:: Some Properties of the Intervals
:: by J\'ozef Bia\las
::
:: Received February 5, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

theorem :: MEASURE6:1

ex F being sequence of [:NAT,NAT:] st
( F is one-to-one & dom F = NAT & rng F = [:NAT,NAT:] )

proof end;

theorem :: MEASURE6:2

for F being sequence of ExtREAL st F is nonnegative holds
0. <= SUM F

proof end;

theorem :: MEASURE6:3

for F being sequence of ExtREAL
for x being R_eal st ex n being Element of NAT st x <= F . n & F is nonnegative holds
x <= SUM F

proof end;

definition

end;

:: deftheorem MEASURE6:def 1 :
canceled;

theorem :: MEASURE6:4

for eps being ExtReal st 0. < eps holds
ex F being sequence of ExtREAL st
( ( for n being Nat holds 0. < F . n ) & SUM F < eps )

proof end;

theorem :: MEASURE6:5

for eps being ExtReal
for X being non empty Subset of ExtREAL st 0. < eps & inf X is Real holds
ex x being ExtReal st
( x in X & x < (inf X) + eps )

proof end;

theorem :: MEASURE6:6

for eps being ExtReal
for X being non empty Subset of ExtREAL st 0. < eps & sup X is Real holds
ex x being ExtReal st
( x in X & (sup X) - eps < x )

proof end;

theorem :: MEASURE6:7

for F being sequence of ExtREAL st F is nonnegative & SUM F < +infty holds
for n being Element of NAT holds F . n in REAL

proof end;

:: PROPERTIES OF THE INTERVALS

registration

cluster non empty complex-membered ext-real-membered real-membered interval for Element of bool REAL;

existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is interval )

proof end;

end;

theorem :: MEASURE6:8

for A being non empty Interval
for a being ExtReal st ex b being ExtReal st
( a <= b & A = ].a,b.[ ) holds
a = inf A by XXREAL_1:28, XXREAL_2:28;

theorem :: MEASURE6:9

for A being non empty Interval
for a being ExtReal st ex b being ExtReal st
( a <= b & A = ].a,b.] ) holds
a = inf A by XXREAL_1:26, XXREAL_2:27;

theorem :: MEASURE6:10

for A being non empty Interval
for a being ExtReal st ex b being ExtReal st
( a <= b & A = [.a,b.] ) holds
a = inf A by XXREAL_2:25;

theorem Th11: :: MEASURE6:11

for A being non empty Interval
for a being ExtReal st ex b being ExtReal st
( a <= b & A = [.a,b.[ ) holds
a = inf A

proof end;

theorem :: MEASURE6:12

for A being non empty Interval
for b being ExtReal st ex a being ExtReal st
( a <= b & A = ].a,b.[ ) holds
b = sup A by XXREAL_1:28, XXREAL_2:32;

theorem Th13: :: MEASURE6:13

for A being non empty Interval
for b being ExtReal st ex a being ExtReal st
( a <= b & A = ].a,b.] ) holds
b = sup A

proof end;

theorem :: MEASURE6:14

for A being non empty Interval
for b being ExtReal st ex a being ExtReal st
( a <= b & A = [.a,b.] ) holds
b = sup A by XXREAL_2:29;

theorem Th15: :: MEASURE6:15

for A being non empty Interval
for b being ExtReal st ex a being ExtReal st
( a <= b & A = [.a,b.[ ) holds
b = sup A

proof end;

theorem :: MEASURE6:16

for A being non empty Interval st A is open_interval holds
A = ].(inf A),(sup A).[

proof end;

theorem :: MEASURE6:17

for A being non empty Interval st A is closed_interval holds
A = [.(inf A),(sup A).]

proof end;

theorem :: MEASURE6:18

for A being non empty Interval st A is right_open_interval holds
A = [.(inf A),(sup A).[

proof end;

theorem :: MEASURE6:19

for A being non empty Interval st A is left_open_interval holds
A = ].(inf A),(sup A).]

proof end;

theorem :: MEASURE6:20

for A, B being non empty Interval
for a, b being Real st a in A & b in B & sup A <= inf B holds
a <= b

proof end;

theorem :: MEASURE6:21

for A, B being real-membered set
for y being Real holds
( y in B ++ A iff ex x, z being Real st
( x in B & z in A & y = x + z ) )

proof end;

theorem :: MEASURE6:22

for A, B being non empty Interval st sup A = inf B & ( sup A in A or inf B in B ) holds
A \/ B is Interval by XXREAL_2:def 5, XXREAL_2:def 6, XXREAL_2:90, XXREAL_2:91;

definition

let A be real-membered set ;
let x be Real;
:: original: ++
redefine func x ++ A -> Subset of REAL;
coherence
x ++ A is Subset of REAL by MEMBERED:3;

end;

Lm1: for A being real-membered set
for x, y being Real holds
( y in x ++ A iff ex z being Real st
( z in A & y = x + z ) )

proof end;

theorem Th23: :: MEASURE6:23

for A being Subset of REAL
for x being Real holds (- x) ++ (x ++ A) = A

proof end;

theorem :: MEASURE6:24

for x being Real
for A being Subset of REAL st A = REAL holds
x ++ A = A

proof end;

theorem :: MEASURE6:25

for x being Real holds x ++ {} = {} ;

theorem Th26: :: MEASURE6:26

for A being Interval
for x being Real holds
( A is open_interval iff x ++ A is open_interval )

proof end;

theorem Th27: :: MEASURE6:27

for A being Interval
for x being Real holds
( A is closed_interval iff x ++ A is closed_interval )

proof end;

theorem Th28: :: MEASURE6:28

for A being Interval
for x being Real holds
( A is right_open_interval iff x ++ A is right_open_interval )

proof end;

theorem Th29: :: MEASURE6:29

for A being Interval
for x being Real holds
( A is left_open_interval iff x ++ A is left_open_interval )

proof end;

theorem :: MEASURE6:30

for A being Interval
for x being Real holds x ++ A is Interval

proof end;

theorem Th31: :: MEASURE6:31

for A being real-membered set
for x being Real
for y being R_eal st x = y holds
sup (x ++ A) = y + (sup A)

proof end;

theorem Th32: :: MEASURE6:32

for A being real-membered set
for x being Real
for y being R_eal st x = y holds
inf (x ++ A) = y + (inf A)

proof end;

theorem :: MEASURE6:33

for A being Interval
for x being Real holds diameter A = diameter (x ++ A)

proof end;

notation

let X be set ;
synonym without_zero X for with_non-empty_elements ;
antonym with_zero X for with_non-empty_elements ;

end;

definition

let X be set ;

redefine attr X is with_non-empty_elements means :Def1: :: MEASURE6:def 2
not 0 in X;

compatibility
( X is without_zero iff not 0 in X ) by SETFAM_1:def 8;

end;

:: deftheorem Def1 defines without_zero MEASURE6:def 2 :
for X being set holds
( X is without_zero iff not 0 in X );

registration

cluster REAL -> with_zero ;

coherence
not REAL is without_zero by XREAL_0:def 1;

cluster omega -> with_zero ;

coherence
not NAT is without_zero ;

end;

registration

cluster non empty without_zero for set ;

existence
ex b₁ being set st
( not b₁ is empty & b₁ is without_zero )

proof end;

cluster non empty with_zero for set ;

existence
ex b₁ being set st
( not b₁ is empty & b₁ is with_zero ) by Def1;

end;

registration

cluster non empty complex-membered ext-real-membered real-membered without_zero for Element of bool REAL;

existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is without_zero )

proof end;

cluster non empty complex-membered ext-real-membered real-membered with_zero for Element of bool REAL;

existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is with_zero )

proof end;

end;

theorem Th34: :: MEASURE6:34

for F being set st not F is empty & F is with_non-empty_elements & F is c=-linear holds
F is centered

proof end;

registration

let F be set ;

cluster c=-linear non empty with_non-empty_elements -> centered for Element of bool (bool F);

coherence
for b₁ being Subset-Family of F st not b₁ is empty & b₁ is with_non-empty_elements & b₁ is c=-linear holds
b₁ is centered by Th34;

end;

registration

let X, Y be non empty set ;
let f be Function of X,Y;

cluster f .: X -> non empty ;

coherence
not f .: X is empty

proof end;

end;

definition

let X, Y be set ;
let f be Function of X,Y;

func " f -> Function of (bool Y),(bool X) means :Def2: :: MEASURE6:def 3
for y being Subset of Y holds it . y = f " y;

existence
ex b₁ being Function of (bool Y),(bool X) st
for y being Subset of Y holds b₁ . y = f " y

proof end;

uniqueness
for b₁, b₂ being Function of (bool Y),(bool X) st ( for y being Subset of Y holds b₁ . y = f " y ) & ( for y being Subset of Y holds b₂ . y = f " y ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines " MEASURE6:def 3 :
for X, Y being set
for f being Function of X,Y
for b₄ being Function of (bool Y),(bool X) holds
( b₄ = " f iff for y being Subset of Y holds b₄ . y = f " y );

theorem :: MEASURE6:35

for X, Y, x being set
for S being Subset-Family of Y
for f being Function of X,Y st x in meet ((" f) .: S) holds
f . x in meet S

proof end;

theorem Th36: :: MEASURE6:36

for r, s being Real st |.r.| + |.s.| = 0 holds
r = 0

proof end;

theorem Th37: :: MEASURE6:37

for r, s, t being Real st r < s & s < t holds
|.s.| < |.r.| + |.t.|

proof end;

theorem Th38: :: MEASURE6:38

for seq being Real_Sequence st seq is convergent & seq is non-zero & lim seq = 0 holds
not seq " is bounded

proof end;

theorem Th39: :: MEASURE6:39

for seq being Real_Sequence holds
( rng seq is real-bounded iff seq is bounded )

proof end;

notation

let X be real-membered set ;
synonym with_max X for right_end ;
synonym with_min X for left_end ;

end;

definition

let X be real-membered set ;

redefine attr X is right_end means :: MEASURE6:def 4
( X is bounded_above & upper_bound X in X );

compatibility
( X is with_max iff ( X is bounded_above & upper_bound X in X ) )

proof end;

redefine attr X is left_end means :: MEASURE6:def 5
( X is bounded_below & lower_bound X in X );

compatibility
( X is with_min iff ( X is bounded_below & lower_bound X in X ) )

proof end;

end;

:: deftheorem defines with_max MEASURE6:def 4 :
for X being real-membered set holds
( X is with_max iff ( X is bounded_above & upper_bound X in X ) );

:: deftheorem defines with_min MEASURE6:def 5 :
for X being real-membered set holds
( X is with_min iff ( X is bounded_below & lower_bound X in X ) );

registration

cluster non empty complex-membered ext-real-membered real-membered real-bounded closed for Element of bool REAL;

existence
ex b₁ being Subset of REAL st
( not b₁ is empty & b₁ is closed & b₁ is real-bounded )

proof end;

end;

definition

let R be Subset-Family of REAL;

attr R is open means :: MEASURE6:def 6
for X being Subset of REAL st X in R holds
X is open ;

attr R is closed means :: MEASURE6:def 7
for X being Subset of REAL st X in R holds
X is closed ;

end;

:: deftheorem defines open MEASURE6:def 6 :
for R being Subset-Family of REAL holds
( R is open iff for X being Subset of REAL st X in R holds
X is open );

:: deftheorem defines closed MEASURE6:def 7 :
for R being Subset-Family of REAL holds
( R is closed iff for X being Subset of REAL st X in R holds
X is closed );

definition

let X be Subset of REAL;
:: original: --
redefine func -- X -> Subset of REAL;
coherence
-- X is Subset of REAL by MEMBERED:3;

end;

theorem :: MEASURE6:40

for r being Real
for X being Subset of REAL holds
( r in X iff - r in -- X ) by MEMBER_1:11;

Lm2: for X being Subset of REAL st X is bounded_above holds
-- X is bounded_below

proof end;

Lm3: for X being Subset of REAL st X is bounded_below holds
-- X is bounded_above

proof end;

theorem Th41: :: MEASURE6:41

for X being Subset of REAL holds
( X is bounded_above iff -- X is bounded_below )

proof end;

theorem :: MEASURE6:42

for X being Subset of REAL holds
( X is bounded_below iff -- X is bounded_above )

proof end;

theorem Th43: :: MEASURE6:43

for X being non empty Subset of REAL st X is bounded_below holds
lower_bound X = - (upper_bound (-- X))

proof end;

theorem Th44: :: MEASURE6:44

for X being non empty Subset of REAL st X is bounded_above holds
upper_bound X = - (lower_bound (-- X))

proof end;

Lm4: for X being Subset of REAL st X is closed holds
-- X is closed

proof end;

theorem :: MEASURE6:45

for X being Subset of REAL holds
( X is closed iff -- X is closed )

proof end;

Lm5: for X being Subset of REAL
for p being Real holds p ++ X = { (p + r3) where r3 is Real : r3 in X }

proof end;

theorem Th46: :: MEASURE6:46

for r being Real
for X being Subset of REAL
for q3 being Real holds
( r in X iff q3 + r in q3 ++ X )

proof end;

theorem :: MEASURE6:47

for X being Subset of REAL holds X = 0 ++ X by MEMBER_1:146;

theorem :: MEASURE6:48

for X being Subset of REAL
for q3, p3 being Real holds q3 ++ (p3 ++ X) = (q3 + p3) ++ X by MEMBER_1:147;

Lm6: for X being Subset of REAL
for s being Real st X is bounded_above holds
s ++ X is bounded_above

proof end;

theorem :: MEASURE6:49

for X being Subset of REAL
for q3 being Real holds
( X is bounded_above iff q3 ++ X is bounded_above )

proof end;

Lm7: for X being Subset of REAL
for p being Real st X is bounded_below holds
p ++ X is bounded_below

proof end;

theorem :: MEASURE6:50

for X being Subset of REAL
for q3 being Real holds
( X is bounded_below iff q3 ++ X is bounded_below )

proof end;

theorem :: MEASURE6:51

for q3 being Real
for X being non empty Subset of REAL st X is bounded_below holds
lower_bound (q3 ++ X) = q3 + (lower_bound X)

proof end;

theorem :: MEASURE6:52

for q3 being Real
for X being non empty Subset of REAL st X is bounded_above holds
upper_bound (q3 ++ X) = q3 + (upper_bound X)

proof end;

Lm8: for q3 being Real
for X being Subset of REAL st X is closed holds
q3 ++ X is closed

proof end;

theorem :: MEASURE6:53

for X being Subset of REAL
for q3 being Real holds
( X is closed iff q3 ++ X is closed )

proof end;

definition

let X be Subset of REAL;

func Inv X -> Subset of REAL equals :: MEASURE6:def 8
{ (1 / r3) where r3 is Real : r3 in X } ;

coherence
{ (1 / r3) where r3 is Real : r3 in X } is Subset of REAL

proof end;

involutiveness
for b₁, b₂ being Subset of REAL st b₁ = { (1 / r3) where r3 is Real : r3 in b₂ } holds
b₂ = { (1 / r3) where r3 is Real : r3 in b₁ }

proof end;

end;

:: deftheorem defines Inv MEASURE6:def 8 :
for X being Subset of REAL holds Inv X = { (1 / r3) where r3 is Real : r3 in X } ;

theorem Th54: :: MEASURE6:54

for r being Real
for X being Subset of REAL holds
( r in X iff 1 / r in Inv X )

proof end;

registration

let X be non empty Subset of REAL;

cluster Inv X -> non empty ;

coherence
not Inv X is empty

proof end;

end;

registration

let X be without_zero Subset of REAL;

cluster Inv X -> without_zero ;

coherence
Inv X is without_zero

proof end;

end;

theorem :: MEASURE6:55

for X being without_zero Subset of REAL st X is closed & X is real-bounded holds
Inv X is closed

proof end;

theorem Th56: :: MEASURE6:56

for Z being Subset-Family of REAL st Z is closed holds
meet Z is closed

proof end;

definition

let X be real-membered set ;

func Cl X -> Subset of REAL equals :: MEASURE6:def 9
meet { A where A is Subset of REAL : ( X c= A & A is closed ) } ;

coherence
meet { A where A is Subset of REAL : ( X c= A & A is closed ) } is Subset of REAL

proof end;

projectivity
for b₁ being Subset of REAL
for b₂ being real-membered set st b₁ = meet { A where A is Subset of REAL : ( b₂ c= A & A is closed ) } holds
b₁ = meet { A where A is Subset of REAL : ( b₁ c= A & A is closed ) }

proof end;

end;

:: deftheorem defines Cl MEASURE6:def 9 :
for X being real-membered set holds Cl X = meet { A where A is Subset of REAL : ( X c= A & A is closed ) } ;

registration

let X be real-membered set ;

cluster Cl X -> closed ;

coherence
Cl X is closed

proof end;

end;

theorem Th57: :: MEASURE6:57

for X being Subset of REAL
for Y being closed Subset of REAL st X c= Y holds
Cl X c= Y

proof end;

theorem Th58: :: MEASURE6:58

for X being real-membered set holds X c= Cl X

proof end;

theorem Th59: :: MEASURE6:59

for X being Subset of REAL holds
( X is closed iff X = Cl X )

proof end;

theorem :: MEASURE6:60

Cl ({} REAL) = {} by Th59;

theorem :: MEASURE6:61

Cl ([#] REAL) = REAL by Th59;

theorem :: MEASURE6:62

for X, Y being real-membered set st X c= Y holds
Cl X c= Cl Y

proof end;

theorem Th63: :: MEASURE6:63

for X being Subset of REAL
for r3 being Real holds
( r3 in Cl X iff for O being open Subset of REAL st r3 in O holds
not O /\ X is empty )

proof end;

theorem :: MEASURE6:64

for X being Subset of REAL
for r3 being Real st r3 in Cl X holds
ex seq being Real_Sequence st
( rng seq c= X & seq is convergent & lim seq = r3 )

proof end;

definition

let X be set ;
let f be Function of X,REAL;

:: original: bounded_below
redefine attr f is bounded_below means :: MEASURE6:def 10
f .: X is bounded_below ;

compatibility
( f is bounded_below iff f .: X is bounded_below )

proof end;

:: original: bounded_above
redefine attr f is bounded_above means :: MEASURE6:def 11
f .: X is bounded_above ;

compatibility
( f is bounded_above iff f .: X is bounded_above )

proof end;

end;

:: deftheorem defines bounded_below MEASURE6:def 10 :
for X being set
for f being Function of X,REAL holds
( f is bounded_below iff f .: X is bounded_below );

:: deftheorem defines bounded_above MEASURE6:def 11 :
for X being set
for f being Function of X,REAL holds
( f is bounded_above iff f .: X is bounded_above );

definition

let X be set ;
let f be Function of X,REAL;

attr f is with_max means :: MEASURE6:def 12
f .: X is with_max ;

attr f is with_min means :: MEASURE6:def 13
f .: X is with_min ;

end;

:: deftheorem defines with_max MEASURE6:def 12 :
for X being set
for f being Function of X,REAL holds
( f is with_max iff f .: X is with_max );

:: deftheorem defines with_min MEASURE6:def 13 :
for X being set
for f being Function of X,REAL holds
( f is with_min iff f .: X is with_min );

theorem Th65: :: MEASURE6:65

for X, A being set
for f being Function of X,REAL holds (- f) .: A = -- (f .: A)

proof end;

Lm9: for X being non empty set
for f being Function of X,REAL st f is with_max holds
- f is with_min

proof end;

Lm10: for X being non empty set
for f being Function of X,REAL st f is with_min holds
- f is with_max

proof end;

theorem Th66: :: MEASURE6:66

for X being non empty set
for f being Function of X,REAL holds
( f is with_min iff - f is with_max )

proof end;

theorem :: MEASURE6:67

for X being non empty set
for f being Function of X,REAL holds
( f is with_max iff - f is with_min )

proof end;

theorem :: MEASURE6:68

for X being set
for A being Subset of REAL
for f being Function of X,REAL holds (- f) " A = f " (-- A)

proof end;

theorem :: MEASURE6:69

for X, A being set
for f being Function of X,REAL
for s being Real holds (s + f) .: A = s ++ (f .: A)

proof end;

theorem :: MEASURE6:70

for X being set
for A being Subset of REAL
for f being Function of X,REAL
for q3 being Real holds (q3 + f) " A = f " ((- q3) ++ A)

proof end;

notation

let f be real-valued Function;
synonym Inv f for f " ;

end;

definition

let C be set ;
let D be real-membered set ;
let f be PartFunc of C,D;
:: original: Inv
redefine func Inv f -> PartFunc of C,REAL;
coherence
Inv f is PartFunc of C,REAL

proof end;

end;

theorem :: MEASURE6:71

for X being set
for A being without_zero Subset of REAL
for f being Function of X,REAL holds (Inv f) " A = f " (Inv A)

proof end;

theorem :: MEASURE6:72

for A being Subset of REAL
for x being Real st x in -- A holds
ex a being Real st
( a in A & x = - a )

proof end;