Article COMPLSP2, MML version 4.99.1005
:: COMPLSP2:funcnot 1 => COMPLSP2:func 1
definition
let a1 be FinSequence of COMPLEX;
func A1 *' -> FinSequence of COMPLEX means
len it = len a1 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a1
holds it . b1 = (a1 . b1) *');
end;
:: COMPLSP2:def 1
theorem
for b1, b2 being FinSequence of COMPLEX holds
b2 = b1 *'
iff
len b2 = len b1 &
(for b3 being natural set
st 1 <= b3 & b3 <= len b1
holds b2 . b3 = (b1 . b3) *');
:: COMPLSP2:th 1
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
st b1 in dom (b2 + b3)
holds (b2 + b3) . b1 = (b2 . b1) + (b3 . b1);
:: COMPLSP2:th 2
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
st b1 in dom (b2 - b3)
holds (b2 - b3) . b1 = (b2 . b1) - (b3 . b1);
:: COMPLSP2:funcnot 2 => COMPLSP2:func 2
definition
let a1 be Element of NAT;
let a2, a3 be Element of a1 -tuples_on COMPLEX;
redefine func a2 - a3 -> Element of a1 -tuples_on COMPLEX;
end;
:: COMPLSP2:funcnot 3 => COMPLSP2:func 3
definition
let a1 be Element of NAT;
let a2, a3 be Element of a1 -tuples_on COMPLEX;
redefine func a2 + a3 -> Element of a1 -tuples_on COMPLEX;
commutativity;
:: for a1 being Element of NAT
:: for a2, a3 being Element of a1 -tuples_on COMPLEX holds
:: a2 + a3 = a3 + a2;
end;
:: COMPLSP2:funcnot 4 => COMPLSP2:func 4
definition
let a1 be Element of NAT;
let a2 be Element of a1 -tuples_on COMPLEX;
let a3 be complex set;
redefine func a3 * a2 -> Element of a1 -tuples_on COMPLEX;
end;
:: COMPLSP2:th 3
theorem
for b1 being complex set
for b2 being FinSequence of COMPLEX holds
len (b1 * b2) = len b2;
:: COMPLSP2:th 4
theorem
for b1 being FinSequence of COMPLEX holds
dom b1 = dom - b1;
:: COMPLSP2:th 5
theorem
for b1 being FinSequence of COMPLEX holds
len - b1 = len b1;
:: COMPLSP2:th 6
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds len (b1 + b2) = len b1;
:: COMPLSP2:th 7
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds len (b1 - b2) = len b1;
:: COMPLSP2:th 8
theorem
for b1 being FinSequence of COMPLEX holds
b1 is Element of COMPLEX len b1;
:: COMPLSP2:th 9
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -tuples_on COMPLEX holds
b2 - b3 = b2 + - b3;
:: COMPLSP2:th 10
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds b1 - b2 = b1 + - b2;
:: COMPLSP2:th 11
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on COMPLEX holds
(- 1) * b2 = - b2;
:: COMPLSP2:th 12
theorem
for b1 being FinSequence of COMPLEX holds
(- 1) * b1 = - b1;
:: COMPLSP2:th 13
theorem
for b1 being Element of NAT
for b2 being FinSequence of COMPLEX holds
(- b2) . b1 = - (b2 . b1);
:: COMPLSP2:funcnot 5 => COMPLSP2:func 5
definition
let a1 be Element of NAT;
let a2 be Element of a1 -tuples_on COMPLEX;
redefine func - a2 -> Element of a1 -tuples_on COMPLEX;
involutiveness;
:: for a1 being Element of NAT
:: for a2 being Element of a1 -tuples_on COMPLEX holds
:: - - a2 = a2;
end;
:: COMPLSP2:th 14
theorem
for b1, b2 being Element of NAT
for b3 being Element of COMPLEX
for b4 being Element of b1 -tuples_on COMPLEX
st b3 = b4 . b2
holds (- b4) . b2 = - b3;
:: COMPLSP2:th 15
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
dom (b2 * b1) = dom b1;
:: COMPLSP2:th 16
theorem
for b1 being FinSequence of COMPLEX
for b2 being natural set
for b3 being complex set holds
(b3 * b1) . b2 = b3 * (b1 . b2);
:: COMPLSP2:th 17
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
(b2 * b1) *' = b2 *' * (b1 *');
:: COMPLSP2:th 18
theorem
for b1, b2 being Element of NAT
for b3, b4 being Element of b1 -tuples_on COMPLEX holds
(b3 + b4) . b2 = (b3 . b2) + (b4 . b2);
:: COMPLSP2:th 19
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds (b1 + b2) *' = b1 *' + (b2 *');
:: COMPLSP2:th 20
theorem
for b1, b2 being Element of NAT
for b3, b4 being Element of b1 -tuples_on COMPLEX holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);
:: COMPLSP2:th 21
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds (b1 - b2) *' = b1 *' - (b2 *');
:: COMPLSP2:th 22
theorem
for b1 being FinSequence of COMPLEX holds
b1 *' *' = b1;
:: COMPLSP2:th 23
theorem
for b1 being FinSequence of COMPLEX holds
(- b1) *' = - (b1 *');
:: COMPLSP2:th 24
theorem
for b1 being complex set holds
b1 + (b1 *') = 2 * Re b1;
:: COMPLSP2:th 25
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds (b2 - b3) . b1 = (b2 . b1) - (b3 . b1);
:: COMPLSP2:th 26
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds (b2 + b3) . b1 = (b2 . b1) + (b3 . b1);
:: COMPLSP2:funcnot 6 => COMPLSP2:func 6
definition
let a1 be FinSequence of COMPLEX;
func Re A1 -> FinSequence of REAL equals
(1 / 2) * (a1 + (a1 *'));
end;
:: COMPLSP2:def 2
theorem
for b1 being FinSequence of COMPLEX holds
Re b1 = (1 / 2) * (b1 + (b1 *'));
:: COMPLSP2:th 27
theorem
for b1 being complex set holds
b1 - (b1 *') = (2 * Im b1) * <i>;
:: COMPLSP2:funcnot 7 => COMPLSP2:func 7
definition
let a1 be FinSequence of COMPLEX;
func Im A1 -> FinSequence of REAL equals
(- ((1 / 2) * <i>)) * (a1 - (a1 *'));
end;
:: COMPLSP2:def 3
theorem
for b1 being FinSequence of COMPLEX holds
Im b1 = (- ((1 / 2) * <i>)) * (b1 - (b1 *'));
:: COMPLSP2:funcnot 8 => COMPLSP2:func 8
definition
let a1, a2 be FinSequence of COMPLEX;
func |(A1,A2)| -> Element of COMPLEX equals
((|(Re a1,Re a2)| - (<i> * |(Re a1,Im a2)|)) + (<i> * |(Im a1,Re a2)|)) + |(Im a1,Im a2)|;
end;
:: COMPLSP2:def 4
theorem
for b1, b2 being FinSequence of COMPLEX holds
|(b1,b2)| = ((|(Re b1,Re b2)| - (<i> * |(Re b1,Im b2)|)) + (<i> * |(Im b1,Re b2)|)) + |(Im b1,Im b2)|;
:: COMPLSP2:th 28
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds b1 + (b2 + b3) = (b1 + b2) + b3;
:: COMPLSP2:th 29
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds b1 + b2 = b2 + b1;
:: COMPLSP2:th 30
theorem
for b1 being complex set
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);
:: COMPLSP2:th 31
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds b1 - b2 = b1 + - b2;
:: COMPLSP2:funcnot 9 => COMPLSP2:func 9
definition
let a1 be Element of NAT;
let a2 be Element of COMPLEX;
redefine func a1 |-> a2 -> Element of a1 -tuples_on COMPLEX;
end;
:: COMPLSP2:th 32
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2 & b1 + b2 = 0c len b1
holds b1 = - b2 & b2 = - b1;
:: COMPLSP2:th 33
theorem
for b1 being FinSequence of COMPLEX holds
b1 + 0c len b1 = b1;
:: COMPLSP2:th 34
theorem
for b1 being FinSequence of COMPLEX holds
b1 + - b1 = 0c len b1;
:: COMPLSP2:th 35
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds - (b1 + b2) = (- b1) + - b2;
:: COMPLSP2:th 36
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds (b1 - b2) - b3 = b1 - (b2 + b3);
:: COMPLSP2:th 37
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds b1 + (b2 - b3) = (b1 + b2) - b3;
:: COMPLSP2:th 38
theorem
for b1 being FinSequence of COMPLEX holds
- - b1 = b1;
:: COMPLSP2:th 39
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds - (b1 - b2) = (- b1) + b2;
:: COMPLSP2:th 40
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds b1 - (b2 - b3) = (b1 - b2) + b3;
:: COMPLSP2:th 41
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
b2 * 0c len b1 = 0c len b1;
:: COMPLSP2:th 42
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
- (b2 * b1) = b2 * - b1;
:: COMPLSP2:th 43
theorem
for b1 being complex set
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds b1 * (b2 - b3) = (b1 * b2) - (b1 * b3);
:: COMPLSP2:th 44
theorem
for b1, b2 being Element of COMPLEX
for b3, b4 being Element of REAL
st b1 = b3 & b2 = b4
holds addcomplex .(b1,b2) = addreal .(b3,b4);
:: COMPLSP2:th 45
theorem
for b1 being Function-like quasi_total Relation of [:COMPLEX,COMPLEX:],COMPLEX
for b2 being Function-like quasi_total Relation of [:REAL,REAL:],REAL
for b3, b4 being FinSequence of COMPLEX
for b5, b6 being FinSequence of REAL
st b3 = b5 &
b4 = b6 &
len b3 = len b6 &
(for b7 being Element of NAT
st b7 in dom b3
holds b1 .(b3 . b7,b4 . b7) = b2 .(b5 . b7,b6 . b7))
holds b1 .:(b3,b4) = b2 .:(b5,b6);
:: COMPLSP2:th 46
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b1 = len b4
holds addcomplex .:(b1,b2) = addreal .:(b3,b4);
:: COMPLSP2:th 47
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b1 = len b4
holds b1 + b2 = b3 + b4;
:: COMPLSP2:th 48
theorem
for b1 being FinSequence of COMPLEX holds
len Re b1 = len b1 & len Im b1 = len b1;
:: COMPLSP2:th 49
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds Re (b1 + b2) = (Re b1) + Re b2 &
Im (b1 + b2) = (Im b1) + Im b2;
:: COMPLSP2:th 50
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b1 = len b4
holds diffcomplex .:(b1,b2) = diffreal .:(b3,b4);
:: COMPLSP2:th 51
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b1 = len b4
holds b1 - b2 = b3 - b4;
:: COMPLSP2:th 52
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds Re (b1 - b2) = (Re b1) - Re b2 &
Im (b1 - b2) = (Im b1) - Im b2;
:: COMPLSP2:th 53
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
b2 * (b3 * b1) = (b2 * b3) * b1;
:: COMPLSP2:th 54
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
(- b2) * b1 = - (b2 * b1);
:: COMPLSP2:th 55
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX
for b2 being Function-like quasi_total Relation of REAL,REAL
for b3 being FinSequence of COMPLEX
for b4 being FinSequence of REAL
st len b3 = len b4 &
(for b5 being Element of NAT
st b5 in dom b3
holds b1 . (b3 . b5) = b2 . (b4 . b5))
holds b1 * b3 = b2 * b4;
:: COMPLSP2:th 56
theorem
for b1 being FinSequence of COMPLEX
for b2 being FinSequence of REAL
st b1 = b2 & len b1 = len b2
holds compcomplex * b1 = compreal * b2;
:: COMPLSP2:th 57
theorem
for b1 being FinSequence of COMPLEX
for b2 being FinSequence of REAL
st b1 = b2 & len b1 = len b2
holds - b1 = - b2;
:: COMPLSP2:th 58
theorem
for b1 being FinSequence of COMPLEX holds
Re (<i> * b1) = - Im b1 &
Im (<i> * b1) = Re b1;
:: COMPLSP2:th 59
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(<i> * b1,b2)| = <i> * |(b1,b2)|;
:: COMPLSP2:th 60
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(b1,<i> * b2)| = - (<i> * |(b1,b2)|);
:: COMPLSP2:th 61
theorem
for b1 being Element of COMPLEX
for b2 being FinSequence of COMPLEX
for b3 being Element of REAL
for b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b2 = len b4
holds b1 multcomplex * b2 = b3 multreal * b4;
:: COMPLSP2:th 62
theorem
for b1 being complex set
for b2 being FinSequence of COMPLEX
for b3 being Element of REAL
for b4 being FinSequence of REAL
st b1 = b3 & b2 = b4 & len b2 = len b4
holds b1 * b2 = b3 * b4;
:: COMPLSP2:th 63
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
(b2 + b3) * b1 = (b2 * b1) + (b3 * b1);
:: COMPLSP2:th 64
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
(b2 - b3) * b1 = (b2 * b1) - (b3 * b1);
:: COMPLSP2:th 65
theorem
for b1 being Element of COMPLEX
for b2 being FinSequence of COMPLEX holds
Re (b1 * b2) = ((Re b1) * Re b2) - ((Im b1) * Im b2) &
Im (b1 * b2) = ((Im b1) * Re b2) + ((Re b1) * Im b2);
:: COMPLSP2:th 66
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds |(b1 + b2,b3)| = |(b1,b3)| + |(b2,b3)|;
:: COMPLSP2:th 67
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(- b1,b2)| = - |(b1,b2)|;
:: COMPLSP2:th 68
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(b1,- b2)| = - |(b1,b2)|;
:: COMPLSP2:th 69
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(- b1,- b2)| = |(b1,b2)|;
:: COMPLSP2:th 70
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds |(b1 - b2,b3)| = |(b1,b3)| - |(b2,b3)|;
:: COMPLSP2:th 71
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds |(b1,b2 + b3)| = |(b1,b2)| + |(b1,b3)|;
:: COMPLSP2:th 72
theorem
for b1, b2, b3 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3
holds |(b1,b2 - b3)| = |(b1,b2)| - |(b1,b3)|;
:: COMPLSP2:th 73
theorem
for b1, b2, b3, b4 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
holds |(b1 + b2,b3 + b4)| = ((|(b1,b3)| + |(b1,b4)|) + |(b2,b3)|) + |(b2,b4)|;
:: COMPLSP2:th 74
theorem
for b1, b2, b3, b4 being FinSequence of COMPLEX
st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
holds |(b1 - b2,b3 - b4)| = ((|(b1,b3)| - |(b1,b4)|) - |(b2,b3)|) + |(b2,b4)|;
:: COMPLSP2:th 75
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(b1,b2)| = |(b2,b1)| *';
:: COMPLSP2:th 76
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(b1 + b2,b1 + b2)| = (|(b1,b1)| + (2 * Re |(b1,b2)|)) + |(b2,b2)|;
:: COMPLSP2:th 77
theorem
for b1, b2 being FinSequence of COMPLEX
st len b1 = len b2
holds |(b1 - b2,b1 - b2)| = (|(b1,b1)| - (2 * Re |(b1,b2)|)) + |(b2,b2)|;
:: COMPLSP2:th 78
theorem
for b1 being Element of COMPLEX
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds |(b1 * b2,b3)| = b1 * |(b2,b3)|;
:: COMPLSP2:th 79
theorem
for b1 being Element of COMPLEX
for b2, b3 being FinSequence of COMPLEX
st len b2 = len b3
holds |(b2,b1 * b3)| = b1 *' * |(b2,b3)|;
:: COMPLSP2:th 80
theorem
for b1, b2 being Element of COMPLEX
for b3, b4, b5 being FinSequence of COMPLEX
st len b3 = len b4 & len b4 = len b5
holds |((b1 * b3) + (b2 * b4),b5)| = (b1 * |(b3,b5)|) + (b2 * |(b4,b5)|);
:: COMPLSP2:th 81
theorem
for b1, b2 being Element of COMPLEX
for b3, b4, b5 being FinSequence of COMPLEX
st len b3 = len b4 & len b4 = len b5
holds |(b3,(b1 * b4) + (b2 * b5))| = (b1 *' * |(b3,b4)|) + (b2 *' * |(b3,b5)|);