :
for b
1, b
2 being
ext-real number holds
( ( b
1 in REAL+ & b
2 in REAL+ implies ( b
1 <= b
2 iff ex b
3, b
4 being
Element of
REAL+ st
( b
1 = b
3 & b
2 = b
4 & b
3 <=' b
4 ) ) ) & ( b
1 in [:{0},REAL+ :] & b
2 in [:{0},REAL+ :] implies ( b
1 <= b
2 iff ex b
3, b
4 being
Element of
REAL+ st
( b
1 = [0,b3] & b
2 = [0,b4] & b
4 <=' b
3 ) ) ) & ( not ( b
1 in REAL+ & b
2 in REAL+ ) & not ( b
1 in [:{0},REAL+ :] & b
2 in [:{0},REAL+ :] ) implies ( b
1 <= b
2 iff not ( not ( b
2 in REAL+ & b
1 in [:{0},REAL+ :] ) & not b
1 = -infty & not b
2 = +infty ) ) ) );
Lemma3:
+infty <> [0,0]
Lemma4:
not +infty in REAL+
by ARYTM_0:1, ORDINAL1:7;
Lemma5:
not -infty in REAL+
Lemma6:
not +infty in [:{0},REAL+ :]
Lemma7:
not -infty in [:{0},REAL+ :]
Lemma8:
-infty < +infty
theorem Th1: :: XXREAL_0:1
Lemma10:
for b1 being ext-real number holds
( -infty >= b1 implies b1 = -infty )
Lemma11:
for b1 being ext-real number holds
( +infty <= b1 implies b1 = +infty )
theorem Th2: :: XXREAL_0:2
theorem Th3: :: XXREAL_0:3
theorem Th4: :: XXREAL_0:4
theorem Th5: :: XXREAL_0:5
theorem Th6: :: XXREAL_0:6
theorem Th7: :: XXREAL_0:7
theorem Th8: :: XXREAL_0:8
Lemma15:
for b1 being ext-real number holds
not ( not b1 in REAL & not b1 = +infty & not b1 = -infty )
theorem Th9: :: XXREAL_0:9
theorem Th10: :: XXREAL_0:10
theorem Th11: :: XXREAL_0:11
theorem Th12: :: XXREAL_0:12
theorem Th13: :: XXREAL_0:13
theorem Th14: :: XXREAL_0:14
:: deftheorem Def6 defines positive XXREAL_0:def 6 :
:: deftheorem Def7 defines negative XXREAL_0:def 7 :
:: deftheorem Def8 defines min XXREAL_0:def 8 :
:: deftheorem Def9 defines max XXREAL_0:def 9 :
theorem Th15: :: XXREAL_0:15
theorem Th16: :: XXREAL_0:16
theorem Th17: :: XXREAL_0:17
theorem Th18: :: XXREAL_0:18
theorem Th19: :: XXREAL_0:19
theorem Th20: :: XXREAL_0:20
theorem Th21: :: XXREAL_0:21
theorem Th22: :: XXREAL_0:22
theorem Th23: :: XXREAL_0:23
theorem Th24: :: XXREAL_0:24
theorem Th25: :: XXREAL_0:25
theorem Th26: :: XXREAL_0:26
theorem Th27: :: XXREAL_0:27
theorem Th28: :: XXREAL_0:28
theorem Th29: :: XXREAL_0:29
theorem Th30: :: XXREAL_0:30
theorem Th31: :: XXREAL_0:31
theorem Th32: :: XXREAL_0:32
theorem Th33: :: XXREAL_0:33
theorem Th34: :: XXREAL_0:34
theorem Th35: :: XXREAL_0:35
theorem Th36: :: XXREAL_0:36
theorem Th37: :: XXREAL_0:37
theorem Th38: :: XXREAL_0:38
theorem Th39: :: XXREAL_0:39
theorem Th40: :: XXREAL_0:40
for b
1, b
2, b
3 being
ext-real number holds
max (max (min b1,b2),(min b2,b3)),
(min b3,b1) = min (min (max b1,b2),(max b2,b3)),
(max b3,b1)
theorem Th41: :: XXREAL_0:41
theorem Th42: :: XXREAL_0:42
theorem Th43: :: XXREAL_0:43
theorem Th44: :: XXREAL_0:44