Article RAT_1, MML version 4.99.1005
:: RAT_1:funcnot 1 => NUMBERS:func 3
definition
func RAT -> set means
for b1 being set holds
b1 in it
iff
ex b2, b3 being integer set st
b1 = b2 / b3;
end;
:: RAT_1:def 1
theorem
for b1 being set holds
b1 = RAT
iff
for b2 being set holds
b2 in b1
iff
ex b3, b4 being integer set st
b2 = b3 / b4;
:: RAT_1:attrnot 1 => RAT_1:attr 1
definition
let a1 be set;
attr a1 is rational means
a1 in RAT;
end;
:: RAT_1:dfs 2
definiens
let a1 be set;
To prove
a1 is rational
it is sufficient to prove
thus a1 in RAT;
:: RAT_1:def 2
theorem
for b1 being set holds
b1 is rational
iff
b1 in RAT;
:: RAT_1:exreg 1
registration
cluster rational Element of REAL;
end;
:: RAT_1:exreg 2
registration
cluster rational set;
end;
:: RAT_1:modenot 1
definition
mode Rational is rational set;
end;
:: RAT_1:th 1
theorem
for b1 being set
st b1 in RAT
holds ex b2, b3 being integer set st
b3 <> 0 & b1 = b2 / b3;
:: RAT_1:th 3
theorem
for b1 being set
st b1 is rational set
holds ex b2, b3 being integer set st
b3 <> 0 & b1 = b2 / b3;
:: RAT_1:condreg 1
registration
cluster rational -> real (set);
end;
:: RAT_1:th 6
theorem
for b1 being set
st ex b2, b3 being integer set st
b1 = b2 / b3
holds b1 is rational;
:: RAT_1:th 7
theorem
for b1 being integer set holds
b1 is rational set;
:: RAT_1:condreg 2
registration
cluster integer -> rational (set);
end;
:: RAT_1:funcreg 1
registration
let a1, a2 be rational set;
cluster a1 * a2 -> rational;
end;
:: RAT_1:funcreg 2
registration
let a1, a2 be rational set;
cluster a1 + a2 -> rational;
end;
:: RAT_1:funcreg 3
registration
let a1, a2 be rational set;
cluster a1 - a2 -> rational;
end;
:: RAT_1:funcreg 4
registration
let a1 be rational set;
let a2 be integer set;
cluster a1 + a2 -> rational;
end;
:: RAT_1:funcreg 5
registration
let a1 be rational set;
let a2 be integer set;
cluster a1 - a2 -> rational;
end;
:: RAT_1:funcreg 6
registration
let a1 be rational set;
let a2 be integer set;
cluster a1 * a2 -> rational;
end;
:: RAT_1:funcreg 7
registration
let a1 be integer set;
let a2 be rational set;
cluster a1 + a2 -> rational;
end;
:: RAT_1:funcreg 8
registration
let a1 be integer set;
let a2 be rational set;
cluster a1 - a2 -> rational;
end;
:: RAT_1:funcreg 9
registration
let a1 be integer set;
let a2 be rational set;
cluster a1 * a2 -> rational;
end;
:: RAT_1:funcreg 10
registration
let a1 be rational set;
let a2 be Element of NAT;
cluster a1 + a2 -> rational;
end;
:: RAT_1:funcreg 11
registration
let a1 be rational set;
let a2 be Element of NAT;
cluster a1 - a2 -> rational;
end;
:: RAT_1:funcreg 12
registration
let a1 be rational set;
let a2 be Element of NAT;
cluster a1 * a2 -> rational;
end;
:: RAT_1:funcreg 13
registration
let a1 be Element of NAT;
let a2 be rational set;
cluster a1 + a2 -> rational;
end;
:: RAT_1:funcreg 14
registration
let a1 be Element of NAT;
let a2 be rational set;
cluster a1 - a2 -> rational;
end;
:: RAT_1:funcreg 15
registration
let a1 be Element of NAT;
let a2 be rational set;
cluster a1 * a2 -> rational;
end;
:: RAT_1:funcreg 16
registration
let a1 be rational set;
cluster - a1 -> complex rational;
end;
:: RAT_1:th 16
theorem
for b1, b2 being rational set holds
b1 / b2 is rational set;
:: RAT_1:funcreg 17
registration
let a1, a2 be rational set;
cluster a1 / a2 -> rational;
end;
:: RAT_1:th 21
theorem
for b1 being rational set holds
b1 " is rational set;
:: RAT_1:funcreg 18
registration
let a1 be rational set;
cluster a1 " -> complex rational;
end;
:: RAT_1:th 22
theorem
for b1, b2 being real set
st b1 < b2
holds ex b3 being rational set st
b1 < b3 & b3 < b2;
:: RAT_1:th 24
theorem
for b1 being rational set holds
ex b2 being integer set st
ex b3 being Element of NAT st
b3 <> 0 & b1 = b2 / b3;
:: RAT_1:th 25
theorem
for b1 being rational set holds
ex b2 being integer set st
ex b3 being Element of NAT st
b3 <> 0 &
b1 = b2 / b3 &
(for b4 being integer set
for b5 being Element of NAT
st b5 <> 0 & b1 = b4 / b5
holds b3 <= b5);
:: RAT_1:funcnot 2 => RAT_1:func 1
definition
let a1 be rational set;
func denominator A1 -> Element of NAT means
it <> 0 &
(ex b1 being integer set st
a1 = b1 / it) &
(for b1 being integer set
for b2 being Element of NAT
st b2 <> 0 & a1 = b1 / b2
holds it <= b2);
end;
:: RAT_1:def 3
theorem
for b1 being rational set
for b2 being Element of NAT holds
b2 = denominator b1
iff
b2 <> 0 &
(ex b3 being integer set st
b1 = b3 / b2) &
(for b3 being integer set
for b4 being Element of NAT
st b4 <> 0 & b1 = b3 / b4
holds b2 <= b4);
:: RAT_1:funcnot 3 => RAT_1:func 2
definition
let a1 be rational set;
func numerator A1 -> integer set equals
(denominator a1) * a1;
end;
:: RAT_1:def 4
theorem
for b1 being rational set holds
numerator b1 = (denominator b1) * b1;
:: RAT_1:th 27
theorem
for b1 being rational set holds
0 < denominator b1;
:: RAT_1:th 29
theorem
for b1 being rational set holds
1 <= denominator b1;
:: RAT_1:th 30
theorem
for b1 being rational set holds
0 < (denominator b1) ";
:: RAT_1:th 34
theorem
for b1 being rational set holds
(denominator b1) " <= 1;
:: RAT_1:th 36
theorem
for b1 being rational set holds
numerator b1 = 0
iff
b1 = 0;
:: RAT_1:th 37
theorem
for b1 being rational set holds
b1 = (numerator b1) / denominator b1 & b1 = (numerator b1) * ((denominator b1) ") & b1 = (denominator b1) " * numerator b1;
:: RAT_1:th 38
theorem
for b1 being rational set
st b1 <> 0
holds denominator b1 = (numerator b1) / b1;
:: RAT_1:th 40
theorem
for b1 being rational set
st b1 is integer set
holds denominator b1 = 1 & numerator b1 = b1;
:: RAT_1:th 41
theorem
for b1 being rational set
st (numerator b1 = b1 or denominator b1 = 1)
holds b1 is integer set;
:: RAT_1:th 42
theorem
for b1 being rational set holds
numerator b1 = b1
iff
denominator b1 = 1;
:: RAT_1:th 44
theorem
for b1 being rational set
st (numerator b1 = b1 or denominator b1 = 1) & 0 <= b1
holds b1 is Element of NAT;
:: RAT_1:th 45
theorem
for b1 being rational set holds
1 < denominator b1
iff
b1 is not integer;
:: RAT_1:th 46
theorem
for b1 being rational set holds
(denominator b1) " < 1
iff
b1 is not integer;
:: RAT_1:th 47
theorem
for b1 being rational set holds
numerator b1 = denominator b1
iff
b1 = 1;
:: RAT_1:th 48
theorem
for b1 being rational set holds
numerator b1 = - denominator b1
iff
b1 = - 1;
:: RAT_1:th 49
theorem
for b1 being rational set holds
- numerator b1 = denominator b1
iff
b1 = - 1;
:: RAT_1:th 50
theorem
for b1 being integer set
for b2 being rational set
st b1 <> 0
holds b2 = ((numerator b2) * b1) / ((denominator b2) * b1);
:: RAT_1:th 60
theorem
for b1 being Element of NAT
for b2 being integer set
for b3 being rational set
st b1 <> 0 & b3 = b2 / b1
holds ex b4 being Element of NAT st
b2 = (numerator b3) * b4 & b1 = (denominator b3) * b4;
:: RAT_1:th 61
theorem
for b1, b2 being integer set
for b3 being rational set
st b3 = b1 / b2 & b2 <> 0
holds ex b4 being integer set st
b1 = (numerator b3) * b4 & b2 = (denominator b3) * b4;
:: RAT_1:th 62
theorem
for b1 being rational set
for b2 being Element of NAT
st 1 < b2
for b3 being integer set
for b4 being Element of NAT
st numerator b1 = b3 * b2
holds denominator b1 <> b4 * b2;
:: RAT_1:th 63
theorem
for b1 being Element of NAT
for b2 being integer set
for b3 being rational set
st b3 = b2 / b1 &
b1 <> 0 &
(for b4 being Element of NAT
st 1 < b4
for b5 being integer set
for b6 being Element of NAT
st b2 = b5 * b4
holds b1 <> b6 * b4)
holds b1 = denominator b3 & b2 = numerator b3;
:: RAT_1:th 64
theorem
for b1 being rational set holds
b1 < - 1
iff
numerator b1 < - denominator b1;
:: RAT_1:th 65
theorem
for b1 being rational set holds
b1 <= - 1
iff
numerator b1 <= - denominator b1;
:: RAT_1:th 66
theorem
for b1 being rational set holds
b1 < - 1
iff
denominator b1 < - numerator b1;
:: RAT_1:th 67
theorem
for b1 being rational set holds
b1 <= - 1
iff
denominator b1 <= - numerator b1;
:: RAT_1:th 72
theorem
for b1 being rational set holds
b1 < 1
iff
numerator b1 < denominator b1;
:: RAT_1:th 73
theorem
for b1 being rational set holds
b1 <= 1
iff
numerator b1 <= denominator b1;
:: RAT_1:th 76
theorem
for b1 being rational set holds
b1 < 0
iff
numerator b1 < 0;
:: RAT_1:th 77
theorem
for b1 being rational set holds
b1 <= 0
iff
numerator b1 <= 0;
:: RAT_1:th 80
theorem
for b1 being real set
for b2 being rational set holds
b1 < b2
iff
b1 * denominator b2 < numerator b2;
:: RAT_1:th 81
theorem
for b1 being real set
for b2 being rational set holds
b1 <= b2
iff
b1 * denominator b2 <= numerator b2;
:: RAT_1:th 84
theorem
for b1, b2 being rational set
st denominator b1 = denominator b2 & numerator b1 = numerator b2
holds b1 = b2;
:: RAT_1:th 86
theorem
for b1, b2 being rational set holds
b1 < b2
iff
(numerator b1) * denominator b2 < (numerator b2) * denominator b1;
:: RAT_1:th 87
theorem
for b1 being rational set holds
denominator - b1 = denominator b1 & numerator - b1 = - numerator b1;
:: RAT_1:th 88
theorem
for b1, b2 being rational set holds
0 < b1 & b2 = 1 / b1
iff
numerator b2 = denominator b1 & denominator b2 = numerator b1;
:: RAT_1:th 89
theorem
for b1, b2 being rational set holds
b1 < 0 & b2 = 1 / b1
iff
numerator b2 = - denominator b1 & denominator b2 = - numerator b1;