:: 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;