:: 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 x be
ext-real number ,
y be
ext-real number ;
pred c1 <= c2 means :
Def5:
:: XXREAL_0:def 5
ex
x',
y' being
Element of
REAL+ st
(
x = x' &
y = y' &
x' <=' y' )
if (
x in REAL+ &
y in REAL+ )
ex
x',
y' being
Element of
REAL+ st
(
x = [0,x'] &
y = [0,y'] &
y' <=' x' )
if (
x in [:{0},REAL+ :] &
y in [:{0},REAL+ :] )
otherwise ( (
y in REAL+ &
x in [:{0},REAL+ :] ) or
x = -infty or
y = +infty );
consistency
( x in REAL+ & y in REAL+ & x in [:{0},REAL+ :] & y in [:{0},REAL+ :] implies ( ex x', y' being Element of REAL+ st
( x = x' & y = y' & x' <=' y' ) iff ex x', y' being Element of REAL+ st
( x = [0,x'] & y = [0,y'] & y' <=' x' ) ) )
by ARYTM_0:5, XBOOLE_0:3;
reflexivity
for x being ext-real number holds
( ( x in REAL+ & x in REAL+ implies ex x', y' being Element of REAL+ st
( x = x' & x = y' & x' <=' y' ) ) & ( x in [:{0},REAL+ :] & x in [:{0},REAL+ :] implies ex x', y' being Element of REAL+ st
( x = [0,x'] & x = [0,y'] & y' <=' x' ) ) & ( ( not x in REAL+ or not x in REAL+ ) & ( not x in [:{0},REAL+ :] or not x in [:{0},REAL+ :] ) & not ( x in REAL+ & x in [:{0},REAL+ :] ) & not x = -infty implies x = +infty ) )
connectedness
for x, y being ext-real number st ( ( x in REAL+ & y in REAL+ & ( for x', y' being Element of REAL+ holds
( not x = x' or not y = y' or not x' <=' y' ) ) ) or ( x in [:{0},REAL+ :] & y in [:{0},REAL+ :] & ( for x', y' being Element of REAL+ holds
( not x = [0,x'] or not y = [0,y'] or not y' <=' x' ) ) ) or ( ( not x in REAL+ or not y in REAL+ ) & ( not x in [:{0},REAL+ :] or not y in [:{0},REAL+ :] ) & not ( y in REAL+ & x in [:{0},REAL+ :] ) & not x = -infty & not y = +infty ) ) holds
( ( y in REAL+ & x in REAL+ implies ex x', y' being Element of REAL+ st
( y = x' & x = y' & x' <=' y' ) ) & ( y in [:{0},REAL+ :] & x in [:{0},REAL+ :] implies ex x', y' being Element of REAL+ st
( y = [0,x'] & x = [0,y'] & y' <=' x' ) ) & ( ( not y in REAL+ or not x in REAL+ ) & ( not y in [:{0},REAL+ :] or not x in [:{0},REAL+ :] ) & not ( x in REAL+ & y in [:{0},REAL+ :] ) & not y = -infty implies x = +infty ) )
end;
:: deftheorem Def5 defines <= XXREAL_0:def 5 :
for
x,
y being
ext-real number holds
( (
x in REAL+ &
y in REAL+ implies (
x <= y iff ex
x',
y' being
Element of
REAL+ st
(
x = x' &
y = y' &
x' <=' y' ) ) ) & (
x in [:{0},REAL+ :] &
y in [:{0},REAL+ :] implies (
x <= y iff ex
x',
y' being
Element of
REAL+ st
(
x = [0,x'] &
y = [0,y'] &
y' <=' x' ) ) ) & ( ( not
x in REAL+ or not
y in REAL+ ) & ( not
x in [:{0},REAL+ :] or not
y in [:{0},REAL+ :] ) implies (
x <= y iff ( (
y in REAL+ &
x in [:{0},REAL+ :] ) or
x = -infty or
y = +infty ) ) ) );
Lemma47:
+infty <> [0,0]
Lemma48:
not +infty in REAL+
by ARYTM_0:1, ORDINAL1:7;
Lemma49:
not -infty in REAL+
Lemma50:
not +infty in [:{0},REAL+ :]
Lemma51:
not -infty in [:{0},REAL+ :]
Lemma52:
-infty < +infty
theorem Th1: :: XXREAL_0:1
Lemma59:
for a being ext-real number st -infty >= a holds
a = -infty
Lemma60:
for a being ext-real number st +infty <= a holds
a = +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
Lemma72:
for a being ext-real number holds
( a in REAL or a = +infty or a = -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
a,
b,
c being
ext-real number holds
max (max (min a,b),(min b,c)),
(min c,a) = min (min (max a,b),(max b,c)),
(max c,a)
theorem Th41: :: XXREAL_0:41
theorem Th42: :: XXREAL_0:42
theorem Th43: :: XXREAL_0:43
theorem Th44: :: XXREAL_0:44