:: Basic Properties of Rational Numbers
:: by Andrzej Kondracki
::
:: Received July 10, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users


begin

Lm1: omega c= ( { [c,d] where c, d is Element of omega : ( c,d are_relative_prime & d <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega
by XBOOLE_1:7;

Lm2: for i1, j1 being Element of NAT holds quotient (i1,j1) = i1 / j1
proof end;

0 in omega
;

then reconsider 09 = 0 as Element of REAL+ by ARYTM_2:2;

Lm3: for a being real number
for x9 being Element of REAL+ st x9 = a holds
09 - x9 = - a
proof end;

Lm4: for x being set st x in RAT holds
ex m, n being Integer st x = m / n
proof end;

Lm5: for x being set
for l being Element of NAT
for m being Integer st x = m / l holds
x in RAT
proof end;

Lm6: for x being set
for m, n being Integer st x = m / n holds
x in RAT
proof end;

definition
redefine func RAT means :Def1: :: RAT_1:def 1
for x being set holds
( x in it iff ex m, n being Integer st x = m / n );
compatibility
for b1 being set holds
( b1 = RAT iff for x being set holds
( x in b1 iff ex m, n being Integer st x = m / n ) )
proof end;
end;

:: deftheorem Def1 defines RAT RAT_1:def 1 :
for b1 being set holds
( b1 = RAT iff for x being set holds
( x in b1 iff ex m, n being Integer st x = m / n ) );

definition
let r be number ;
attr r is rational means :Def2: :: RAT_1:def 2
r in RAT ;
end;

:: deftheorem Def2 defines rational RAT_1:def 2 :
for r being number holds
( r is rational iff r in RAT );

registration
cluster rational Element of REAL ;
existence
ex b1 being Real st b1 is rational
proof end;
end;

registration
cluster rational set ;
existence
ex b1 being number st b1 is rational
proof end;
end;

definition
mode Rational is rational number ;
end;

theorem Th1: :: RAT_1:1
for x being set st x in RAT holds
ex m, n being Integer st
( n <> 0 & x = m / n )
proof end;

theorem :: RAT_1:2
canceled;

theorem Th3: :: RAT_1:3
for x being set st x is Rational holds
ex m, n being Integer st
( n <> 0 & x = m / n )
proof end;

registration
cluster rational -> real set ;
coherence
for b1 being number st b1 is rational holds
b1 is real
proof end;
end;

theorem :: RAT_1:4
canceled;

theorem :: RAT_1:5
canceled;

theorem Th6: :: RAT_1:6
for x being set st ex m, n being Integer st x = m / n holds
x is rational
proof end;

theorem Th7: :: RAT_1:7
for x being Integer holds x is Rational
proof end;

registration
cluster integer -> rational set ;
coherence
for b1 being number st b1 is integer holds
b1 is rational
by Th7;
end;

registration
let p, q be Rational;
cluster p * q -> rational ;
coherence
p * q is rational
proof end;
cluster p + q -> rational ;
coherence
p + q is rational
proof end;
cluster p - q -> rational ;
coherence
p - q is rational
proof end;
end;

registration
let p be Rational;
let m be Integer;
cluster p + m -> rational ;
coherence
p + m is rational
;
cluster p - m -> rational ;
coherence
p - m is rational
;
cluster p * m -> rational ;
coherence
p * m is rational
;
end;

registration
let m be Integer;
let p be Rational;
cluster m + p -> rational ;
coherence
m + p is rational
;
cluster m - p -> rational ;
coherence
m - p is rational
;
cluster m * p -> rational ;
coherence
m * p is rational
;
end;

registration
let p be Rational;
let k be Element of NAT ;
cluster p + k -> rational ;
coherence
p + k is rational
;
cluster p - k -> rational ;
coherence
p - k is rational
;
cluster p * k -> rational ;
coherence
p * k is rational
;
end;

registration
let k be Element of NAT ;
let p be Rational;
cluster k + p -> rational ;
coherence
k + p is rational
;
cluster k - p -> rational ;
coherence
k - p is rational
;
cluster k * p -> rational ;
coherence
k * p is rational
;
end;

registration
let p be Rational;
cluster - p -> rational ;
coherence
- p is rational
proof end;
end;

theorem :: RAT_1:8
canceled;

theorem :: RAT_1:9
canceled;

theorem :: RAT_1:10
canceled;

theorem :: RAT_1:11
canceled;

theorem :: RAT_1:12
canceled;

theorem :: RAT_1:13
canceled;

theorem :: RAT_1:14
canceled;

theorem :: RAT_1:15
canceled;

theorem Th16: :: RAT_1:16
for p, q being Rational holds p / q is Rational
proof end;

registration
let p, q be rational number ;
cluster p / q -> rational ;
coherence
p / q is rational
by Th16;
end;

theorem :: RAT_1:17
canceled;

theorem :: RAT_1:18
canceled;

theorem :: RAT_1:19
canceled;

theorem :: RAT_1:20
canceled;

theorem Th21: :: RAT_1:21
for p being Rational holds p " is Rational
proof end;

registration
let p be rational number ;
cluster p " -> rational ;
coherence
p " is rational
by Th21;
end;

theorem :: RAT_1:22
for a, b being real number st a < b holds
ex p being Rational st
( a < p & p < b )
proof end;

theorem :: RAT_1:23
canceled;

theorem Th24: :: RAT_1:24
for p being Rational ex m being Integer ex k being Element of NAT st
( k <> 0 & p = m / k )
proof end;

theorem Th25: :: RAT_1:25
for p being Rational ex m being Integer ex k being Element of NAT st
( k <> 0 & p = m / k & ( for n being Integer
for l being Element of NAT st l <> 0 & p = n / l holds
k <= l ) )
proof end;

definition
let p be Rational;
func denominator p -> Element of NAT means :Def3: :: RAT_1:def 3
( it <> 0 & ex m being Integer st p = m / it & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
it <= k ) );
existence
ex b1 being Element of NAT st
( b1 <> 0 & ex m being Integer st p = m / b1 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b1 <= k ) )
proof end;
uniqueness
for b1, b2 being Element of NAT st b1 <> 0 & ex m being Integer st p = m / b1 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b1 <= k ) & b2 <> 0 & ex m being Integer st p = m / b2 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b2 <= k ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines denominator RAT_1:def 3 :
for p being Rational
for b2 being Element of NAT holds
( b2 = denominator p iff ( b2 <> 0 & ex m being Integer st p = m / b2 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b2 <= k ) ) );

definition
let p be Rational;
func numerator p -> Integer equals :: RAT_1:def 4
(denominator p) * p;
coherence
(denominator p) * p is Integer
proof end;
end;

:: deftheorem defines numerator RAT_1:def 4 :
for p being Rational holds numerator p = (denominator p) * p;

theorem :: RAT_1:26
canceled;

theorem Th27: :: RAT_1:27
for p being Rational holds 0 < denominator p
proof end;

theorem :: RAT_1:28
canceled;

theorem Th29: :: RAT_1:29
for p being Rational holds 1 <= denominator p
proof end;

theorem Th30: :: RAT_1:30
for p being Rational holds 0 < (denominator p) "
proof end;

theorem :: RAT_1:31
canceled;

theorem :: RAT_1:32
canceled;

theorem :: RAT_1:33
canceled;

theorem Th34: :: RAT_1:34
for p being Rational holds 1 >= (denominator p) "
proof end;

theorem :: RAT_1:35
canceled;

theorem Th36: :: RAT_1:36
for p being Rational holds
( numerator p = 0 iff p = 0 )
proof end;

theorem Th37: :: RAT_1:37
for p being Rational holds
( p = (numerator p) / (denominator p) & p = (numerator p) * ((denominator p) ") & p = ((denominator p) ") * (numerator p) )
proof end;

theorem :: RAT_1:38
for p being Rational st p <> 0 holds
denominator p = (numerator p) / p
proof end;

theorem :: RAT_1:39
canceled;

theorem Th40: :: RAT_1:40
for p being Rational st p is Integer holds
( denominator p = 1 & numerator p = p )
proof end;

theorem Th41: :: RAT_1:41
for p being Rational st ( numerator p = p or denominator p = 1 ) holds
p is Integer
proof end;

theorem :: RAT_1:42
for p being Rational holds
( numerator p = p iff denominator p = 1 ) by Th40;

theorem :: RAT_1:43
canceled;

theorem :: RAT_1:44
for p being Rational st ( numerator p = p or denominator p = 1 ) & 0 <= p holds
p is Element of NAT
proof end;

theorem :: RAT_1:45
for p being Rational holds
( 1 < denominator p iff not p is integer )
proof end;

Lm7: 1 " = 1
;

theorem :: RAT_1:46
for p being Rational holds
( 1 > (denominator p) " iff not p is integer )
proof end;

theorem Th47: :: RAT_1:47
for p being Rational holds
( numerator p = denominator p iff p = 1 )
proof end;

theorem Th48: :: RAT_1:48
for p being Rational holds
( numerator p = - (denominator p) iff p = - 1 )
proof end;

theorem :: RAT_1:49
for p being Rational holds
( - (numerator p) = denominator p iff p = - 1 )
proof end;

theorem Th50: :: RAT_1:50
for m being Integer
for p being Rational st m <> 0 holds
p = ((numerator p) * m) / ((denominator p) * m)
proof end;

theorem :: RAT_1:51
canceled;

theorem :: RAT_1:52
canceled;

theorem :: RAT_1:53
canceled;

theorem :: RAT_1:54
canceled;

theorem :: RAT_1:55
canceled;

theorem :: RAT_1:56
canceled;

theorem :: RAT_1:57
canceled;

theorem :: RAT_1:58
canceled;

theorem :: RAT_1:59
canceled;

theorem Th60: :: RAT_1:60
for k being Element of NAT
for m being Integer
for p being Rational st k <> 0 & p = m / k holds
ex l being Element of NAT st
( m = (numerator p) * l & k = (denominator p) * l )
proof end;

theorem :: RAT_1:61
for m, n being Integer
for p being Rational st p = m / n & n <> 0 holds
ex m1 being Integer st
( m = (numerator p) * m1 & n = (denominator p) * m1 )
proof end;

theorem Th62: :: RAT_1:62
for p being Rational
for l being Element of NAT holds
( not 1 < l or for m being Integer
for k being Element of NAT holds
( not numerator p = m * l or not denominator p = k * l ) )
proof end;

theorem Th63: :: RAT_1:63
for k being Element of NAT
for m being Integer
for p being Rational st p = m / k & k <> 0 & ( for l being Element of NAT holds
( not 1 < l or for m1 being Integer
for k1 being Element of NAT holds
( not m = m1 * l or not k = k1 * l ) ) ) holds
( k = denominator p & m = numerator p )
proof end;

theorem Th64: :: RAT_1:64
for p being Rational holds
( p < - 1 iff numerator p < - (denominator p) )
proof end;

theorem Th65: :: RAT_1:65
for p being Rational holds
( p <= - 1 iff numerator p <= - (denominator p) )
proof end;

theorem :: RAT_1:66
for p being Rational holds
( p < - 1 iff denominator p < - (numerator p) )
proof end;

theorem :: RAT_1:67
for p being Rational holds
( p <= - 1 iff denominator p <= - (numerator p) )
proof end;

theorem :: RAT_1:68
canceled;

theorem :: RAT_1:69
canceled;

theorem :: RAT_1:70
canceled;

theorem :: RAT_1:71
canceled;

theorem Th72: :: RAT_1:72
for p being Rational holds
( p < 1 iff numerator p < denominator p )
proof end;

theorem :: RAT_1:73
for p being Rational holds
( p <= 1 iff numerator p <= denominator p )
proof end;

theorem :: RAT_1:74
canceled;

theorem :: RAT_1:75
canceled;

theorem :: RAT_1:76
for p being Rational holds
( p < 0 iff numerator p < 0 )
proof end;

theorem Th77: :: RAT_1:77
for p being Rational holds
( p <= 0 iff numerator p <= 0 )
proof end;

theorem :: RAT_1:78
canceled;

theorem :: RAT_1:79
canceled;

theorem Th80: :: RAT_1:80
for a being real number
for p being Rational holds
( a < p iff a * (denominator p) < numerator p )
proof end;

theorem :: RAT_1:81
for a being real number
for p being Rational holds
( a <= p iff a * (denominator p) <= numerator p )
proof end;

theorem :: RAT_1:82
canceled;

theorem :: RAT_1:83
canceled;

theorem :: RAT_1:84
for p, q being Rational st denominator p = denominator q & numerator p = numerator q holds
p = q
proof end;

theorem :: RAT_1:85
canceled;

theorem :: RAT_1:86
for p, q being Rational holds
( p < q iff (numerator p) * (denominator q) < (numerator q) * (denominator p) )
proof end;

theorem Th87: :: RAT_1:87
for p being Rational holds
( denominator (- p) = denominator p & numerator (- p) = - (numerator p) )
proof end;

theorem Th88: :: RAT_1:88
for p, q being Rational holds
( 0 < p & q = 1 / p iff ( numerator q = denominator p & denominator q = numerator p ) )
proof end;

theorem :: RAT_1:89
for p, q being Rational holds
( p < 0 & q = 1 / p iff ( numerator q = - (denominator p) & denominator q = - (numerator p) ) )
proof end;