:: Introduction to Arithmetics
:: by Andrzej Trybulec
::
:: Copyright (c) 2003-2017 Association of Mizar Users

Lm1: for x being Element of REAL+ holds [0,x] in
proof end;

theorem Th1: :: ARYTM_0:1
proof end;

theorem Th2: :: ARYTM_0:2
for x being Element of REAL+ st x <> {} holds
[{},x] in REAL
proof end;

theorem Th3: :: ARYTM_0:3
for y being set st [{},y] in REAL holds
y <> {}
proof end;

theorem Th4: :: ARYTM_0:4
for x, y being Element of REAL+ holds x - y in REAL
proof end;

theorem Th5: :: ARYTM_0:5
proof end;

registration
let x, y be object ;
cluster [x,y] -> non zero ;
coherence
not [x,y] is zero
;
end;

theorem Th6: :: ARYTM_0:6
for x, y being Element of REAL+ st x - y = {} holds
x = y
proof end;

Lm2: for a, b being set holds not 1 = [a,b]
proof end;

theorem Th7: :: ARYTM_0:7
for x, y, z being Element of REAL+ st x <> {} & x *' y = x *' z holds
y = z
proof end;

Lm3:
by NUMBERS:19;

definition
let x, y be Element of REAL ;
func + (x,y) -> Element of REAL means :Def1: :: ARYTM_0:def 1
ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & it = x9 + y9 ) if ( x in REAL+ & y in REAL+ )
ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & it = x9 - y9 ) if ( x in REAL+ & y in )
ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & it = y9 - x9 ) if ( y in REAL+ & x in )
otherwise ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & it = [0,(x9 + y9)] );
existence
( ( x in REAL+ & y in REAL+ implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 + y9 ) ) & ( x in REAL+ & y in implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = x9 - y9 ) ) & ( y in REAL+ & x in implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = y9 - x9 ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in ) & ( not y in REAL+ or not x in ) implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = [0,(x9 + y9)] ) ) )
proof end;
uniqueness
for b1, b2 being Element of REAL holds
( ( x in REAL+ & y in REAL+ & ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 + y9 ) & ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b2 = x9 + y9 ) implies b1 = b2 ) & ( x in REAL+ & y in & ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = x9 - y9 ) & ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b2 = x9 - y9 ) implies b1 = b2 ) & ( y in REAL+ & x in & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = y9 - x9 ) & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b2 = y9 - x9 ) implies b1 = b2 ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in ) & ( not y in REAL+ or not x in ) & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = [0,(x9 + y9)] ) & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b2 = [0,(x9 + y9)] ) implies b1 = b2 ) )
proof end;
consistency
for b1 being Element of REAL holds
( ( x in REAL+ & y in REAL+ & x in REAL+ & y in implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 + y9 ) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = x9 - y9 ) ) ) & ( x in REAL+ & y in REAL+ & y in REAL+ & x in implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 + y9 ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = y9 - x9 ) ) ) & ( x in REAL+ & y in & y in REAL+ & x in implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = x9 - y9 ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = y9 - x9 ) ) ) )
by ;
commutativity
for b1, x, y being Element of REAL st ( x in REAL+ & y in REAL+ implies ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 + y9 ) ) & ( x in REAL+ & y in implies ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = x9 - y9 ) ) & ( y in REAL+ & x in implies ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = y9 - x9 ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in ) & ( not y in REAL+ or not x in ) implies ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = [0,(x9 + y9)] ) ) holds
( ( y in REAL+ & x in REAL+ implies ex x9, y9 being Element of REAL+ st
( y = x9 & x = y9 & b1 = x9 + y9 ) ) & ( y in REAL+ & x in implies ex x9, y9 being Element of REAL+ st
( y = x9 & x = [0,y9] & b1 = x9 - y9 ) ) & ( x in REAL+ & y in implies ex x9, y9 being Element of REAL+ st
( y = [0,x9] & x = y9 & b1 = y9 - x9 ) ) & ( ( not y in REAL+ or not x in REAL+ ) & ( not y in REAL+ or not x in ) & ( not x in REAL+ or not y in ) implies ex x9, y9 being Element of REAL+ st
( y = [0,x9] & x = [0,y9] & b1 = [0,(x9 + y9)] ) ) )
;
func * (x,y) -> Element of REAL means :Def2: :: ARYTM_0:def 2
ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & it = x9 *' y9 ) if ( x in REAL+ & y in REAL+ )
ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & it = [0,(x9 *' y9)] ) if ( x in REAL+ & y in & x <> 0 )
ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & it = [0,(y9 *' x9)] ) if ( y in REAL+ & x in & y <> 0 )
ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & it = y9 *' x9 ) if ( x in & y in )
otherwise it = 0 ;
existence
( ( x in REAL+ & y in REAL+ implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) ) & ( x in REAL+ & y in & x <> 0 implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) ) & ( y in REAL+ & x in & y <> 0 implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) ) & ( x in & y in implies ex b1 being Element of REAL ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in or not x <> 0 ) & ( not y in REAL+ or not x in or not y <> 0 ) & ( not x in or not y in ) implies ex b1 being Element of REAL st b1 = 0 ) )
proof end;
uniqueness
for b1, b2 being Element of REAL holds
( ( x in REAL+ & y in REAL+ & ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) & ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b2 = x9 *' y9 ) implies b1 = b2 ) & ( x in REAL+ & y in & x <> 0 & ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) & ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b2 = [0,(x9 *' y9)] ) implies b1 = b2 ) & ( y in REAL+ & x in & y <> 0 & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b2 = [0,(y9 *' x9)] ) implies b1 = b2 ) & ( x in & y in & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) & ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b2 = y9 *' x9 ) implies b1 = b2 ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in or not x <> 0 ) & ( not y in REAL+ or not x in or not y <> 0 ) & ( not x in or not y in ) & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof end;
consistency
for b1 being Element of REAL holds
( ( x in REAL+ & y in REAL+ & x in REAL+ & y in & x <> 0 implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) ) ) & ( x in REAL+ & y in REAL+ & y in REAL+ & x in & y <> 0 implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) ) ) & ( x in REAL+ & y in REAL+ & x in & y in implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) ) ) & ( x in REAL+ & y in & x <> 0 & y in REAL+ & x in & y <> 0 implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) ) ) & ( x in REAL+ & y in & x <> 0 & x in & y in implies ( ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) ) ) & ( y in REAL+ & x in & y <> 0 & x in & y in implies ( ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) ) ) )
by ;
commutativity
for b1, x, y being Element of REAL st ( x in REAL+ & y in REAL+ implies ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b1 = x9 *' y9 ) ) & ( x in REAL+ & y in & x <> 0 implies ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b1 = [0,(x9 *' y9)] ) ) & ( y in REAL+ & x in & y <> 0 implies ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b1 = [0,(y9 *' x9)] ) ) & ( x in & y in implies ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b1 = y9 *' x9 ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in or not x <> 0 ) & ( not y in REAL+ or not x in or not y <> 0 ) & ( not x in or not y in ) implies b1 = 0 ) holds
( ( y in REAL+ & x in REAL+ implies ex x9, y9 being Element of REAL+ st
( y = x9 & x = y9 & b1 = x9 *' y9 ) ) & ( y in REAL+ & x in & y <> 0 implies ex x9, y9 being Element of REAL+ st
( y = x9 & x = [0,y9] & b1 = [0,(x9 *' y9)] ) ) & ( x in REAL+ & y in & x <> 0 implies ex x9, y9 being Element of REAL+ st
( y = [0,x9] & x = y9 & b1 = [0,(y9 *' x9)] ) ) & ( y in & x in implies ex x9, y9 being Element of REAL+ st
( y = [0,x9] & x = [0,y9] & b1 = y9 *' x9 ) ) & ( ( not y in REAL+ or not x in REAL+ ) & ( not y in REAL+ or not x in or not y <> 0 ) & ( not x in REAL+ or not y in or not x <> 0 ) & ( not y in or not x in ) implies b1 = 0 ) )
;
end;

:: deftheorem Def1 defines + ARYTM_0:def 1 :
for x, y, b3 being Element of REAL holds
( ( x in REAL+ & y in REAL+ implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b3 = x9 + y9 ) ) ) & ( x in REAL+ & y in implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b3 = x9 - y9 ) ) ) & ( y in REAL+ & x in implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b3 = y9 - x9 ) ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in ) & ( not y in REAL+ or not x in ) implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b3 = [0,(x9 + y9)] ) ) ) );

:: deftheorem Def2 defines * ARYTM_0:def 2 :
for x, y, b3 being Element of REAL holds
( ( x in REAL+ & y in REAL+ implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = y9 & b3 = x9 *' y9 ) ) ) & ( x in REAL+ & y in & x <> 0 implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st
( x = x9 & y = [0,y9] & b3 = [0,(x9 *' y9)] ) ) ) & ( y in REAL+ & x in & y <> 0 implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = y9 & b3 = [0,(y9 *' x9)] ) ) ) & ( x in & y in implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st
( x = [0,x9] & y = [0,y9] & b3 = y9 *' x9 ) ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in or not x <> 0 ) & ( not y in REAL+ or not x in or not y <> 0 ) & ( not x in or not y in ) implies ( b3 = * (x,y) iff b3 = 0 ) ) );

reconsider jj = 1 as Element of REAL by NUMBERS:19;

definition
let x be Element of REAL ;
func opp x -> Element of REAL means :Def3: :: ARYTM_0:def 3
+ (x,it) = 0 ;
existence
ex b1 being Element of REAL st + (x,b1) = 0
proof end;
uniqueness
for b1, b2 being Element of REAL st + (x,b1) = 0 & + (x,b2) = 0 holds
b1 = b2
proof end;
involutiveness
for b1, b2 being Element of REAL st + (b2,b1) = 0 holds
+ (b1,b2) = 0
;
func inv x -> Element of REAL means :Def4: :: ARYTM_0:def 4
* (x,it) = 1 if x <> 0
otherwise it = 0 ;
existence
( ( x <> 0 implies ex b1 being Element of REAL st * (x,b1) = 1 ) & ( not x <> 0 implies ex b1 being Element of REAL st b1 = 0 ) )
proof end;
uniqueness
for b1, b2 being Element of REAL holds
( ( x <> 0 & * (x,b1) = 1 & * (x,b2) = 1 implies b1 = b2 ) & ( not x <> 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof end;
consistency
for b1 being Element of REAL holds verum
;
involutiveness
for b1, b2 being Element of REAL st ( b2 <> 0 implies * (b2,b1) = 1 ) & ( not b2 <> 0 implies b1 = 0 ) holds
( ( b1 <> 0 implies * (b1,b2) = 1 ) & ( not b1 <> 0 implies b2 = 0 ) )
proof end;
end;

:: deftheorem Def3 defines opp ARYTM_0:def 3 :
for x, b2 being Element of REAL holds
( b2 = opp x iff + (x,b2) = 0 );

:: deftheorem Def4 defines inv ARYTM_0:def 4 :
for x, b2 being Element of REAL holds
( ( x <> 0 implies ( b2 = inv x iff * (x,b2) = 1 ) ) & ( not x <> 0 implies ( b2 = inv x iff b2 = 0 ) ) );

Lm4: for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )

proof end;

theorem Th8: :: ARYTM_0:8
for a, b being Element of REAL holds not (0,1) --> (a,b) in REAL
proof end;

definition
let x, y be Element of REAL ;
func [*x,y*] -> Element of COMPLEX equals :Def5: :: ARYTM_0:def 5
x if y = 0
otherwise (0,1) --> (x,y);
consistency
for b1 being Element of COMPLEX holds verum
;
coherence
( ( y = 0 implies x is Element of COMPLEX ) & ( not y = 0 implies (0,1) --> (x,y) is Element of COMPLEX ) )
proof end;
end;

:: deftheorem Def5 defines [* ARYTM_0:def 5 :
for x, y being Element of REAL holds
( ( y = 0 implies [*x,y*] = x ) & ( not y = 0 implies [*x,y*] = (0,1) --> (x,y) ) );

theorem :: ARYTM_0:9
for c being Element of COMPLEX ex r, s being Element of REAL st c = [*r,s*]
proof end;

theorem :: ARYTM_0:10
for x1, x2, y1, y2 being Element of REAL st [*x1,x2*] = [*y1,y2*] holds
( x1 = y1 & x2 = y2 )
proof end;

set RR = \ ;

reconsider o = 0 as Element of REAL by Lm3;

theorem Th11: :: ARYTM_0:11
for x, o being Element of REAL st o = 0 holds
+ (x,o) = x
proof end;

theorem Th12: :: ARYTM_0:12
for x, o being Element of REAL st o = 0 holds
* (x,o) = 0
proof end;

theorem Th13: :: ARYTM_0:13
for x, y, z being Element of REAL holds * (x,(* (y,z))) = * ((* (x,y)),z)
proof end;

theorem Th14: :: ARYTM_0:14
for x, y, z being Element of REAL holds * (x,(+ (y,z))) = + ((* (x,y)),(* (x,z)))
proof end;

theorem :: ARYTM_0:15
for x, y being Element of REAL holds * ((opp x),y) = opp (* (x,y))
proof end;

theorem Th16: :: ARYTM_0:16
for x being Element of REAL holds * (x,x) in REAL+
proof end;

theorem :: ARYTM_0:17
for x, y being Element of REAL st + ((* (x,x)),(* (y,y))) = 0 holds
x = 0
proof end;

theorem Th18: :: ARYTM_0:18
for x, y, z being Element of REAL st x <> 0 & * (x,y) = 1 & * (x,z) = 1 holds
y = z
proof end;

theorem Th19: :: ARYTM_0:19
for x, y being Element of REAL st y = 1 holds
* (x,y) = x
proof end;

theorem :: ARYTM_0:20
for x, y being Element of REAL st y <> 0 holds
* ((* (x,y)),(inv y)) = x
proof end;

theorem Th21: :: ARYTM_0:21
for x, y being Element of REAL holds
( not * (x,y) = 0 or x = 0 or y = 0 )
proof end;

theorem :: ARYTM_0:22
for x, y being Element of REAL holds inv (* (x,y)) = * ((inv x),(inv y))
proof end;

theorem Th23: :: ARYTM_0:23
for x, y, z being Element of REAL holds + (x,(+ (y,z))) = + ((+ (x,y)),z)
proof end;

theorem :: ARYTM_0:24
for x, y being Element of REAL st [*x,y*] in REAL holds
y = 0
proof end;

theorem :: ARYTM_0:25
for x, y being Element of REAL holds opp (+ (x,y)) = + ((opp x),(opp y))
proof end;