:: 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 c1,
c2 be
ext-real number ;
pred c1 <= c2 means :
Def5:
:: XXREAL_0:def 5
ex
b1,
b2 being
Element of
REAL+ st
(
a1 = b1 &
a2 = b2 &
b1 <=' b2 )
if (
a1 in REAL+ &
a2 in REAL+ )
ex
b1,
b2 being
Element of
REAL+ st
(
a1 = [0,b1] &
a2 = [0,b2] &
b2 <=' b1 )
if (
a1 in [:{0},REAL+ :] &
a2 in [:{0},REAL+ :] )
otherwise ( (
a2 in REAL+ &
a1 in [:{0},REAL+ :] ) or
a1 = -infty or
a2 = +infty );
consistency
( c1 in REAL+ & c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] implies ( ex b1, b2 being Element of REAL+ st
( c1 = b1 & c2 = b2 & b1 <=' b2 ) iff ex b1, b2 being Element of REAL+ st
( c1 = [0,b1] & c2 = [0,b2] & b2 <=' b1 ) ) )
by ARYTM_0:5, XBOOLE_0:3;
reflexivity
for b1 being ext-real number holds
( ( b1 in REAL+ & b1 in REAL+ implies ex b2, b3 being Element of REAL+ st
( b1 = b2 & b1 = b3 & b2 <=' b3 ) ) & ( b1 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] implies ex b2, b3 being Element of REAL+ st
( b1 = [0,b2] & b1 = [0,b3] & b3 <=' b2 ) ) & ( ( not b1 in REAL+ or not b1 in REAL+ ) & ( not b1 in [:{0},REAL+ :] or not b1 in [:{0},REAL+ :] ) & not ( b1 in REAL+ & b1 in [:{0},REAL+ :] ) & not b1 = -infty implies b1 = +infty ) )
connectedness
for b1, b2 being ext-real number st ( ( b1 in REAL+ & b2 in REAL+ & ( for b3, b4 being Element of REAL+ holds
( not b1 = b3 or not b2 = b4 or not b3 <=' b4 ) ) ) or ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ( for b3, b4 being Element of REAL+ holds
( not b1 = [0,b3] or not b2 = [0,b4] or not b4 <=' b3 ) ) ) or ( ( not b1 in REAL+ or not b2 in REAL+ ) & ( not b1 in [:{0},REAL+ :] or not b2 in [:{0},REAL+ :] ) & not ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) & not b1 = -infty & not b2 = +infty ) ) holds
( ( b2 in REAL+ & b1 in REAL+ implies ex b3, b4 being Element of REAL+ st
( b2 = b3 & b1 = b4 & b3 <=' b4 ) ) & ( b2 in [:{0},REAL+ :] & b1 in [:{0},REAL+ :] implies ex b3, b4 being Element of REAL+ st
( b2 = [0,b3] & b1 = [0,b4] & b4 <=' b3 ) ) & ( ( not b2 in REAL+ or not b1 in REAL+ ) & ( not b2 in [:{0},REAL+ :] or not b1 in [:{0},REAL+ :] ) & not ( b1 in REAL+ & b2 in [:{0},REAL+ :] ) & not b2 = -infty implies b1 = +infty ) )
end;
:: deftheorem Def5 defines <= XXREAL_0:def 5 :
for
b1,
b2 being
ext-real number holds
( (
b1 in REAL+ &
b2 in REAL+ implies (
b1 <= b2 iff ex
b3,
b4 being
Element of
REAL+ st
(
b1 = b3 &
b2 = b4 &
b3 <=' b4 ) ) ) & (
b1 in [:{0},REAL+ :] &
b2 in [:{0},REAL+ :] implies (
b1 <= b2 iff ex
b3,
b4 being
Element of
REAL+ st
(
b1 = [0,b3] &
b2 = [0,b4] &
b4 <=' b3 ) ) ) & ( ( not
b1 in REAL+ or not
b2 in REAL+ ) & ( not
b1 in [:{0},REAL+ :] or not
b2 in [:{0},REAL+ :] ) implies (
b1 <= b2 iff ( (
b2 in REAL+ &
b1 in [:{0},REAL+ :] ) or
b1 = -infty or
b2 = +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 st -infty >= b1 holds
b1 = -infty
Lemma11:
for b1 being ext-real number st +infty <= b1 holds
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
( b1 in REAL or b1 = +infty or 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
b1,
b2,
b3 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