Article COMSEQ_1, MML version 4.99.1005
:: COMSEQ_1:modenot 1
definition
mode Complex_Sequence is Function-like quasi_total Relation of NAT,COMPLEX;
end;
:: COMSEQ_1:th 1
theorem
for b1 being Relation-like Function-like set holds
b1 is Function-like quasi_total Relation of NAT,COMPLEX
iff
proj1 b1 = NAT &
(for b2 being set
st b2 in NAT
holds b1 . b2 is Element of COMPLEX);
:: COMSEQ_1:th 2
theorem
for b1 being Relation-like Function-like set holds
b1 is Function-like quasi_total Relation of NAT,COMPLEX
iff
proj1 b1 = NAT &
(for b2 being Element of NAT holds
b1 . b2 is Element of COMPLEX);
:: COMSEQ_1:sch 1
scheme COMSEQ_1:sch 1
{F1 -> Element of COMPLEX}:
ex b1 being Function-like quasi_total Relation of NAT,COMPLEX st
for b2 being Element of NAT holds
b1 . b2 = F1(b2)
:: COMSEQ_1:attrnot 1 => COMSEQ_1:attr 1
definition
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
attr a1 is non-zero means
rng a1 c= COMPLEX \ {0c};
end;
:: COMSEQ_1:dfs 1
definiens
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
a1 is non-zero
it is sufficient to prove
thus rng a1 c= COMPLEX \ {0c};
:: COMSEQ_1:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is non-zero
iff
rng b1 c= COMPLEX \ {0c};
:: COMSEQ_1:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is non-zero
iff
for b2 being set
st b2 in NAT
holds b1 . b2 <> 0c;
:: COMSEQ_1:exreg 1
registration
cluster Relation-like Function-like non empty total quasi_total complex-valued non-zero Relation of NAT,COMPLEX;
end;
:: COMSEQ_1:th 4
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is non-zero
iff
for b2 being Element of NAT holds
b1 . b2 <> 0c;
:: COMSEQ_1:th 6
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st for b3 being Element of NAT holds
b1 . b3 = b2 . b3
holds b1 = b2;
:: COMSEQ_1:th 7
theorem
for b1 being Element of COMPLEX holds
ex b2 being Function-like quasi_total Relation of NAT,COMPLEX st
rng b2 = {b1};
:: COMSEQ_1:th 9
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 + b2) + b3 = b1 + (b2 + b3);
:: COMSEQ_1:th 11
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 (#) b2) (#) b3 = b1 (#) (b2 (#) b3);
:: COMSEQ_1:th 12
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 + b2) (#) b3 = (b1 (#) b3) + (b2 (#) b3);
:: COMSEQ_1:th 13
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 + b3) = (b1 (#) b2) + (b1 (#) b3);
:: COMSEQ_1:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
- b1 = (- 1r) (#) b1;
:: COMSEQ_1:th 15
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 (#) b3) = (b1 (#) b2) (#) b3;
:: COMSEQ_1:th 16
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 (#) b3) = b2 (#) (b1 (#) b3);
:: COMSEQ_1:th 17
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 - b2) (#) b3 = (b1 (#) b3) - (b2 (#) b3);
:: COMSEQ_1:th 18
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 (#) b2) - (b1 (#) b3) = b1 (#) (b2 - b3);
:: COMSEQ_1:th 19
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 + b3) = (b1 (#) b2) + (b1 (#) b3);
:: COMSEQ_1:th 20
theorem
for b1, b2 being Element of COMPLEX
for b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 * b2) (#) b3 = b1 (#) (b2 (#) b3);
:: COMSEQ_1:th 21
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 - b3) = (b1 (#) b2) - (b1 (#) b3);
:: COMSEQ_1:th 22
theorem
for b1 being Element of COMPLEX
for b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 /" b3) = (b1 (#) b2) /" b3;
:: COMSEQ_1:th 23
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 - (b2 + b3) = (b1 - b2) - b3;
:: COMSEQ_1:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
1r (#) b1 = b1;
:: COMSEQ_1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
- - b1 = b1;
:: COMSEQ_1:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 - - b2 = b1 + b2;
:: COMSEQ_1:th 27
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 - (b2 - b3) = (b1 - b2) + b3;
:: COMSEQ_1:th 28
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 + (b2 - b3) = (b1 + b2) - b3;
:: COMSEQ_1:th 29
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(- b1) (#) b2 = - (b1 (#) b2) & b1 (#) - b2 = - (b1 (#) b2);
:: COMSEQ_1:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds b1 " is non-zero;
:: COMSEQ_1:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 " " = b1;
:: COMSEQ_1:th 32
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is non-zero & b2 is non-zero
iff
b1 (#) b2 is non-zero;
:: COMSEQ_1:th 33
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds b1 " (#) (b2 ") = (b1 (#) b2) ";
:: COMSEQ_1:th 34
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds (b2 /" b1) (#) b1 = b2;
:: COMSEQ_1:th 35
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds (b3 /" b1) (#) (b4 /" b2) = (b3 (#) b4) /" (b1 (#) b2);
:: COMSEQ_1:th 36
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds b1 /" b2 is non-zero;
:: COMSEQ_1:th 37
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds (b1 /" b2) " = b2 /" b1;
:: COMSEQ_1:th 38
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 (#) (b2 /" b3) = (b1 (#) b2) /" b3;
:: COMSEQ_1:th 39
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds b3 /" (b1 /" b2) = (b3 (#) b2) /" b1;
:: COMSEQ_1:th 40
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds b3 /" b1 = (b3 (#) b2) /" (b1 (#) b2);
:: COMSEQ_1:th 41
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 <> 0c & b2 is non-zero
holds b1 (#) b2 is non-zero;
:: COMSEQ_1:th 42
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds - b1 is non-zero;
:: COMSEQ_1:th 43
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 (#) b2) " = b1 " (#) (b2 ");
:: COMSEQ_1:th 44
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds (- b1) " = (- 1r) (#) (b1 ");
:: COMSEQ_1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds - (b2 /" b1) = (- b2) /" b1 & b2 /" - b1 = - (b2 /" b1);
:: COMSEQ_1:th 46
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b1 /" b2) + (b3 /" b2) = (b1 + b3) /" b2 &
(b1 /" b2) - (b3 /" b2) = (b1 - b3) /" b2;
:: COMSEQ_1:th 47
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero
holds (b3 /" b1) + (b4 /" b2) = ((b3 (#) b2) + (b4 (#) b1)) /" (b1 (#) b2) &
(b3 /" b1) - (b4 /" b2) = ((b3 (#) b2) - (b4 (#) b1)) /" (b1 (#) b2);
:: COMSEQ_1:th 48
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero & b2 is non-zero & b3 is non-zero
holds (b4 /" b1) /" (b2 /" b3) = (b4 (#) b3) /" (b1 (#) b2);
:: COMSEQ_1:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
|.b1 (#) b2.| = |.b1.| (#) |.b2.|;
:: COMSEQ_1:th 50
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is non-zero
holds |.b1.| is non-empty;
:: COMSEQ_1:th 51
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
|.b1.| " = |.b1 ".|;
:: COMSEQ_1:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
|.b1 /" b2.| = |.b1.| /" |.b2.|;
:: COMSEQ_1:th 53
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX holds
|.b1 (#) b2.| = |.b1.| (#) |.b2.|;