:: Binary Operations
:: by Czes{\l}aw Byli\'nski
::
:: Received April 14, 1989
:: Copyright (c) 1990 Association of Mizar Users

begin

definition

let f be Function;
let a, b be set ;
func f . a,b -> set equals :: BINOP_1:def 1
f . [a,b];
correctness
coherence
f . [a,b] is set ;
;

end;

:: deftheorem defines . BINOP_1:def 1 :

for f being Function
for a, b being set holds f . a,b = f . [a,b];

definition

let A, B be non empty set ;
let C be set ;
let f be Function of [:A,B:],C;
let a be Element of A;
let b be Element of B;
:: original: .
redefine func f . a,b -> Element of C;
coherence
f . a,b is Element of C

proof end;

end;

theorem Th1: :: BINOP_1:1

for X, Y, Z being set
for f1, f2 being Function of [:X,Y:],Z st ( for x, y being set st x in X & y in Y holds
f1 . x,y = f2 . x,y ) holds
f1 = f2

proof end;

theorem :: BINOP_1:2

for X, Y, Z being set
for f1, f2 being Function of [:X,Y:],Z st ( for a being Element of X
for b being Element of Y holds f1 . a,b = f2 . a,b ) holds
f1 = f2

proof end;

definition

let A be set ;
mode UnOp of A is Function of A,A;
mode BinOp of A is Function of [:A,A:],A;

end;

scheme :: BINOP_1:sch 1
FuncEx2{ F₁() -> set , F₂() -> set , F₃() -> set , P₁[ set , set , set ] } :

ex f being Function of [:F₁(),F₂():],F₃() st
for x, y being set st x in F₁() & y in F₂() holds
P₁[x,y,f . x,y]

provided

A1: for x, y being set st x in F₁() & y in F₂() holds
ex z being set st
( z in F₃() & P₁[x,y,z] )

proof end;

scheme :: BINOP_1:sch 2
Lambda2{ F₁() -> set , F₂() -> set , F₃() -> set , F₄( set , set ) -> set } :

ex f being Function of [:F₁(),F₂():],F₃() st
for x, y being set st x in F₁() & y in F₂() holds
f . x,y = F₄(x,y)

provided

A1: for x, y being set st x in F₁() & y in F₂() holds
F₄(x,y) in F₃()

proof end;

scheme :: BINOP_1:sch 3
FuncEx2D{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> non empty set , P₁[ set , set , set ] } :

ex f being Function of [:F₁(),F₂():],F₃() st
for x being Element of F₁()
for y being Element of F₂() holds P₁[x,y,f . x,y]

provided

A1: for x being Element of F₁()
for y being Element of F₂() ex z being Element of F₃() st P₁[x,y,z]

proof end;

scheme :: BINOP_1:sch 4
Lambda2D{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> non empty set , F₄( Element of F₁(), Element of F₂()) -> Element of F₃() } :

ex f being Function of [:F₁(),F₂():],F₃() st
for x being Element of F₁()
for y being Element of F₂() holds f . x,y = F₄(x,y)

proof end;

definition

let A be set ;
let o be BinOp of A;
attr o is commutative means :Def2: :: BINOP_1:def 2
for a, b being Element of A holds o . a,b = o . b,a;
attr o is associative means :Def3: :: BINOP_1:def 3
for a, b, c being Element of A holds o . a,(o . b,c) = o . (o . a,b),c;
attr o is idempotent means :Def4: :: BINOP_1:def 4
for a being Element of A holds o . a,a = a;

end;

:: deftheorem Def2 defines commutative BINOP_1:def 2 :

for A being set
for o being BinOp of A holds
( o is commutative iff for a, b being Element of A holds o . a,b = o . b,a );

:: deftheorem Def3 defines associative BINOP_1:def 3 :

for A being set
for o being BinOp of A holds
( o is associative iff for a, b, c being Element of A holds o . a,(o . b,c) = o . (o . a,b),c );

:: deftheorem Def4 defines idempotent BINOP_1:def 4 :

for A being set
for o being BinOp of A holds
( o is idempotent iff for a being Element of A holds o . a,a = a );

registration

cluster Function-like V15([:{} ,{} :], {} ) -> empty commutative associative Element of bool [:[:{} ,{} :],{} :];
coherence
for b₁ being BinOp of {} holds
( b₁ is empty & b₁ is associative & b₁ is commutative )

proof end;

end;

definition

let A be set ;
let e be Element of A;
let o be BinOp of A;
pred e is_a_left_unity_wrt o means :Def5: :: BINOP_1:def 5
for a being Element of A holds o . e,a = a;
pred e is_a_right_unity_wrt o means :Def6: :: BINOP_1:def 6
for a being Element of A holds o . a,e = a;

end;

:: deftheorem Def5 defines is_a_left_unity_wrt BINOP_1:def 5 :

for A being set
for e being Element of A
for o being BinOp of A holds
( e is_a_left_unity_wrt o iff for a being Element of A holds o . e,a = a );

:: deftheorem Def6 defines is_a_right_unity_wrt BINOP_1:def 6 :

for A being set
for e being Element of A
for o being BinOp of A holds
( e is_a_right_unity_wrt o iff for a being Element of A holds o . a,e = a );

definition

let A be set ;
let e be Element of A;
let o be BinOp of A;
pred e is_a_unity_wrt o means :: BINOP_1:def 7
( e is_a_left_unity_wrt o & e is_a_right_unity_wrt o );

end;

:: deftheorem defines is_a_unity_wrt BINOP_1:def 7 :

for A being set
for e being Element of A
for o being BinOp of A holds
( e is_a_unity_wrt o iff ( e is_a_left_unity_wrt o & e is_a_right_unity_wrt o ) );

theorem :: BINOP_1:3

canceled;

theorem :: BINOP_1:4

canceled;

theorem :: BINOP_1:5

canceled;

theorem :: BINOP_1:6

canceled;

theorem :: BINOP_1:7

canceled;

theorem :: BINOP_1:8

canceled;

theorem :: BINOP_1:9

canceled;

theorem :: BINOP_1:10

canceled;

theorem Th11: :: BINOP_1:11

for A being set
for o being BinOp of A
for e being Element of A holds
( e is_a_unity_wrt o iff for a being Element of A holds
( o . e,a = a & o . a,e = a ) )

proof end;

theorem Th12: :: BINOP_1:12

for A being set
for o being BinOp of A
for e being Element of A st o is commutative holds
( e is_a_unity_wrt o iff for a being Element of A holds o . e,a = a )

proof end;

theorem Th13: :: BINOP_1:13

for A being set
for o being BinOp of A
for e being Element of A st o is commutative holds
( e is_a_unity_wrt o iff for a being Element of A holds o . a,e = a )

proof end;

theorem Th14: :: BINOP_1:14

for A being set
for o being BinOp of A
for e being Element of A st o is commutative holds
( e is_a_unity_wrt o iff e is_a_left_unity_wrt o )

proof end;

theorem Th15: :: BINOP_1:15

for A being set
for o being BinOp of A
for e being Element of A st o is commutative holds
( e is_a_unity_wrt o iff e is_a_right_unity_wrt o )

proof end;

theorem :: BINOP_1:16

for A being set
for o being BinOp of A
for e being Element of A st o is commutative holds
( e is_a_left_unity_wrt o iff e is_a_right_unity_wrt o )

proof end;

theorem Th17: :: BINOP_1:17

for A being set
for o being BinOp of A
for e1, e2 being Element of A st e1 is_a_left_unity_wrt o & e2 is_a_right_unity_wrt o holds
e1 = e2

proof end;

theorem Th18: :: BINOP_1:18

for A being set
for o being BinOp of A
for e1, e2 being Element of A st e1 is_a_unity_wrt o & e2 is_a_unity_wrt o holds
e1 = e2

proof end;

definition

let A be set ;
let o be BinOp of A;
assume A1: ex e being Element of A st e is_a_unity_wrt o ;
func the_unity_wrt o -> Element of A means :: BINOP_1:def 8
it is_a_unity_wrt o;
existence
ex b₁ being Element of A st b₁ is_a_unity_wrt o by A1;
uniqueness
for b₁, b₂ being Element of A st b₁ is_a_unity_wrt o & b₂ is_a_unity_wrt o holds
b₁ = b₂ by Th18;

end;

:: deftheorem defines the_unity_wrt BINOP_1:def 8 :

for A being set
for o being BinOp of A st ex e being Element of A st e is_a_unity_wrt o holds
for b₃ being Element of A holds
( b₃ = the_unity_wrt o iff b₃ is_a_unity_wrt o );

definition

let A be set ;
let o9, o be BinOp of A;
pred o9 is_left_distributive_wrt o means :Def9: :: BINOP_1:def 9
for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c);
pred o9 is_right_distributive_wrt o means :Def10: :: BINOP_1:def 10
for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c);

end;

:: deftheorem Def9 defines is_left_distributive_wrt BINOP_1:def 9 :

for A being set
for o9, o being BinOp of A holds
( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c) );

:: deftheorem Def10 defines is_right_distributive_wrt BINOP_1:def 10 :

for A being set
for o9, o being BinOp of A holds
( o9 is_right_distributive_wrt o iff for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c) );

definition

let A be set ;
let o9, o be BinOp of A;
pred o9 is_distributive_wrt o means :: BINOP_1:def 11
( o9 is_left_distributive_wrt o & o9 is_right_distributive_wrt o );

end;

:: deftheorem defines is_distributive_wrt BINOP_1:def 11 :

for A being set
for o9, o being BinOp of A holds
( o9 is_distributive_wrt o iff ( o9 is_left_distributive_wrt o & o9 is_right_distributive_wrt o ) );

theorem :: BINOP_1:19

canceled;

theorem :: BINOP_1:20

canceled;

theorem :: BINOP_1:21

canceled;

theorem :: BINOP_1:22

canceled;

theorem Th23: :: BINOP_1:23

for A being set
for o9, o being BinOp of A holds
( o9 is_distributive_wrt o iff for a, b, c being Element of A holds
( o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c) & o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c) ) )

proof end;

theorem Th24: :: BINOP_1:24

for A being non empty set
for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_distributive_wrt o iff for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c) )

proof end;

theorem Th25: :: BINOP_1:25

for A being non empty set
for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_distributive_wrt o iff for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c) )

proof end;

theorem Th26: :: BINOP_1:26

for A being non empty set
for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o )

proof end;

theorem Th27: :: BINOP_1:27

for A being non empty set
for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_distributive_wrt o iff o9 is_right_distributive_wrt o )

proof end;

theorem :: BINOP_1:28

for A being non empty set
for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_right_distributive_wrt o iff o9 is_left_distributive_wrt o )

proof end;

definition

let A be set ;
let u be UnOp of A;
let o be BinOp of A;
pred u is_distributive_wrt o means :Def12: :: BINOP_1:def 12
for a, b being Element of A holds u . (o . a,b) = o . (u . a),(u . b);

end;

:: deftheorem Def12 defines is_distributive_wrt BINOP_1:def 12 :

for A being set
for u being UnOp of A
for o being BinOp of A holds
( u is_distributive_wrt o iff for a, b being Element of A holds u . (o . a,b) = o . (u . a),(u . b) );

definition

let A be non empty set ;
let o be BinOp of A;
redefine attr o is commutative means :: BINOP_1:def 13
for a, b being Element of A holds o . a,b = o . b,a;
correctness
compatibility
( o is commutative iff for a, b being Element of A holds o . a,b = o . b,a );
by Def2;
redefine attr o is associative means :: BINOP_1:def 14
for a, b, c being Element of A holds o . a,(o . b,c) = o . (o . a,b),c;
correctness
compatibility
( o is associative iff for a, b, c being Element of A holds o . a,(o . b,c) = o . (o . a,b),c );
by Def3;
redefine attr o is idempotent means :: BINOP_1:def 15
for a being Element of A holds o . a,a = a;
correctness
compatibility
( o is idempotent iff for a being Element of A holds o . a,a = a );
by Def4;

end;

:: deftheorem defines commutative BINOP_1:def 13 :

for A being non empty set
for o being BinOp of A holds
( o is commutative iff for a, b being Element of A holds o . a,b = o . b,a );

:: deftheorem defines associative BINOP_1:def 14 :

for A being non empty set
for o being BinOp of A holds
( o is associative iff for a, b, c being Element of A holds o . a,(o . b,c) = o . (o . a,b),c );

:: deftheorem defines idempotent BINOP_1:def 15 :

for A being non empty set
for o being BinOp of A holds
( o is idempotent iff for a being Element of A holds o . a,a = a );

definition

let A be non empty set ;
let e be Element of A;
let o be BinOp of A;
redefine pred e is_a_left_unity_wrt o means :: BINOP_1:def 16
for a being Element of A holds o . e,a = a;
correctness
compatibility
( e is_a_left_unity_wrt o iff for a being Element of A holds o . e,a = a );
by Def5;
redefine pred e is_a_right_unity_wrt o means :: BINOP_1:def 17
for a being Element of A holds o . a,e = a;
correctness
compatibility
( e is_a_right_unity_wrt o iff for a being Element of A holds o . a,e = a );
by Def6;

end;

:: deftheorem defines is_a_left_unity_wrt BINOP_1:def 16 :

for A being non empty set
for e being Element of A
for o being BinOp of A holds
( e is_a_left_unity_wrt o iff for a being Element of A holds o . e,a = a );

:: deftheorem defines is_a_right_unity_wrt BINOP_1:def 17 :

for A being non empty set
for e being Element of A
for o being BinOp of A holds
( e is_a_right_unity_wrt o iff for a being Element of A holds o . a,e = a );

definition

let A be non empty set ;
let o9, o be BinOp of A;
redefine pred o9 is_left_distributive_wrt o means :: BINOP_1:def 18
for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c);
correctness
compatibility
( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c) );
by Def9;
redefine pred o9 is_right_distributive_wrt o means :: BINOP_1:def 19
for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c);
correctness
compatibility
( o9 is_right_distributive_wrt o iff for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c) );
by Def10;

end;

:: deftheorem defines is_left_distributive_wrt BINOP_1:def 18 :

for A being non empty set
for o9, o being BinOp of A holds
( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . a,(o . b,c) = o . (o9 . a,b),(o9 . a,c) );

:: deftheorem defines is_right_distributive_wrt BINOP_1:def 19 :

for A being non empty set
for o9, o being BinOp of A holds
( o9 is_right_distributive_wrt o iff for a, b, c being Element of A holds o9 . (o . a,b),c = o . (o9 . a,c),(o9 . b,c) );

definition

let A be non empty set ;
let u be UnOp of A;
let o be BinOp of A;
redefine pred u is_distributive_wrt o means :: BINOP_1:def 20
for a, b being Element of A holds u . (o . a,b) = o . (u . a),(u . b);
correctness
compatibility
( u is_distributive_wrt o iff for a, b being Element of A holds u . (o . a,b) = o . (u . a),(u . b) );
by Def12;

end;

:: deftheorem defines is_distributive_wrt BINOP_1:def 20 :

for A being non empty set
for u being UnOp of A
for o being BinOp of A holds
( u is_distributive_wrt o iff for a, b being Element of A holds u . (o . a,b) = o . (u . a),(u . b) );

theorem :: BINOP_1:29

for X, Y, Z, x, y being set
for f being Function of [:X,Y:],Z st x in X & y in Y & Z <> {} holds
f . x,y in Z

proof end;

theorem :: BINOP_1:30

for x, y, X, Y, Z being set
for f being Function of [:X,Y:],Z
for g being Function st Z <> {} & x in X & y in Y holds
(g * f) . x,y = g . (f . x,y)

proof end;

theorem :: BINOP_1:31

for X, Y being set
for f being Function st dom f = [:X,Y:] holds
( f is constant iff for x1, x2, y1, y2 being set st x1 in X & x2 in X & y1 in Y & y2 in Y holds
f . x1,y1 = f . x2,y2 )

proof end;

theorem :: BINOP_1:32

for X, Y, Z being set
for f1, f2 being PartFunc of [:X,Y:],Z st dom f1 = dom f2 & ( for x, y being set st [x,y] in dom f1 holds
f1 . x,y = f2 . x,y ) holds
f1 = f2

proof end;

scheme :: BINOP_1:sch 5
PartFuncEx2{ F₁() -> set , F₂() -> set , F₃() -> set , P₁[ set , set , set ] } :

ex f being PartFunc of [:F₁(),F₂():],F₃() st
( ( for x, y being set holds
( [x,y] in dom f iff ( x in F₁() & y in F₂() & ex z being set st P₁[x,y,z] ) ) ) & ( for x, y being set st [x,y] in dom f holds
P₁[x,y,f . x,y] ) )

provided

A1: for x, y, z being set st x in F₁() & y in F₂() & P₁[x,y,z] holds
z in F₃() and
A2: for x, y, z1, z2 being set st x in F₁() & y in F₂() & P₁[x,y,z1] & P₁[x,y,z2] holds
z1 = z2

proof end;

scheme :: BINOP_1:sch 6
LambdaR2{ F₁() -> set , F₂() -> set , F₃() -> set , F₄( set , set ) -> set , P₁[ set , set ] } :

ex f being PartFunc of [:F₁(),F₂():],F₃() st
( ( for x, y being set holds
( [x,y] in dom f iff ( x in F₁() & y in F₂() & P₁[x,y] ) ) ) & ( for x, y being set st [x,y] in dom f holds
f . x,y = F₄(x,y) ) )

provided

A1: for x, y being set st P₁[x,y] holds
F₄(x,y) in F₃()

proof end;

scheme :: BINOP_1:sch 7
PartLambda2{ F₁() -> set , F₂() -> set , F₃() -> set , F₄( set , set ) -> set , P₁[ set , set ] } :

ex f being PartFunc of [:F₁(),F₂():],F₃() st
( ( for x, y being set holds
( [x,y] in dom f iff ( x in F₁() & y in F₂() & P₁[x,y] ) ) ) & ( for x, y being set st [x,y] in dom f holds
f . x,y = F₄(x,y) ) )

provided

A1: for x, y being set st x in F₁() & y in F₂() & P₁[x,y] holds
F₄(x,y) in F₃()

proof end;

scheme :: BINOP_1:sch 8
Sch8{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> set , F₄( set , set ) -> set , P₁[ set , set ] } :

ex f being PartFunc of [:F₁(),F₂():],F₃() st
( ( for x being Element of F₁()
for y being Element of F₂() holds
( [x,y] in dom f iff P₁[x,y] ) ) & ( for x being Element of F₁()
for y being Element of F₂() st [x,y] in dom f holds
f . x,y = F₄(x,y) ) )

provided

A1: for x being Element of F₁()
for y being Element of F₂() st P₁[x,y] holds
F₄(x,y) in F₃()

proof end;

definition

let A be set ;
let f be BinOp of A;
let x, y be Element of A;
:: original: .
redefine func f . x,y -> Element of A;
coherence
f . x,y is Element of A

proof end;

end;