:: Real Functions Spaces
:: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski
::
:: Received March 23, 1990
:: Copyright (c) 1990 Association of Mizar Users

begin

definition

let A be set ;
let B be non empty set ;
let F be BinOp of (Funcs (A,B));
let f, g be Element of Funcs (A,B);
:: original: .
redefine func F . (f,g) -> Element of Funcs (A,B);
coherence
F . (f,g) is Element of Funcs (A,B)

proof end;

end;

definition

let A, B, C, D be non empty set ;
let F be Function of [:C,D:],(Funcs (A,B));
let cd be Element of [:C,D:];
:: original: .
redefine func F . cd -> Element of Funcs (A,B);
coherence
F . cd is Element of Funcs (A,B)

proof end;

end;

definition

let X be non empty set ;
let Z be set ;
let F be BinOp of X;
let f, g be Function of Z,X;
canceled;
:: original: .:
redefine func F .: (f,g) -> Element of Funcs (Z,X);
coherence
F .: (f,g) is Element of Funcs (Z,X)

proof end;

end;

:: deftheorem FUNCSDOM:def 1 :
canceled;

definition

let X be non empty set ;
let Z be set ;
let F be BinOp of X;
let a be Element of X;
let f be Function of Z,X;
:: original: [;]
redefine func F [;] (a,f) -> Element of Funcs (Z,X);
coherence
F [;] (a,f) is Element of Funcs (Z,X)

proof end;

end;

definition

let A be set ;
func RealFuncAdd A -> BinOp of (Funcs (A,REAL)) means :Def2: :: FUNCSDOM:def 2
for f, g being Element of Funcs (A,REAL) holds it . (f,g) = addreal .: (f,g);
existence
ex b₁ being BinOp of (Funcs (A,REAL)) st
for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = addreal .: (f,g)

proof end;

uniqueness
for b₁, b₂ being BinOp of (Funcs (A,REAL)) st ( for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = addreal .: (f,g) ) & ( for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = addreal .: (f,g) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines RealFuncAdd FUNCSDOM:def 2 :
for A being set
for b₂ being BinOp of (Funcs (A,REAL)) holds
( b₂ = RealFuncAdd A iff for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = addreal .: (f,g) );

definition

let A be set ;
func RealFuncMult A -> BinOp of (Funcs (A,REAL)) means :Def3: :: FUNCSDOM:def 3
for f, g being Element of Funcs (A,REAL) holds it . (f,g) = multreal .: (f,g);
existence
ex b₁ being BinOp of (Funcs (A,REAL)) st
for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = multreal .: (f,g)

proof end;

uniqueness
for b₁, b₂ being BinOp of (Funcs (A,REAL)) st ( for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = multreal .: (f,g) ) & ( for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = multreal .: (f,g) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines RealFuncMult FUNCSDOM:def 3 :
for A being set
for b₂ being BinOp of (Funcs (A,REAL)) holds
( b₂ = RealFuncMult A iff for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = multreal .: (f,g) );

definition

let A be set ;
func RealFuncExtMult A -> Function of [:REAL,(Funcs (A,REAL)):],(Funcs (A,REAL)) means :Def4: :: FUNCSDOM:def 4
for a being Real
for f being Element of Funcs (A,REAL) holds it . (a,f) = multreal [;] (a,f);
existence
ex b₁ being Function of [:REAL,(Funcs (A,REAL)):],(Funcs (A,REAL)) st
for a being Real
for f being Element of Funcs (A,REAL) holds b₁ . (a,f) = multreal [;] (a,f)

proof end;

uniqueness
for b₁, b₂ being Function of [:REAL,(Funcs (A,REAL)):],(Funcs (A,REAL)) st ( for a being Real
for f being Element of Funcs (A,REAL) holds b₁ . (a,f) = multreal [;] (a,f) ) & ( for a being Real
for f being Element of Funcs (A,REAL) holds b₂ . (a,f) = multreal [;] (a,f) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines RealFuncExtMult FUNCSDOM:def 4 :
for A being set
for b₂ being Function of [:REAL,(Funcs (A,REAL)):],(Funcs (A,REAL)) holds
( b₂ = RealFuncExtMult A iff for a being Real
for f being Element of Funcs (A,REAL) holds b₂ . (a,f) = multreal [;] (a,f) );

definition

let A be set ;
func RealFuncZero A -> Element of Funcs (A,REAL) equals :: FUNCSDOM:def 5
A --> 0;
coherence
A --> 0 is Element of Funcs (A,REAL)

proof end;

end;

:: deftheorem defines RealFuncZero FUNCSDOM:def 5 :
for A being set holds RealFuncZero A = A --> 0;

definition

let A be set ;
func RealFuncUnit A -> Element of Funcs (A,REAL) equals :: FUNCSDOM:def 6
A --> 1;
coherence
A --> 1 is Element of Funcs (A,REAL)

proof end;

end;

:: deftheorem defines RealFuncUnit FUNCSDOM:def 6 :
for A being set holds RealFuncUnit A = A --> 1;

Lm1: for A, B being non empty set
for x being Element of A
for f being Function of A,B holds x in dom f

proof end;

theorem :: FUNCSDOM:1

canceled;

theorem :: FUNCSDOM:2

canceled;

theorem :: FUNCSDOM:3

canceled;

theorem :: FUNCSDOM:4

canceled;

theorem :: FUNCSDOM:5

canceled;

theorem :: FUNCSDOM:6

canceled;

theorem :: FUNCSDOM:7

canceled;

theorem :: FUNCSDOM:8

canceled;

theorem :: FUNCSDOM:9

canceled;

theorem Th10: :: FUNCSDOM:10

for A being non empty set
for h, f, g being Element of Funcs (A,REAL) holds
( h = (RealFuncAdd A) . (f,g) iff for x being Element of A holds h . x = (f . x) + (g . x) )

proof end;

theorem Th11: :: FUNCSDOM:11

for A being non empty set
for h, f, g being Element of Funcs (A,REAL) holds
( h = (RealFuncMult A) . (f,g) iff for x being Element of A holds h . x = (f . x) * (g . x) )

proof end;

theorem :: FUNCSDOM:12

canceled;

theorem :: FUNCSDOM:13

canceled;

theorem :: FUNCSDOM:14

for A being non empty set holds RealFuncZero A <> RealFuncUnit A

proof end;

theorem Th15: :: FUNCSDOM:15

for A being non empty set
for h, f being Element of Funcs (A,REAL)
for a being Real holds
( h = (RealFuncExtMult A) . [a,f] iff for x being Element of A holds h . x = a * (f . x) )

proof end;

theorem Th16: :: FUNCSDOM:16

for A being set
for f, g being Element of Funcs (A,REAL) holds (RealFuncAdd A) . (f,g) = (RealFuncAdd A) . (g,f)

proof end;

theorem Th17: :: FUNCSDOM:17

for A being set
for f, g, h being Element of Funcs (A,REAL) holds (RealFuncAdd A) . (f,((RealFuncAdd A) . (g,h))) = (RealFuncAdd A) . (((RealFuncAdd A) . (f,g)),h)

proof end;

theorem Th18: :: FUNCSDOM:18

for A being set
for f, g being Element of Funcs (A,REAL) holds (RealFuncMult A) . (f,g) = (RealFuncMult A) . (g,f)

proof end;

theorem Th19: :: FUNCSDOM:19

for A being set
for f, g, h being Element of Funcs (A,REAL) holds (RealFuncMult A) . (f,((RealFuncMult A) . (g,h))) = (RealFuncMult A) . (((RealFuncMult A) . (f,g)),h)

proof end;

theorem Th20: :: FUNCSDOM:20

for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncMult A) . ((RealFuncUnit A),f) = f

proof end;

theorem Th21: :: FUNCSDOM:21

for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncAdd A) . ((RealFuncZero A),f) = f

proof end;

theorem Th22: :: FUNCSDOM:22

for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncAdd A) . (f,((RealFuncExtMult A) . [(- 1),f])) = RealFuncZero A

proof end;

theorem Th23: :: FUNCSDOM:23

for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncExtMult A) . (1,f) = f

proof end;

theorem Th24: :: FUNCSDOM:24

for a, b being Real
for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)

proof end;

theorem Th25: :: FUNCSDOM:25

for a, b being Real
for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncAdd A) . (((RealFuncExtMult A) . (a,f)),((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a + b),f)

proof end;

Lm2: for a being Real
for A being set
for f, g being Element of Funcs (A,REAL) holds (RealFuncAdd A) . (((RealFuncExtMult A) . (a,f)),((RealFuncExtMult A) . (a,g))) = (RealFuncExtMult A) . (a,((RealFuncAdd A) . (f,g)))

proof end;

theorem Th26: :: FUNCSDOM:26

for A being set
for f, g, h being Element of Funcs (A,REAL) holds (RealFuncMult A) . (f,((RealFuncAdd A) . (g,h))) = (RealFuncAdd A) . (((RealFuncMult A) . (f,g)),((RealFuncMult A) . (f,h)))

proof end;

theorem Th27: :: FUNCSDOM:27

for A being set
for f, g, h being Element of Funcs (A,REAL)
for a being Real holds (RealFuncMult A) . (((RealFuncExtMult A) . (a,f)),g) = (RealFuncExtMult A) . (a,((RealFuncMult A) . (f,g)))

proof end;

theorem Th28: :: FUNCSDOM:28

for x1 being set
for A being non empty set ex f, g being Element of Funcs (A,REAL) st
( ( for z being set st z in A holds
( ( z = x1 implies f . z = 1 ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1 ) ) ) )

proof end;

theorem Th29: :: FUNCSDOM:29

for x1, x2 being set
for A being non empty set
for f, g being Element of Funcs (A,REAL) st x1 in A & x2 in A & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1 ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1 ) ) ) holds
for a, b being Real st (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) = RealFuncZero A holds
( a = 0 & b = 0 )

proof end;

theorem :: FUNCSDOM:30

for x1, x2 being set
for A being non empty set st x1 in A & x2 in A & x1 <> x2 holds
ex f, g being Element of Funcs (A,REAL) st
for a, b being Real st (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) = RealFuncZero A holds
( a = 0 & b = 0 )

proof end;

theorem Th31: :: FUNCSDOM:31

for x1, x2 being set
for A being non empty set
for f, g being Element of Funcs (A,REAL) st A = {x1,x2} & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1 ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1 ) ) ) holds
for h being Element of Funcs (A,REAL) ex a, b being Real st h = (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))

proof end;

theorem :: FUNCSDOM:32

for x1, x2 being set
for A being non empty set st A = {x1,x2} & x1 <> x2 holds
ex f, g being Element of Funcs (A,REAL) st
for h being Element of Funcs (A,REAL) ex a, b being Real st h = (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))

proof end;

theorem Th33: :: FUNCSDOM:33

for x1, x2 being set
for A being non empty set st A = {x1,x2} & x1 <> x2 holds
ex f, g being Element of Funcs (A,REAL) st
( ( for a, b being Real st (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs (A,REAL) ex a, b being Real st h = (RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) ) )

proof end;

definition

let A be set ;
func RealVectSpace A -> strict RealLinearSpace equals :: FUNCSDOM:def 7
RLSStruct(# (Funcs (A,REAL)),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #);
coherence
RLSStruct(# (Funcs (A,REAL)),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #) is strict RealLinearSpace

proof end;

end;

:: deftheorem defines RealVectSpace FUNCSDOM:def 7 :
for A being set holds RealVectSpace A = RLSStruct(# (Funcs (A,REAL)),(RealFuncZero A),(RealFuncAdd A),(RealFuncExtMult A) #);

theorem :: FUNCSDOM:34

canceled;

theorem :: FUNCSDOM:35

canceled;

theorem :: FUNCSDOM:36

canceled;

theorem :: FUNCSDOM:37

ex V being strict RealLinearSpace ex u, v being Element of V st
( ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being Element of V ex a, b being Real st w = (a * u) + (b * v) ) )

proof end;

definition

let A be non empty set ;
canceled;
canceled;
canceled;
canceled;
func RRing A -> strict doubleLoopStr equals :: FUNCSDOM:def 12
doubleLoopStr(# (Funcs (A,REAL)),(RealFuncAdd A),(RealFuncMult A),(RealFuncUnit A),(RealFuncZero A) #);
correctness
coherence
doubleLoopStr(# (Funcs (A,REAL)),(RealFuncAdd A),(RealFuncMult A),(RealFuncUnit A),(RealFuncZero A) #) is strict doubleLoopStr ;
;

end;

:: deftheorem FUNCSDOM:def 8 :
canceled;

:: deftheorem FUNCSDOM:def 9 :
canceled;

:: deftheorem FUNCSDOM:def 10 :
canceled;

:: deftheorem FUNCSDOM:def 11 :
canceled;

:: deftheorem defines RRing FUNCSDOM:def 12 :
for A being non empty set holds RRing A = doubleLoopStr(# (Funcs (A,REAL)),(RealFuncAdd A),(RealFuncMult A),(RealFuncUnit A),(RealFuncZero A) #);

registration

let A be non empty set ;
cluster RRing A -> non empty strict ;
coherence
not RRing A is empty ;

end;

Lm3: now

let A be non empty set ; :: thesis: for h, a being Element of (RRing A) st a = RealFuncUnit A holds
( h * a = h & a * h = h )
set FR = RRing A;
let h, a be Element of (RRing A); :: thesis: ( a = RealFuncUnit A implies ( h * a = h & a * h = h ) )
reconsider g = h as Element of Funcs (A,REAL) ;
assume A1: a = RealFuncUnit A ; :: thesis: ( h * a = h & a * h = h )
hence h * a = (RealFuncMult A) . ((RealFuncUnit A),g) by Th18
.= h by Th20 ;
:: thesis: a * h = h
thus a * h = h by A1, Th20; :: thesis: verum

end;

registration

let A be non empty set ;
cluster RRing A -> strict unital ;
coherence
RRing A is unital

proof end;

end;

theorem :: FUNCSDOM:38

canceled;

theorem :: FUNCSDOM:39

canceled;

theorem :: FUNCSDOM:40

canceled;

theorem :: FUNCSDOM:41

for A being non empty set holds 1. (RRing A) = RealFuncUnit A ;

theorem Th42: :: FUNCSDOM:42

for A being non empty set
for x, y, z being Element of (RRing A) holds
( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. (RRing A)) = x & ex t being Element of (RRing A) st x + t = 0. (RRing A) & x * y = y * x & (x * y) * z = x * (y * z) & x * (1. (RRing A)) = x & (1. (RRing A)) * x = x & x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) )

proof end;

registration

cluster non empty right_complementable strict associative commutative right-distributive right_unital Abelian add-associative right_zeroed doubleLoopStr ;
existence
ex b₁ being non empty doubleLoopStr st
( b₁ is strict & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is associative & b₁ is commutative & b₁ is right_unital & b₁ is right-distributive )

proof end;

end;

definition

mode Ring is non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ;

end;

theorem :: FUNCSDOM:43

for A being non empty set holds RRing A is commutative Ring

proof end;

definition

attr c₁ is strict ;
struct AlgebraStr -> doubleLoopStr , RLSStruct ;
aggr AlgebraStr(# carrier, multF, addF, Mult, OneF, ZeroF #) -> AlgebraStr ;

end;

registration

cluster non empty AlgebraStr ;
existence
not for b₁ being AlgebraStr holds b₁ is empty

proof end;

end;

definition

let A be set ;
canceled;
canceled;
canceled;
canceled;
canceled;
canceled;
func RAlgebra A -> strict AlgebraStr equals :: FUNCSDOM:def 19
AlgebraStr(# (Funcs (A,REAL)),(RealFuncMult A),(RealFuncAdd A),(RealFuncExtMult A),(RealFuncUnit A),(RealFuncZero A) #);
correctness
coherence
AlgebraStr(# (Funcs (A,REAL)),(RealFuncMult A),(RealFuncAdd A),(RealFuncExtMult A),(RealFuncUnit A),(RealFuncZero A) #) is strict AlgebraStr ;
;

end;

:: deftheorem FUNCSDOM:def 13 :
canceled;

:: deftheorem FUNCSDOM:def 14 :
canceled;

:: deftheorem FUNCSDOM:def 15 :
canceled;

:: deftheorem FUNCSDOM:def 16 :
canceled;

:: deftheorem FUNCSDOM:def 17 :
canceled;

:: deftheorem FUNCSDOM:def 18 :
canceled;

:: deftheorem defines RAlgebra FUNCSDOM:def 19 :
for A being set holds RAlgebra A = AlgebraStr(# (Funcs (A,REAL)),(RealFuncMult A),(RealFuncAdd A),(RealFuncExtMult A),(RealFuncUnit A),(RealFuncZero A) #);

registration

let A be non empty set ;
cluster RAlgebra A -> non empty strict ;
coherence
not RAlgebra A is empty

proof end;

end;

Lm4: now

let A be non empty set ; :: thesis: for x, e being Element of (RAlgebra A) st e = RealFuncUnit A holds
( x * e = x & e * x = x )
set F = RAlgebra A;
let x, e be Element of (RAlgebra A); :: thesis: ( e = RealFuncUnit A implies ( x * e = x & e * x = x ) )
reconsider f = x as Element of Funcs (A,REAL) ;
assume A1: e = RealFuncUnit A ; :: thesis: ( x * e = x & e * x = x )
hence x * e = (RealFuncMult A) . ((RealFuncUnit A),f) by Th18
.= x by Th20 ;
:: thesis: e * x = x
thus e * x = x by A1, Th20; :: thesis: verum

end;

registration

let A be non empty set ;
cluster RAlgebra A -> unital strict ;
coherence
RAlgebra A is unital

proof end;

end;

theorem :: FUNCSDOM:44

for A being non empty set holds 1. (RAlgebra A) = RealFuncUnit A ;

definition

let IT be non empty AlgebraStr ;
attr IT is vector-associative means :: FUNCSDOM:def 20
for x, y being Element of IT
for a being Real holds a * (x * y) = (a * x) * y;

end;

:: deftheorem defines vector-associative FUNCSDOM:def 20 :
for IT being non empty AlgebraStr holds
( IT is vector-associative iff for x, y being Element of IT
for a being Real holds a * (x * y) = (a * x) * y );

registration

let A be set ;
cluster RAlgebra A -> non empty strict ;
coherence
not RAlgebra A is empty

proof end;

end;

registration

let A be set ;
cluster RAlgebra A -> right_complementable associative commutative right-distributive right_unital Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative strict vector-associative ;
coherence
( RAlgebra A is strict & RAlgebra A is Abelian & RAlgebra A is add-associative & RAlgebra A is right_zeroed & RAlgebra A is right_complementable & RAlgebra A is commutative & RAlgebra A is associative & RAlgebra A is right_unital & RAlgebra A is right-distributive & RAlgebra A is vector-associative & RAlgebra A is scalar-associative & RAlgebra A is vector-distributive & RAlgebra A is scalar-distributive )

proof end;

end;

registration

cluster non empty right_complementable associative commutative right-distributive right_unital Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative strict vector-associative AlgebraStr ;
existence
ex b₁ being non empty AlgebraStr st
( b₁ is strict & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is commutative & b₁ is associative & b₁ is right_unital & b₁ is right-distributive & b₁ is vector-associative & b₁ is scalar-associative & b₁ is vector-distributive & b₁ is scalar-distributive )

proof end;

end;

definition

mode Algebra is non empty right_complementable associative commutative right-distributive right_unital Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative vector-associative AlgebraStr ;

end;