:: Binary Operations
:: by Czes{\l}aw Byli\'nski
::
:: Received April 14, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let f be Function;
let a, b be object ;

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 object 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[ object , object , object ] } :

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

provided

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

proof end;

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

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

provided

A1: for x, y being object 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[ object , object , object ] } :

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 :: BINOP_1:def 2
for a, b being Element of A holds o . (a,b) = o . (b,a);

attr o is associative means :: 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 :: BINOP_1:def 4
for a being Element of A holds o . (a,a) = a;

end;

:: deftheorem 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 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 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 V18([:{},{}:], {} ) -> empty commutative associative for 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 object ;
let o be BinOp of A;

pred e is_a_left_unity_wrt o means :: BINOP_1:def 5
for a being Element of A holds o . (e,a) = a;

pred e is_a_right_unity_wrt o means :: BINOP_1:def 6
for a being Element of A holds o . (a,e) = a;

end;

:: deftheorem defines is_a_left_unity_wrt BINOP_1:def 5 :
for A being set
for e being object
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 6 :
for A being set
for e being object
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 object ;
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 object
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 Th3: :: BINOP_1:3

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 Th4: :: BINOP_1:4

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 Th5: :: BINOP_1:5

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 Th6: :: BINOP_1:6

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 ) by Th4;

theorem Th7: :: BINOP_1:7

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 ) by Th5;

theorem :: BINOP_1:8

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 Th9: :: BINOP_1:9

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 Th10: :: BINOP_1:10

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 by Th9;

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 Th10;

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 :: 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 :: 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 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 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 Th11: :: BINOP_1:11

for A being set
for o, o9 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 Th12: :: BINOP_1:12

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 Th13: :: BINOP_1:13

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 Th14: :: BINOP_1:14

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 ) by Th12;

theorem Th15: :: BINOP_1:15

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 ) by Th13;

theorem :: BINOP_1:16

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 :: BINOP_1:def 12
for a, b being Element of A holds u . (o . (a,b)) = o . ((u . a),(u . b));

end;

:: deftheorem 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 e be object ;
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 );
;

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 );
;

end;

:: deftheorem BINOP_1:def 13 :
canceled;

:: deftheorem BINOP_1:def 14 :
canceled;

:: deftheorem BINOP_1:def 15 :
canceled;

:: deftheorem defines is_a_left_unity_wrt BINOP_1:def 16 :
for A being non empty set
for e being object
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 object
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))) );
;

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))) );
;

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)) );
;

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)) );

:: FUNCT_2, 2005.12.13, A.T.

theorem :: BINOP_1:17

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

proof end;

:: from TOPALG_3, 2005.12.14, A.T.

theorem :: BINOP_1:18

for x, y being object
for 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)) by FUNCT_2:15, ZFMISC_1:87;

:: missing, 2005.12.17, A.T.

theorem :: BINOP_1:19

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 object st x1 in X & x2 in X & y1 in Y & y2 in Y holds
f . (x1,y1) = f . (x2,y2) )

proof end;

:: from PARTFUN1, 2006.12.05, AT

theorem :: BINOP_1:20

for X, Y, Z being set
for f1, f2 being PartFunc of [:X,Y:],Z st dom f1 = dom f2 & ( for x, y being object 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[ object , object , object ] } :

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

provided

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

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

provided

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

proof end;

:: from GRCAT_1, 2006.12.05, AT

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

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

provided

A1: for x, y being object 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( object , object ) -> object , P₁[ object , object ] } :

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 a, b be Element of A;
:: original: .
redefine func f . (a,b) -> Element of A;
coherence
f . (a,b) is Element of A

proof end;

end;

definition

let X, Y, Z be set ;
let f1, f2 be Function of [:X,Y:],Z;

:: original: =
redefine pred f1 = f2 means :: BINOP_1:def 21
for x, y being set st x in X & y in Y holds
f1 . (x,y) = f2 . (x,y);

compatibility
( f1 = f2 iff for x, y being set st x in X & y in Y holds
f1 . (x,y) = f2 . (x,y) ) by Th1;

end;

:: deftheorem defines = BINOP_1:def 21 :
for X, Y, Z being set
for f1, f2 being Function of [:X,Y:],Z holds
( f1 = f2 iff for x, y being set st x in X & y in Y holds
f1 . (x,y) = f2 . (x,y) );

scheme :: BINOP_1:sch 9
Sch9{ F₁() -> set , F₂() -> set , F₃() -> set , P₁[ object , object , object ] } :

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;