Article TOPRNS_1, MML version 4.99.1005
:: TOPRNS_1:modenot 1
definition
let a1 be Element of NAT;
mode Real_Sequence of a1 is Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
end;
:: TOPRNS_1:th 2
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like set holds
b2 is Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
iff
proj1 b2 = NAT &
(for b3 being Element of NAT holds
b2 . b3 is Element of the carrier of TOP-REAL b1);
:: TOPRNS_1:attrnot 1 => TOPRNS_1:attr 1
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
attr a2 is non-zero means
proj2 a2 c= (the carrier of TOP-REAL a1) \ {0.REAL a1};
end;
:: TOPRNS_1:dfs 1
definiens
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
a2 is non-zero
it is sufficient to prove
thus proj2 a2 c= (the carrier of TOP-REAL a1) \ {0.REAL a1};
:: TOPRNS_1:def 1
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 is non-zero(b1)
iff
proj2 b2 c= (the carrier of TOP-REAL b1) \ {0.REAL b1};
:: TOPRNS_1:th 3
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 is non-zero(b1)
iff
for b3 being set
st b3 in NAT
holds b2 . b3 <> 0.REAL b1;
:: TOPRNS_1:th 4
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 is non-zero(b1)
iff
for b3 being Element of NAT holds
b2 . b3 <> 0.REAL b1;
:: TOPRNS_1:th 5
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st for b4 being set
st b4 in NAT
holds b2 . b4 = b3 . b4
holds b2 = b3;
:: TOPRNS_1:th 6
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st for b4 being Element of NAT holds
b2 . b4 = b3 . b4
holds b2 = b3;
:: TOPRNS_1:sch 1
scheme TOPRNS_1:sch 1
{F1 -> Element of NAT,
F2 -> Element of the carrier of TOP-REAL F1()}:
ex b1 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL F1() st
for b2 being Element of NAT holds
b1 . b2 = F2(b2)
:: TOPRNS_1:funcnot 1 => TOPRNS_1:func 1
definition
let a1 be Element of NAT;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
func A2 + A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) + (a3 . b1);
end;
:: TOPRNS_1:def 2
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b4 = b2 + b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) + (b3 . b5);
:: TOPRNS_1:funcnot 2 => TOPRNS_1:func 2
definition
let a1 be Element of REAL;
let a2 be Element of NAT;
let a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a2;
func A1 * A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a2 means
for b1 being Element of NAT holds
it . b1 = a1 * (a3 . b1);
end;
:: TOPRNS_1:def 3
theorem
for b1 being Element of REAL
for b2 being Element of NAT
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b2 holds
b4 = b1 * b3
iff
for b5 being Element of NAT holds
b4 . b5 = b1 * (b3 . b5);
:: TOPRNS_1:funcnot 3 => TOPRNS_1:func 3
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
func - A2 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 means
for b1 being Element of NAT holds
it . b1 = - (a2 . b1);
end;
:: TOPRNS_1:def 4
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b3 = - b2
iff
for b4 being Element of NAT holds
b3 . b4 = - (b2 . b4);
:: TOPRNS_1:funcnot 4 => TOPRNS_1:func 4
definition
let a1 be Element of NAT;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
func A2 - A3 -> Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1 equals
a2 + - a3;
end;
:: TOPRNS_1:def 5
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - b3 = b2 + - b3;
:: TOPRNS_1:funcnot 5 => TOPRNS_1:func 5
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
func |.A2.| -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = |.a2 . b1.|;
end;
:: TOPRNS_1:def 7
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = |.b2.|
iff
for b4 being Element of NAT holds
b3 . b4 = |.b2 . b4.|;
:: TOPRNS_1:th 8
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Element of the carrier of TOP-REAL b1 holds
(abs b2) * |.b3.| = |.b2 * b3.|;
:: TOPRNS_1:th 9
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
|.b2 * b3.| = (abs b2) (#) |.b3.|;
:: TOPRNS_1:th 10
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 + b3 = b3 + b2;
:: TOPRNS_1:th 11
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);
:: TOPRNS_1:th 12
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
- b2 = (- 1) * b2;
:: TOPRNS_1:th 13
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);
:: TOPRNS_1:th 14
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4);
:: TOPRNS_1:th 15
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 * (b3 - b4) = (b2 * b3) - (b2 * b4);
:: TOPRNS_1:th 16
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - (b3 + b4) = (b2 - b3) - b4;
:: TOPRNS_1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
1 * b2 = b2;
:: TOPRNS_1:th 18
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
- - b2 = b2;
:: TOPRNS_1:th 19
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - - b3 = b2 + b3;
:: TOPRNS_1:th 20
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 - (b3 - b4) = (b2 - b3) + b4;
:: TOPRNS_1:th 21
theorem
for b1 being Element of NAT
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 + (b3 - b4) = (b2 + b3) - b4;
:: TOPRNS_1:th 22
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 <> 0 & b3 is non-zero(b1)
holds b2 * b3 is non-zero(b1);
:: TOPRNS_1:th 23
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is non-zero(b1)
holds - b2 is non-zero(b1);
:: TOPRNS_1:th 24
theorem
for b1 being Element of NAT holds
|.0.REAL b1.| = 0;
:: TOPRNS_1:th 25
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
st |.b2.| = 0
holds b2 = 0.REAL b1;
:: TOPRNS_1:th 26
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
0 <= |.b2.|;
:: TOPRNS_1:th 27
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
|.- b2.| = |.b2.|;
:: TOPRNS_1:th 28
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| = |.b3 - b2.|;
:: TOPRNS_1:th 29
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| = 0
iff
b2 = b3;
:: TOPRNS_1:th 30
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| <= |.b2.| + |.b3.|;
:: TOPRNS_1:th 31
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| <= |.b2.| + |.b3.|;
:: TOPRNS_1:th 32
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2.| - |.b3.| <= |.b2 + b3.|;
:: TOPRNS_1:th 33
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2.| - |.b3.| <= |.b2 - b3.|;
:: TOPRNS_1:th 34
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2 <> b3
holds 0 < |.b2 - b3.|;
:: TOPRNS_1:th 35
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| <= |.b2 - b4.| + |.b4 - b3.|;
:: TOPRNS_1:th 36
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
for b4, b5 being Element of the carrier of TOP-REAL b1
st 0 <= b2 & |.b4.| < |.b5.| & b2 < b3
holds |.b4.| * b2 < |.b5.| * b3;
:: TOPRNS_1:th 38
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Element of the carrier of TOP-REAL b1 holds
- |.b3.| < b2 & b2 < |.b3.|
iff
abs b2 < |.b3.|;
:: TOPRNS_1:attrnot 2 => TOPRNS_1:attr 2
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
attr a2 is bounded means
ex b1 being Element of REAL st
for b2 being Element of NAT holds
|.a2 . b2.| < b1;
end;
:: TOPRNS_1:dfs 7
definiens
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
a2 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
for b2 being Element of NAT holds
|.a2 . b2.| < b1;
:: TOPRNS_1:def 8
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
for b4 being Element of NAT holds
|.b2 . b4.| < b3;
:: TOPRNS_1:th 39
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Element of NAT holds
ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of NAT
st b5 <= b3
holds |.b2 . b5.| < b4);
:: TOPRNS_1:attrnot 3 => TOPRNS_1:attr 3
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
attr a2 is convergent means
ex b1 being Element of the carrier of TOP-REAL a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds |.(a2 . b4) - b1.| < b2;
end;
:: TOPRNS_1:dfs 8
definiens
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
To prove
a2 is convergent
it is sufficient to prove
thus ex b1 being Element of the carrier of TOP-REAL a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds |.(a2 . b4) - b1.| < b2;
:: TOPRNS_1:def 9
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 holds
b2 is convergent(b1)
iff
ex b3 being Element of the carrier of TOP-REAL b1 st
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds |.(b2 . b6) - b3.| < b4;
:: TOPRNS_1:funcnot 6 => TOPRNS_1:func 6
definition
let a1 be Element of NAT;
let a2 be Function-like quasi_total Relation of NAT,the carrier of TOP-REAL a1;
assume a2 is convergent(a1);
func lim A2 -> Element of the carrier of TOP-REAL a1 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds |.(a2 . b3) - it.| < b1;
end;
:: TOPRNS_1:def 10
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1)
for b3 being Element of the carrier of TOP-REAL b1 holds
b3 = lim b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds |.(b2 . b6) - b3.| < b4;
:: TOPRNS_1:th 41
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 + b3 is convergent(b1);
:: TOPRNS_1:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 + b3) = (lim b2) + lim b3;
:: TOPRNS_1:th 43
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b3 is convergent(b1)
holds b2 * b3 is convergent(b1);
:: TOPRNS_1:th 44
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b3 is convergent(b1)
holds lim (b2 * b3) = b2 * lim b3;
:: TOPRNS_1:th 45
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1)
holds - b2 is convergent(b1);
:: TOPRNS_1:th 46
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1)
holds lim - b2 = - lim b2;
:: TOPRNS_1:th 47
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 - b3 is convergent(b1);
:: TOPRNS_1:th 48
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 - b3) = (lim b2) - lim b3;
:: TOPRNS_1:th 50
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1)
holds b2 is bounded(b1);
:: TOPRNS_1:th 51
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
st b2 is convergent(b1) & lim b2 <> 0.REAL b1
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds |.lim b2.| / 2 < |.b2 . b4.|;