Article COMPLEX2, MML version 4.99.1005
:: COMPLEX2:th 1
theorem
for b1, b2 being Element of REAL holds
- (b1 + (b2 * <i>)) = (- b1) + ((- b2) * <i>);
:: COMPLEX2:th 2
theorem
for b1, b2 being real set
st 0 < b2
holds ex b3 being real set st
b3 = (b2 * - [\b1 / b2/]) + b1 &
0 <= b3 &
b3 < b2;
:: COMPLEX2:th 3
theorem
for b1, b2, b3 being real set
st 0 < b1 & 0 <= b2 & 0 <= b3 & b2 < b1 & b3 < b1
for b4 being integer set
st b2 = b3 + (b1 * b4)
holds b2 = b3;
:: COMPLEX2:th 4
theorem
for b1, b2 being real set holds
sin (b1 - b2) = ((sin b1) * cos b2) - ((cos b1) * sin b2) &
cos (b1 - b2) = ((cos b1) * cos b2) + ((sin b1) * sin b2);
:: COMPLEX2:th 5
theorem
for b1 being real set holds
sin . (b1 - PI) = - (sin . b1) &
cos . (b1 - PI) = - (cos . b1);
:: COMPLEX2:th 6
theorem
for b1 being real set holds
sin (b1 - PI) = - sin b1 & cos (b1 - PI) = - cos b1;
:: COMPLEX2:th 7
theorem
for b1, b2 being real set
st b1 in ].0,PI / 2.[ &
b2 in ].0,PI / 2.[
holds b1 < b2
iff
sin b1 < sin b2;
:: COMPLEX2:th 8
theorem
for b1, b2 being real set
st b1 in ].PI / 2,PI.[ &
b2 in ].PI / 2,PI.[
holds b1 < b2
iff
sin b2 < sin b1;
:: COMPLEX2:th 9
theorem
for b1 being real set
for b2 being integer set holds
sin b1 = sin (((2 * PI) * b2) + b1);
:: COMPLEX2:th 10
theorem
for b1 being real set
for b2 being integer set holds
cos b1 = cos (((2 * PI) * b2) + b1);
:: COMPLEX2:th 11
theorem
for b1 being real set
st sin b1 = 0
holds cos b1 <> 0;
:: COMPLEX2:th 12
theorem
for b1, b2 being real set
st 0 <= b1 & b1 < 2 * PI & 0 <= b2 & b2 < 2 * PI & sin b1 = sin b2 & cos b1 = cos b2
holds b1 = b2;
:: COMPLEX2:th 19
theorem
for b1 being complex set holds
b1 = [*|.b1.| * cos Arg b1,|.b1.| * sin Arg b1*];
:: COMPLEX2:th 21
theorem
for b1 being complex set
for b2 being Element of REAL
st b1 <> 0 &
b1 = [*|.b1.| * cos b2,|.b1.| * sin b2*] &
0 <= b2 &
b2 < 2 * PI
holds b2 = Arg b1;
:: COMPLEX2:th 22
theorem
for b1 being complex set
st b1 <> 0
holds (PI <= Arg b1 or Arg - b1 = (Arg b1) + PI) &
(PI <= Arg b1 implies Arg - b1 = (Arg b1) - PI);
:: COMPLEX2:th 23
theorem
for b1 being Element of REAL
st 0 <= b1
holds Arg [*b1,0*] = 0;
:: COMPLEX2:th 24
theorem
for b1 being complex set holds
Arg b1 = 0
iff
b1 = [*|.b1.|,0*];
:: COMPLEX2:th 25
theorem
for b1 being complex set
st b1 <> 0
holds Arg b1 < PI
iff
PI <= Arg - b1;
:: COMPLEX2:th 26
theorem
for b1, b2 being complex set
st (b1 = b2 implies b1 - b2 <> 0)
holds Arg (b1 - b2) < PI
iff
PI <= Arg (b2 - b1);
:: COMPLEX2:th 27
theorem
for b1 being complex set holds
Arg b1 in ].0,PI.[
iff
0 < Im b1;
:: COMPLEX2:th 28
theorem
for b1 being complex set
st Arg b1 <> 0
holds Arg b1 < PI
iff
0 < sin Arg b1;
:: COMPLEX2:th 29
theorem
for b1, b2 being complex set
st Arg b1 < PI & Arg b2 < PI
holds Arg (b1 + b2) < PI;
:: COMPLEX2:th 34
theorem
for b1 being complex set holds
Arg b1 = 0
iff
0 <= Re b1 & Im b1 = 0;
:: COMPLEX2:th 35
theorem
for b1 being complex set holds
Arg b1 = PI
iff
Re b1 < 0 & Im b1 = 0;
:: COMPLEX2:th 36
theorem
for b1 being complex set holds
Im b1 = 0
iff
(Arg b1 = 0 or Arg b1 = PI);
:: COMPLEX2:th 37
theorem
for b1 being complex set
st Arg b1 <= PI
holds 0 <= Im b1;
:: COMPLEX2:th 38
theorem
for b1 being Element of COMPLEX
st b1 <> 0
holds cos Arg - b1 = - cos Arg b1 &
sin Arg - b1 = - sin Arg b1;
:: COMPLEX2:th 39
theorem
for b1 being complex set
st b1 <> 0
holds cos Arg b1 = (Re b1) / |.b1.| &
sin Arg b1 = (Im b1) / |.b1.|;
:: COMPLEX2:th 40
theorem
for b1 being complex set
for b2 being Element of REAL
st 0 < b2
holds Arg (b1 * [*b2,0*]) = Arg b1;
:: COMPLEX2:th 41
theorem
for b1 being complex set
for b2 being Element of REAL
st b2 < 0
holds Arg (b1 * [*b2,0*]) = Arg - b1;
:: COMPLEX2:funcnot 1 => COMPLEX2:func 1
definition
let a1, a2 be complex set;
func A1 .|. A2 -> Element of COMPLEX equals
a1 * (a2 *');
end;
:: COMPLEX2:def 3
theorem
for b1, b2 being complex set holds
b1 .|. b2 = b1 * (b2 *');
:: COMPLEX2:th 42
theorem
for b1, b2 being Element of COMPLEX holds
b1 .|. b2 = [*((Re b1) * Re b2) + ((Im b1) * Im b2),(- ((Re b1) * Im b2)) + ((Im b1) * Re b2)*];
:: COMPLEX2:th 43
theorem
for b1 being Element of COMPLEX holds
b1 .|. b1 = [*((Re b1) * Re b1) + ((Im b1) * Im b1),0*] &
b1 .|. b1 = [*(Re b1) ^2 + ((Im b1) ^2),0*];
:: COMPLEX2:th 44
theorem
for b1 being Element of COMPLEX holds
b1 .|. b1 = [*|.b1.| ^2,0*] &
|.b1.| ^2 = Re (b1 .|. b1);
:: COMPLEX2:th 45
theorem
for b1, b2 being Element of COMPLEX holds
|.b1 .|. b2.| = |.b1.| * |.b2.|;
:: COMPLEX2:th 46
theorem
for b1 being Element of COMPLEX
st b1 .|. b1 = 0
holds b1 = 0;
:: COMPLEX2:th 47
theorem
for b1, b2 being Element of COMPLEX holds
b1 .|. b2 = (b2 .|. b1) *';
:: COMPLEX2:th 48
theorem
for b1, b2, b3 being Element of COMPLEX holds
(b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3);
:: COMPLEX2:th 49
theorem
for b1, b2, b3 being Element of COMPLEX holds
b1 .|. (b2 + b3) = (b1 .|. b2) + (b1 .|. b3);
:: COMPLEX2:th 50
theorem
for b1, b2, b3 being Element of COMPLEX holds
(b1 * b2) .|. b3 = b1 * (b2 .|. b3);
:: COMPLEX2:th 51
theorem
for b1, b2, b3 being Element of COMPLEX holds
b1 .|. (b2 * b3) = b2 *' * (b1 .|. b3);
:: COMPLEX2:th 52
theorem
for b1, b2, b3 being Element of COMPLEX holds
(b1 * b2) .|. b3 = b2 .|. (b1 *' * b3);
:: COMPLEX2:th 53
theorem
for b1, b2, b3, b4, b5 being Element of COMPLEX holds
((b1 * b2) + (b3 * b4)) .|. b5 = (b1 * (b2 .|. b5)) + (b3 * (b4 .|. b5));
:: COMPLEX2:th 54
theorem
for b1, b2, b3, b4, b5 being Element of COMPLEX holds
b1 .|. ((b2 * b3) + (b4 * b5)) = (b2 *' * (b1 .|. b3)) + (b4 *' * (b1 .|. b5));
:: COMPLEX2:th 55
theorem
for b1, b2 being Element of COMPLEX holds
(- b1) .|. b2 = b1 .|. - b2;
:: COMPLEX2:th 56
theorem
for b1, b2 being Element of COMPLEX holds
(- b1) .|. b2 = - (b1 .|. b2);
:: COMPLEX2:th 57
theorem
for b1, b2 being Element of COMPLEX holds
- (b1 .|. b2) = b1 .|. - b2;
:: COMPLEX2:th 58
theorem
for b1, b2 being Element of COMPLEX holds
(- b1) .|. - b2 = b1 .|. b2;
:: COMPLEX2:th 59
theorem
for b1, b2, b3 being Element of COMPLEX holds
(b1 - b2) .|. b3 = (b1 .|. b3) - (b2 .|. b3);
:: COMPLEX2:th 60
theorem
for b1, b2, b3 being Element of COMPLEX holds
b1 .|. (b2 - b3) = (b1 .|. b2) - (b1 .|. b3);
:: COMPLEX2:th 62
theorem
for b1, b2 being Element of COMPLEX holds
(b1 + b2) .|. (b1 + b2) = (((b1 .|. b1) + (b1 .|. b2)) + (b2 .|. b1)) + (b2 .|. b2);
:: COMPLEX2:th 63
theorem
for b1, b2 being Element of COMPLEX holds
(b1 - b2) .|. (b1 - b2) = (((b1 .|. b1) - (b1 .|. b2)) - (b2 .|. b1)) + (b2 .|. b2);
:: COMPLEX2:th 64
theorem
for b1, b2 being Element of COMPLEX holds
Re (b1 .|. b2) = 0
iff
(Im (b1 .|. b2) = |.b1.| * |.b2.| or Im (b1 .|. b2) = - (|.b1.| * |.b2.|));
:: COMPLEX2:funcnot 2 => COMPLEX2:func 2
definition
let a1 be complex set;
let a2 be Element of REAL;
func Rotate(A1,A2) -> Element of COMPLEX equals
[*|.a1.| * cos (a2 + Arg a1),|.a1.| * sin (a2 + Arg a1)*];
end;
:: COMPLEX2:def 4
theorem
for b1 being complex set
for b2 being Element of REAL holds
Rotate(b1,b2) = [*|.b1.| * cos (b2 + Arg b1),|.b1.| * sin (b2 + Arg b1)*];
:: COMPLEX2:th 65
theorem
for b1 being Element of COMPLEX holds
Rotate(b1,0) = b1;
:: COMPLEX2:th 66
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL holds
Rotate(b1,b2) = 0
iff
b1 = 0;
:: COMPLEX2:th 67
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL holds
|.Rotate(b1,b2).| = |.b1.|;
:: COMPLEX2:th 68
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL
st b1 <> 0
holds ex b3 being integer set st
Arg Rotate(b1,b2) = ((2 * PI) * b3) + (b2 + Arg b1);
:: COMPLEX2:th 69
theorem
for b1 being Element of COMPLEX holds
Rotate(b1,- Arg b1) = [*|.b1.|,0*];
:: COMPLEX2:th 70
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL holds
Re Rotate(b1,b2) = ((Re b1) * cos b2) - ((Im b1) * sin b2) &
Im Rotate(b1,b2) = ((Re b1) * sin b2) + ((Im b1) * cos b2);
:: COMPLEX2:th 71
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL holds
Rotate(b1 + b2,b3) = (Rotate(b1,b3)) + Rotate(b2,b3);
:: COMPLEX2:th 72
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL holds
Rotate(- b1,b2) = - Rotate(b1,b2);
:: COMPLEX2:th 73
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL holds
Rotate(b1 - b2,b3) = (Rotate(b1,b3)) - Rotate(b2,b3);
:: COMPLEX2:th 74
theorem
for b1 being Element of COMPLEX holds
Rotate(b1,PI) = - b1;
:: COMPLEX2:funcnot 3 => COMPLEX2:func 3
definition
let a1, a2 be Element of COMPLEX;
func angle(A1,A2) -> Element of REAL equals
Arg Rotate(a2,- Arg a1)
if (Arg a1 <> 0 implies a2 <> 0)
otherwise (2 * PI) - Arg a1;
end;
:: COMPLEX2:def 5
theorem
for b1, b2 being Element of COMPLEX holds
(Arg b1 <> 0 & b2 = 0 or angle(b1,b2) = Arg Rotate(b2,- Arg b1)) &
(Arg b1 <> 0 & b2 = 0 implies angle(b1,b2) = (2 * PI) - Arg b1);
:: COMPLEX2:th 75
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL
st 0 <= b2
holds angle([*b2,0*],b1) = Arg b1;
:: COMPLEX2:th 76
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL
st Arg b1 = Arg b2 & b1 <> 0 & b2 <> 0
holds Arg Rotate(b1,b3) = Arg Rotate(b2,b3);
:: COMPLEX2:th 77
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL
st 0 < b3
holds angle(b1,b2) = angle(b1 * [*b3,0*],b2 * [*b3,0*]);
:: COMPLEX2:th 78
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & b2 <> 0 & Arg b1 = Arg b2
holds Arg - b1 = Arg - b2;
:: COMPLEX2:th 79
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL
st b1 <> 0 & b2 <> 0
holds angle(b1,b2) = angle(Rotate(b1,b3),Rotate(b2,b3));
:: COMPLEX2:th 80
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of REAL
st b3 < 0 & b1 <> 0 & b2 <> 0
holds angle(b1,b2) = angle(b1 * [*b3,0*],b2 * [*b3,0*]);
:: COMPLEX2:th 81
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & b2 <> 0
holds angle(b1,b2) = angle(- b1,- b2);
:: COMPLEX2:th 82
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & angle(b2,b1) = 0
holds angle(b2,- b1) = PI;
:: COMPLEX2:th 83
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & b2 <> 0
holds cos angle(b1,b2) = (Re (b1 .|. b2)) / (|.b1.| * |.b2.|) &
sin angle(b1,b2) = - ((Im (b1 .|. b2)) / (|.b1.| * |.b2.|));
:: COMPLEX2:funcnot 4 => COMPLEX2:func 4
definition
let a1, a2, a3 be complex set;
func angle(A1,A2,A3) -> real set equals
(Arg (a3 - a2)) - Arg (a1 - a2)
if 0 <= (Arg (a3 - a2)) - Arg (a1 - a2)
otherwise (2 * PI) + ((Arg (a3 - a2)) - Arg (a1 - a2));
end;
:: COMPLEX2:def 6
theorem
for b1, b2, b3 being complex set holds
(0 <= (Arg (b3 - b2)) - Arg (b1 - b2) implies angle(b1,b2,b3) = (Arg (b3 - b2)) - Arg (b1 - b2)) &
(0 <= (Arg (b3 - b2)) - Arg (b1 - b2) or angle(b1,b2,b3) = (2 * PI) + ((Arg (b3 - b2)) - Arg (b1 - b2)));
:: COMPLEX2:th 84
theorem
for b1, b2, b3 being Element of COMPLEX holds
0 <= angle(b1,b2,b3) & angle(b1,b2,b3) < 2 * PI;
:: COMPLEX2:th 85
theorem
for b1, b2, b3 being Element of COMPLEX holds
angle(b1,b2,b3) = angle(b1 - b2,0,b3 - b2);
:: COMPLEX2:th 86
theorem
for b1, b2, b3, b4 being Element of COMPLEX holds
angle(b1,b2,b3) = angle(b1 + b4,b2 + b4,b3 + b4);
:: COMPLEX2:th 87
theorem
for b1, b2 being Element of COMPLEX holds
angle(b1,b2) = angle(b1,0,b2);
:: COMPLEX2:th 88
theorem
for b1, b2, b3 being Element of COMPLEX
st angle(b1,b2,b3) = 0
holds Arg (b1 - b2) = Arg (b3 - b2) & angle(b3,b2,b1) = 0;
:: COMPLEX2:th 89
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & b2 <> 0
holds Re (b1 .|. b2) = 0
iff
(angle(b1,0,b2) = PI / 2 or angle(b1,0,b2) = (3 / 2) * PI);
:: COMPLEX2:th 90
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & b2 <> 0
holds (Im (b1 .|. b2) = |.b1.| * |.b2.| or Im (b1 .|. b2) = - (|.b1.| * |.b2.|))
iff
(angle(b1,0,b2) = PI / 2 or angle(b1,0,b2) = (3 / 2) * PI);
:: COMPLEX2:th 91
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 &
b3 <> b2 &
(angle(b1,b2,b3) = PI / 2 or angle(b1,b2,b3) = (3 / 2) * PI)
holds |.b1 - b2.| ^2 + (|.b3 - b2.| ^2) = |.b1 - b3.| ^2;
:: COMPLEX2:th 92
theorem
for b1, b2, b3 being Element of COMPLEX
for b4 being Element of REAL
st b1 <> b2 & b2 <> b3
holds angle(b1,b2,b3) = angle(Rotate(b1,b4),Rotate(b2,b4),Rotate(b3,b4));
:: COMPLEX2:th 93
theorem
for b1, b2 being Element of COMPLEX holds
angle(b1,b2,b1) = 0;
:: COMPLEX2:th 94
theorem
for b1, b2, b3 being Element of COMPLEX holds
angle(b1,b2,b3) <> 0
iff
(angle(b1,b2,b3)) + angle(b3,b2,b1) = 2 * PI;
:: COMPLEX2:th 95
theorem
for b1, b2, b3 being Element of COMPLEX
st angle(b1,b2,b3) <> 0
holds angle(b3,b2,b1) <> 0;
:: COMPLEX2:th 96
theorem
for b1, b2, b3 being Element of COMPLEX
st angle(b1,b2,b3) = PI
holds angle(b3,b2,b1) = PI;
:: COMPLEX2:th 97
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 & b1 <> b3 & b2 <> b3 & angle(b1,b2,b3) = 0 & angle(b2,b3,b1) = 0
holds angle(b3,b1,b2) <> 0;
:: COMPLEX2:th 98
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 & b2 <> b3 & 0 < angle(b1,b2,b3) & angle(b1,b2,b3) < PI
holds ((angle(b1,b2,b3)) + angle(b2,b3,b1)) + angle(b3,b1,b2) = PI &
0 < angle(b2,b3,b1) &
0 < angle(b3,b1,b2);
:: COMPLEX2:th 99
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 & b2 <> b3 & PI < angle(b1,b2,b3)
holds ((angle(b1,b2,b3)) + angle(b2,b3,b1)) + angle(b3,b1,b2) = 5 * PI &
PI < angle(b2,b3,b1) &
PI < angle(b3,b1,b2);
:: COMPLEX2:th 100
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 & b2 <> b3 & angle(b1,b2,b3) = PI
holds angle(b2,b3,b1) = 0 & angle(b3,b1,b2) = 0;
:: COMPLEX2:th 101
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> b2 &
b1 <> b3 &
b2 <> b3 &
angle(b1,b2,b3) = 0 &
(angle(b2,b3,b1) = 0 implies angle(b3,b1,b2) <> PI)
holds angle(b2,b3,b1) = PI & angle(b3,b1,b2) = 0;
:: COMPLEX2:th 102
theorem
for b1, b2, b3 being Element of COMPLEX holds
(((angle(b1,b2,b3)) + angle(b2,b3,b1)) + angle(b3,b1,b2) = PI or ((angle(b1,b2,b3)) + angle(b2,b3,b1)) + angle(b3,b1,b2) = 5 * PI)
iff
b1 <> b2 & b1 <> b3 & b2 <> b3;