Article PREPOWER, MML version 4.99.1005
:: PREPOWER:funcreg 1
registration
let a1 be integer set;
cluster |.a1.| -> natural complex;
end;
:: PREPOWER:th 2
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent &
(for b3 being Element of NAT holds
b1 <= b2 . b3)
holds b1 <= lim b2;
:: PREPOWER:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent &
(for b3 being Element of NAT holds
b2 . b3 <= b1)
holds lim b2 <= b1;
:: PREPOWER:funcnot 1 => PREPOWER:func 1
definition
let a1 be real set;
func A1 GeoSeq -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = a1 |^ b1;
end;
:: PREPOWER:def 1
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 = b1 GeoSeq
iff
for b3 being Element of NAT holds
b2 . b3 = b1 |^ b3;
:: PREPOWER:th 4
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 = b1 GeoSeq
iff
b2 . 0 = 1 &
(for b3 being Element of NAT holds
b2 . (b3 + 1) = (b2 . b3) * b1);
:: PREPOWER:th 5
theorem
for b1 being real set
st b1 <> 0
for b2 being Element of NAT holds
b1 GeoSeq . b2 <> 0;
:: PREPOWER:th 12
theorem
for b1 being real set
for b2 being natural set
st 0 <> b1
holds 0 <> b1 |^ b2;
:: PREPOWER:th 13
theorem
for b1 being real set
for b2 being natural set
st 0 < b1
holds 0 < b1 |^ b2;
:: PREPOWER:th 14
theorem
for b1 being real set
for b2 being natural set holds
(1 / b1) |^ b2 = 1 / (b1 |^ b2);
:: PREPOWER:th 15
theorem
for b1, b2 being real set
for b3 being natural set holds
(b1 / b2) |^ b3 = (b1 |^ b3) / (b2 |^ b3);
:: PREPOWER:th 17
theorem
for b1, b2 being real set
for b3 being natural set
st 0 < b1 & b1 <= b2
holds b1 |^ b3 <= b2 |^ b3;
:: PREPOWER:th 18
theorem
for b1, b2 being real set
for b3 being natural set
st 0 <= b1 & b1 < b2 & 1 <= b3
holds b1 |^ b3 < b2 |^ b3;
:: PREPOWER:th 19
theorem
for b1 being real set
for b2 being natural set
st 1 <= b1
holds 1 <= b1 |^ b2;
:: PREPOWER:th 20
theorem
for b1 being real set
for b2 being natural set
st 1 <= b1 & 1 <= b2
holds b1 <= b1 |^ b2;
:: PREPOWER:th 21
theorem
for b1 being real set
for b2 being natural set
st 1 < b1 & 2 <= b2
holds b1 < b1 |^ b2;
:: PREPOWER:th 22
theorem
for b1 being real set
for b2 being natural set
st 0 < b1 & b1 <= 1 & 1 <= b2
holds b1 |^ b2 <= b1;
:: PREPOWER:th 23
theorem
for b1 being real set
for b2 being natural set
st 0 < b1 & b1 < 1 & 2 <= b2
holds b1 |^ b2 < b1;
:: PREPOWER:th 24
theorem
for b1 being real set
for b2 being natural set
st - 1 < b1
holds 1 + (b2 * b1) <= (1 + b1) |^ b2;
:: PREPOWER:th 25
theorem
for b1 being real set
for b2 being natural set
st 0 < b1 & b1 < 1
holds (1 + b1) |^ b2 <= 1 + ((3 |^ b2) * b1);
:: PREPOWER:th 26
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent &
(for b4 being Element of NAT holds
b3 . b4 = (b2 . b4) |^ b1)
holds b3 is convergent & lim b3 = (lim b2) |^ b1;
:: PREPOWER:funcnot 2 => PREPOWER:func 2
definition
let a1 be natural set;
let a2 be real set;
assume 1 <= a1;
func A1 -Root A2 -> real set means
it |^ a1 = a2 & 0 < it
if 0 < a2
otherwise case a2 = 0;
thus it = 0;
end;
;
end;
:: PREPOWER:def 3
theorem
for b1 being natural set
for b2 being real set
st 1 <= b1
for b3 being real set holds
(b2 <= 0 or (b3 = b1 -Root b2
iff
b3 |^ b1 = b2 & 0 < b3)) &
(b2 = 0 implies (b3 = b1 -Root b2
iff
b3 = 0));
:: PREPOWER:funcnot 3 => PREPOWER:func 3
definition
let a1 be Element of NAT;
let a2 be Element of REAL;
redefine func a1 -Root a2 -> Element of REAL;
end;
:: PREPOWER:th 28
theorem
for b1 being real set
for b2 being natural set
st 0 <= b1 & 1 <= b2
holds (b2 -Root b1) |^ b2 = b1 & b2 -Root (b1 |^ b2) = b1;
:: PREPOWER:th 29
theorem
for b1 being Element of NAT
st 1 <= b1
holds b1 -Root 1 = 1;
:: PREPOWER:th 30
theorem
for b1 being real set
st 0 <= b1
holds 1 -Root b1 = b1;
:: PREPOWER:th 31
theorem
for b1, b2 being real set
for b3 being Element of NAT
st 0 <= b1 & 0 <= b2 & 1 <= b3
holds b3 -Root (b1 * b2) = (b3 -Root b1) * (b3 -Root b2);
:: PREPOWER:th 32
theorem
for b1 being real set
for b2 being Element of NAT
st 0 < b1 & 1 <= b2
holds b2 -Root (1 / b1) = 1 / (b2 -Root b1);
:: PREPOWER:th 33
theorem
for b1, b2 being real set
for b3 being Element of NAT
st 0 <= b1 & 0 < b2 & 1 <= b3
holds b3 -Root (b1 / b2) = (b3 -Root b1) / (b3 -Root b2);
:: PREPOWER:th 34
theorem
for b1 being real set
for b2, b3 being Element of NAT
st 0 <= b1 & 1 <= b2 & 1 <= b3
holds b2 -Root (b3 -Root b1) = (b2 * b3) -Root b1;
:: PREPOWER:th 35
theorem
for b1 being real set
for b2, b3 being Element of NAT
st 0 <= b1 & 1 <= b2 & 1 <= b3
holds (b2 -Root b1) * (b3 -Root b1) = (b2 * b3) -Root (b1 |^ (b2 + b3));
:: PREPOWER:th 36
theorem
for b1, b2 being real set
for b3 being Element of NAT
st 0 <= b1 & b1 <= b2 & 1 <= b3
holds b3 -Root b1 <= b3 -Root b2;
:: PREPOWER:th 37
theorem
for b1, b2 being real set
for b3 being Element of NAT
st 0 <= b1 & b1 < b2 & 1 <= b3
holds b3 -Root b1 < b3 -Root b2;
:: PREPOWER:th 38
theorem
for b1 being real set
for b2 being Element of NAT
st 1 <= b1 & 1 <= b2
holds 1 <= b2 -Root b1 & b2 -Root b1 <= b1;
:: PREPOWER:th 39
theorem
for b1 being real set
for b2 being Element of NAT
st 0 <= b1 & b1 < 1 & 1 <= b2
holds b1 <= b2 -Root b1 & b2 -Root b1 < 1;
:: PREPOWER:th 40
theorem
for b1 being real set
for b2 being Element of NAT
st 0 < b1 & 1 <= b2
holds (b2 -Root b1) - 1 <= (b1 - 1) / b2;
:: PREPOWER:th 41
theorem
for b1 being real set
st 0 <= b1
holds 2 -Root b1 = sqrt b1;
:: PREPOWER:th 42
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st 0 < b1 &
(for b3 being Element of NAT
st 1 <= b3
holds b2 . b3 = b3 -Root b1)
holds b2 is convergent & lim b2 = 1;
:: PREPOWER:funcnot 4 => PREPOWER:func 4
definition
let a1 be real set;
let a2 be integer set;
func A1 #Z A2 -> set equals
a1 |^ abs a2
if 0 <= a2
otherwise case a2 < 0;
thus (a1 |^ abs a2) ";
end;
;
end;
:: PREPOWER:def 4
theorem
for b1 being real set
for b2 being integer set holds
(0 <= b2 implies b1 #Z b2 = b1 |^ abs b2) &
(0 <= b2 or b1 #Z b2 = (b1 |^ abs b2) ");
:: PREPOWER:funcreg 2
registration
let a1 be real set;
let a2 be integer set;
cluster a1 #Z a2 -> real;
end;
:: PREPOWER:funcnot 5 => PREPOWER:func 5
definition
let a1 be Element of REAL;
let a2 be integer set;
redefine func a1 #Z a2 -> Element of REAL;
end;
:: PREPOWER:th 44
theorem
for b1 being real set holds
b1 #Z 0 = 1;
:: PREPOWER:th 45
theorem
for b1 being real set holds
b1 #Z 1 = b1;
:: PREPOWER:th 46
theorem
for b1 being real set
for b2 being natural set holds
b1 #Z b2 = b1 |^ b2;
:: PREPOWER:th 47
theorem
for b1 being integer set holds
1 #Z b1 = 1;
:: PREPOWER:th 48
theorem
for b1 being real set
for b2 being integer set
st b1 <> 0
holds b1 #Z b2 <> 0;
:: PREPOWER:th 49
theorem
for b1 being real set
for b2 being integer set
st 0 < b1
holds 0 < b1 #Z b2;
:: PREPOWER:th 50
theorem
for b1, b2 being real set
for b3 being integer set holds
(b1 * b2) #Z b3 = (b1 #Z b3) * (b2 #Z b3);
:: PREPOWER:th 51
theorem
for b1 being real set
for b2 being integer set
st b1 <> 0
holds b1 #Z - b2 = 1 / (b1 #Z b2);
:: PREPOWER:th 52
theorem
for b1 being real set
for b2 being integer set holds
(1 / b1) #Z b2 = 1 / (b1 #Z b2);
:: PREPOWER:th 53
theorem
for b1 being real set
for b2, b3 being natural set
st b1 <> 0
holds b1 #Z (b2 - b3) = (b1 |^ b2) / (b1 |^ b3);
:: PREPOWER:th 54
theorem
for b1 being real set
for b2, b3 being integer set
st b1 <> 0
holds b1 #Z (b2 + b3) = (b1 #Z b2) * (b1 #Z b3);
:: PREPOWER:th 55
theorem
for b1 being real set
for b2, b3 being integer set holds
(b1 #Z b2) #Z b3 = b1 #Z (b2 * b3);
:: PREPOWER:th 56
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being integer set
st 0 < b1 & 1 <= b2
holds (b2 -Root b1) #Z b3 = b2 -Root (b1 #Z b3);
:: PREPOWER:funcnot 6 => PREPOWER:func 6
definition
let a1 be real set;
let a2 be rational set;
func A1 #Q A2 -> set equals
(denominator a2) -Root (a1 #Z numerator a2);
end;
:: PREPOWER:def 5
theorem
for b1 being real set
for b2 being rational set holds
b1 #Q b2 = (denominator b2) -Root (b1 #Z numerator b2);
:: PREPOWER:funcreg 3
registration
let a1 be real set;
let a2 be rational set;
cluster a1 #Q a2 -> real;
end;
:: PREPOWER:funcnot 7 => PREPOWER:func 7
definition
let a1 be Element of REAL;
let a2 be rational set;
redefine func a1 #Q a2 -> Element of REAL;
end;
:: PREPOWER:th 58
theorem
for b1 being real set
for b2 being rational set
st 0 < b1 & b2 = 0
holds b1 #Q b2 = 1;
:: PREPOWER:th 59
theorem
for b1 being real set
for b2 being rational set
st 0 < b1 & b2 = 1
holds b1 #Q b2 = b1;
:: PREPOWER:th 60
theorem
for b1 being real set
for b2 being rational set
for b3 being natural set
st 0 < b1 & b2 = b3
holds b1 #Q b2 = b1 |^ b3;
:: PREPOWER:th 61
theorem
for b1 being real set
for b2 being rational set
for b3 being natural set
st 0 < b1 & 1 <= b3 & b2 = b3 "
holds b1 #Q b2 = b3 -Root b1;
:: PREPOWER:th 62
theorem
for b1 being rational set holds
1 #Q b1 = 1;
:: PREPOWER:th 63
theorem
for b1 being real set
for b2 being rational set
st 0 < b1
holds 0 < b1 #Q b2;
:: PREPOWER:th 64
theorem
for b1 being real set
for b2, b3 being rational set
st 0 < b1
holds (b1 #Q b2) * (b1 #Q b3) = b1 #Q (b2 + b3);
:: PREPOWER:th 65
theorem
for b1 being real set
for b2 being rational set
st 0 < b1
holds 1 / (b1 #Q b2) = b1 #Q - b2;
:: PREPOWER:th 66
theorem
for b1 being real set
for b2, b3 being rational set
st 0 < b1
holds (b1 #Q b2) / (b1 #Q b3) = b1 #Q (b2 - b3);
:: PREPOWER:th 67
theorem
for b1, b2 being real set
for b3 being rational set
st 0 < b1 & 0 < b2
holds (b1 * b2) #Q b3 = (b1 #Q b3) * (b2 #Q b3);
:: PREPOWER:th 68
theorem
for b1 being real set
for b2 being rational set
st 0 < b1
holds (1 / b1) #Q b2 = 1 / (b1 #Q b2);
:: PREPOWER:th 69
theorem
for b1, b2 being real set
for b3 being rational set
st 0 < b1 & 0 < b2
holds (b1 / b2) #Q b3 = (b1 #Q b3) / (b2 #Q b3);
:: PREPOWER:th 70
theorem
for b1 being real set
for b2, b3 being rational set
st 0 < b1
holds (b1 #Q b2) #Q b3 = b1 #Q (b2 * b3);
:: PREPOWER:th 71
theorem
for b1 being real set
for b2 being rational set
st 1 <= b1 & 0 <= b2
holds 1 <= b1 #Q b2;
:: PREPOWER:th 72
theorem
for b1 being real set
for b2 being rational set
st 1 <= b1 & b2 <= 0
holds b1 #Q b2 <= 1;
:: PREPOWER:th 73
theorem
for b1 being real set
for b2 being rational set
st 1 < b1 & 0 < b2
holds 1 < b1 #Q b2;
:: PREPOWER:th 74
theorem
for b1 being real set
for b2, b3 being rational set
st 1 <= b1 & b3 <= b2
holds b1 #Q b3 <= b1 #Q b2;
:: PREPOWER:th 75
theorem
for b1 being real set
for b2, b3 being rational set
st 1 < b1 & b3 < b2
holds b1 #Q b3 < b1 #Q b2;
:: PREPOWER:th 76
theorem
for b1 being real set
for b2 being rational set
st 0 < b1 & b1 < 1 & 0 < b2
holds b1 #Q b2 < 1;
:: PREPOWER:th 77
theorem
for b1 being real set
for b2 being rational set
st 0 < b1 & b1 <= 1 & b2 <= 0
holds 1 <= b1 #Q b2;
:: PREPOWER:attrnot 1 => PREPOWER:attr 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is Rational_Sequence-like means
for b1 being Element of NAT holds
a1 . b1 is rational set;
end;
:: PREPOWER:dfs 5
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is Rational_Sequence-like
it is sufficient to prove
thus for b1 being Element of NAT holds
a1 . b1 is rational set;
:: PREPOWER:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is Rational_Sequence-like
iff
for b2 being Element of NAT holds
b1 . b2 is rational set;
:: PREPOWER:exreg 1
registration
cluster Relation-like Function-like non empty quasi_total complex-valued ext-real-valued real-valued total Rational_Sequence-like Relation of NAT,REAL;
end;
:: PREPOWER:modenot 1
definition
mode Rational_Sequence is Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL;
end;
:: PREPOWER:funcnot 8 => PREPOWER:func 8
definition
let a1 be Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL;
let a2 be Element of NAT;
redefine func a1 . a2 -> rational set;
end;
:: PREPOWER:th 79
theorem
for b1 being real set holds
ex b2 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL st
b2 is convergent &
lim b2 = b1 &
(for b3 being Element of NAT holds
b2 . b3 <= b1);
:: PREPOWER:th 80
theorem
for b1 being real set holds
ex b2 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL st
b2 is convergent &
lim b2 = b1 &
(for b3 being Element of NAT holds
b1 <= b2 . b3);
:: PREPOWER:funcnot 9 => PREPOWER:func 9
definition
let a1 be real set;
let a2 be Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL;
func A1 #Q A2 -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = a1 #Q (a2 . b1);
end;
:: PREPOWER:def 7
theorem
for b1 being real set
for b2 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = b1 #Q b2
iff
for b4 being Element of NAT holds
b3 . b4 = b1 #Q (b2 . b4);
:: PREPOWER:th 82
theorem
for b1 being real set
for b2 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL
st b2 is convergent & 0 < b1
holds b1 #Q b2 is convergent;
:: PREPOWER:th 83
theorem
for b1, b2 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL
for b3 being real set
st b1 is convergent & b2 is convergent & lim b1 = lim b2 & 0 < b3
holds b3 #Q b1 is convergent & b3 #Q b2 is convergent & lim (b3 #Q b1) = lim (b3 #Q b2);
:: PREPOWER:funcnot 10 => PREPOWER:func 10
definition
let a1, a2 be real set;
assume 0 < a1;
func A1 #R A2 -> real set means
ex b1 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL st
b1 is convergent & lim b1 = a2 & a1 #Q b1 is convergent & lim (a1 #Q b1) = it;
end;
:: PREPOWER:def 8
theorem
for b1, b2 being real set
st 0 < b1
for b3 being real set holds
b3 = b1 #R b2
iff
ex b4 being Function-like quasi_total Rational_Sequence-like Relation of NAT,REAL st
b4 is convergent & lim b4 = b2 & b1 #Q b4 is convergent & lim (b1 #Q b4) = b3;
:: PREPOWER:funcnot 11 => PREPOWER:func 11
definition
let a1, a2 be Element of REAL;
redefine func a1 #R a2 -> Element of REAL;
end;
:: PREPOWER:th 85
theorem
for b1 being real set
st 0 < b1
holds b1 #R 0 = 1;
:: PREPOWER:th 86
theorem
for b1 being real set
st 0 < b1
holds b1 #R 1 = b1;
:: PREPOWER:th 87
theorem
for b1 being real set holds
1 #R b1 = 1;
:: PREPOWER:th 88
theorem
for b1 being real set
for b2 being rational set
st 0 < b1
holds b1 #R b2 = b1 #Q b2;
:: PREPOWER:th 89
theorem
for b1, b2, b3 being real set
st 0 < b1
holds b1 #R (b2 + b3) = (b1 #R b2) * (b1 #R b3);
:: PREPOWER:th 90
theorem
for b1, b2 being real set
st 0 < b1
holds b1 #R - b2 = 1 / (b1 #R b2);
:: PREPOWER:th 91
theorem
for b1, b2, b3 being real set
st 0 < b1
holds b1 #R (b2 - b3) = (b1 #R b2) / (b1 #R b3);
:: PREPOWER:th 92
theorem
for b1, b2, b3 being real set
st 0 < b1 & 0 < b2
holds (b1 * b2) #R b3 = (b1 #R b3) * (b2 #R b3);
:: PREPOWER:th 93
theorem
for b1, b2 being real set
st 0 < b1
holds (1 / b1) #R b2 = 1 / (b1 #R b2);
:: PREPOWER:th 94
theorem
for b1, b2, b3 being real set
st 0 < b1 & 0 < b2
holds (b1 / b2) #R b3 = (b1 #R b3) / (b2 #R b3);
:: PREPOWER:th 95
theorem
for b1, b2 being real set
st 0 < b1
holds 0 < b1 #R b2;
:: PREPOWER:th 96
theorem
for b1, b2, b3 being real set
st 1 <= b1 & b3 <= b2
holds b1 #R b3 <= b1 #R b2;
:: PREPOWER:th 97
theorem
for b1, b2, b3 being real set
st 1 < b1 & b3 < b2
holds b1 #R b3 < b1 #R b2;
:: PREPOWER:th 98
theorem
for b1, b2, b3 being real set
st 0 < b1 & b1 <= 1 & b3 <= b2
holds b1 #R b2 <= b1 #R b3;
:: PREPOWER:th 99
theorem
for b1, b2 being real set
st 1 <= b1 & 0 <= b2
holds 1 <= b1 #R b2;
:: PREPOWER:th 100
theorem
for b1, b2 being real set
st 1 < b1 & 0 < b2
holds 1 < b1 #R b2;
:: PREPOWER:th 101
theorem
for b1, b2 being real set
st 1 <= b1 & b2 <= 0
holds b1 #R b2 <= 1;
:: PREPOWER:th 102
theorem
for b1, b2 being real set
st 1 < b1 & b2 < 0
holds b1 #R b2 < 1;
:: PREPOWER:th 103
theorem
for b1 being rational set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent &
b3 is convergent &
0 < lim b2 &
(for b4 being Element of NAT holds
0 < b2 . b4 & b3 . b4 = (b2 . b4) #Q b1)
holds lim b3 = (lim b2) #Q b1;
:: PREPOWER:th 104
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
st 0 < b1 &
b2 is convergent &
b3 is convergent &
(for b4 being Element of NAT holds
b3 . b4 = b1 #R (b2 . b4))
holds lim b3 = b1 #R lim b2;
:: PREPOWER:th 105
theorem
for b1, b2, b3 being real set
st 0 < b1
holds (b1 #R b2) #R b3 = b1 #R (b2 * b3);
:: PREPOWER:th 106
theorem
for b1, b2 being real set
st 0 < b1 & 0 < b2
holds ex b3 being Element of NAT st
b2 / (2 |^ b3) <= b1;
:: PREPOWER:th 107
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being natural set
st b1 <= b3 & 1 <= b2
holds b2 |^ b1 <= b2 |^ b3;
:: PREPOWER:th 108
theorem
for b1, b2, b3 being Element of NAT
st b1 divides b2 & b1 divides b3
holds b1 divides b2 - b3;
:: PREPOWER:th 109
theorem
for b1, b2 being Element of NAT holds
b1 divides b2
iff
b1 divides b2;
:: PREPOWER:th 110
theorem
for b1, b2 being Element of NAT holds
b1 hcf b2 = b1 hcf abs (b2 - b1);
:: PREPOWER:th 111
theorem
for b1, b2 being integer set
st 0 <= b1 & 0 <= b2
holds b1 gcd b2 = b1 gcd (b2 - b1);