Article POLYEQ_3, MML version 4.99.1005
:: POLYEQ_3:funcnot 1 => POLYEQ_3:func 1
definition
let a1 be Element of REAL;
let a2 be Element of COMPLEX;
redefine func a1 * a2 -> Element of COMPLEX;
commutativity;
:: for a1 being Element of REAL
:: for a2 being Element of COMPLEX holds
:: a1 * a2 = a2 * a1;
end;
:: POLYEQ_3:funcnot 2 => POLYEQ_3:func 2
definition
let a1 be Element of REAL;
let a2 be Element of COMPLEX;
redefine func a1 + a2 -> Element of COMPLEX;
commutativity;
:: for a1 being Element of REAL
:: for a2 being Element of COMPLEX holds
:: a1 + a2 = a2 + a1;
end;
:: POLYEQ_3:funcnot 3 => POLYEQ_3:func 3
definition
let a1 be Element of COMPLEX;
redefine func A1 ^2 -> Element of COMPLEX equals
((Re a1) ^2 - ((Im a1) ^2)) + ((2 * ((Re a1) * Im a1)) * <i>);
end;
:: POLYEQ_3:def 1
theorem
for b1 being Element of COMPLEX holds
b1 ^2 = ((Re b1) ^2 - ((Im b1) ^2)) + ((2 * ((Re b1) * Im b1)) * <i>);
:: POLYEQ_3:funcnot 4 => POLYEQ_3:func 4
definition
let a1, a2, a3 be Element of REAL;
let a4 be Element of COMPLEX;
redefine func Polynom(a1,a2,a3,a4) -> Element of COMPLEX;
end;
:: POLYEQ_3:th 1
theorem
for b1, b2, b3, b4 being Element of REAL holds
(b1 + (b2 * <i>)) * (b3 + (b4 * <i>)) = ((b1 * b3) - (b2 * b4)) + (((b1 * b4) + (b3 * b2)) * <i>);
:: POLYEQ_3:th 2
theorem
for b1, b2 being Element of REAL holds
(b1 + (b2 * <i>)) ^2 = (b1 ^2 - (b2 ^2)) + (((2 * b1) * b2) * <i>);
:: POLYEQ_3:th 4
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
0 <= delta(b1,b2,b3) &
Polynom(b1,b2,b3,b4) = 0 &
b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) &
b4 <> ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)
holds b4 = - (b2 / (2 * b1));
:: POLYEQ_3:th 5
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
delta(b1,b2,b3) < 0 &
Polynom(b1,b2,b3,b4) = 0 &
b4 <> (- (b2 / (2 * b1))) + (((sqrt - delta(b1,b2,b3)) / (2 * b1)) * <i>)
holds b4 = (- (b2 / (2 * b1))) + ((- ((sqrt - delta(b1,b2,b3)) / (2 * b1))) * <i>);
:: POLYEQ_3:th 6
theorem
for b1, b2 being Element of REAL
for b3 being Element of COMPLEX
st b1 <> 0 &
(for b4 being Element of COMPLEX holds
Polynom(0,b1,b2,b4) = 0)
holds b3 = - (b2 / b1);
:: POLYEQ_3:th 7
theorem
for b1, b2, b3 being Element of REAL
for b4, b5, b6 being complex set
st b1 <> 0 &
(for b7 being complex set holds
Polynom(b1,b2,b3,b7) = Quard(b1,b5,b6,b7))
holds b2 / b1 = - (b5 + b6) & b3 / b1 = b5 * b6;
:: POLYEQ_3:funcnot 5 => POLYEQ_3:func 5
definition
let a1 be complex set;
func A1 ^3 -> Element of COMPLEX equals
a1 ^2 * a1;
end;
:: POLYEQ_3:def 2
theorem
for b1 being complex set holds
b1 ^3 = b1 ^2 * b1;
:: POLYEQ_3:funcnot 6 => POLYEQ_1:func 7
definition
let a1, a2, a3, a4, a5 be complex set;
func Polynom(A1,A2,A3,A4,A5) -> set equals
(((a1 * (a5 ^3)) + (a2 * (a5 ^2))) + (a3 * a5)) + a4;
end;
:: POLYEQ_3:def 3
theorem
for b1, b2, b3, b4, b5 being complex set holds
Polynom(b1,b2,b3,b4,b5) = (((b1 * (b5 ^3)) + (b2 * (b5 ^2))) + (b3 * b5)) + b4;
:: POLYEQ_3:th 8
theorem
0 ^3 = 0;
:: POLYEQ_3:th 9
theorem
1 ^3 = 1;
:: POLYEQ_3:th 10
theorem
(- 1) ^3 = - 1;
:: POLYEQ_3:th 11
theorem
for b1 being Element of COMPLEX holds
Re (b1 ^3) = ((Re b1) |^ 3) - ((3 * Re b1) * ((Im b1) ^2)) &
Im (b1 ^3) = (- ((Im b1) |^ 3)) + ((3 * ((Re b1) ^2)) * Im b1);
:: POLYEQ_3:th 12
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
st for b9 being complex set holds
Polynom(b1,b2,b3,b4,b9) = Polynom(b5,b6,b7,b8,b9)
holds b1 = b5 & b2 = b6 & b3 = b7 & b4 = b8;
:: POLYEQ_3:th 13
theorem
for b1, b2 being Element of COMPLEX holds
(b1 + b2) ^3 = ((b1 ^3 + ((3 * b2) * (b1 ^2))) + ((3 * (b2 ^2)) * b1)) + (b2 ^3);
:: POLYEQ_3:th 14
theorem
for b1, b2 being Element of COMPLEX holds
(b1 * b2) ^3 = b1 ^3 * (b2 ^3);
:: POLYEQ_3:th 15
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
0 <= delta(b1,b2,b3) &
Polynom(0,b1,b2,b3,b4) = 0 &
b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) &
b4 <> ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)
holds b4 = - (b2 / (2 * b1));
:: POLYEQ_3:th 16
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
delta(b1,b2,b3) < 0 &
Polynom(0,b1,b2,b3,b4) = 0 &
b4 <> (- (b2 / (2 * b1))) + (((sqrt - delta(b1,b2,b3)) / (2 * b1)) * <i>)
holds b4 = (- (b2 / (2 * b1))) + ((- ((sqrt - delta(b1,b2,b3)) / (2 * b1))) * <i>);
:: POLYEQ_3:th 17
theorem
for b1, b2 being Element of REAL
for b3 being Element of COMPLEX
st b1 <> 0 &
(4 * b1) * b2 <= 0 &
Polynom(b1,0,b2,0,b3) = 0 &
b3 <> (sqrt - ((4 * b1) * b2)) / (2 * b1) &
b3 <> (- sqrt - ((4 * b1) * b2)) / (2 * b1)
holds b3 = 0;
:: POLYEQ_3:th 18
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
0 <= delta(b1,b2,b3) &
Polynom(b1,b2,b3,0,b4) = 0 &
b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) &
b4 <> ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) &
b4 <> - (b2 / (2 * b1))
holds b4 = 0;
:: POLYEQ_3:th 19
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 &
delta(b1,b2,b3) < 0 &
Polynom(b1,b2,b3,0,b4) = 0 &
b4 <> (- (b2 / (2 * b1))) + (((sqrt - delta(b1,b2,b3)) / (2 * b1)) * <i>) &
b4 <> (- (b2 / (2 * b1))) + ((- ((sqrt - delta(b1,b2,b3)) / (2 * b1))) * <i>)
holds b4 = 0;
:: POLYEQ_3:th 20
theorem
for b1, b2 being Element of REAL
st b1 ^2 = b2 & b1 <> sqrt b2
holds b1 = - sqrt b2;
:: POLYEQ_3:th 21
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 & Im b4 = 0 & Polynom(b1,0,b2,b3,b4) = 0
for b5, b6 being Element of REAL
st Re b4 = b5 + b6 &
((3 * b6) * b5) + (b2 / b1) = 0 &
b4 <> (3 -root ((- (b3 / (2 * b1))) + sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3)))) + (3 -root ((- (b3 / (2 * b1))) - sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3)))) &
b4 <> (3 -root ((- (b3 / (2 * b1))) + sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3)))) + (3 -root ((- (b3 / (2 * b1))) + sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3))))
holds b4 = (3 -root ((- (b3 / (2 * b1))) - sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3)))) + (3 -root ((- (b3 / (2 * b1))) - sqrt ((b3 ^2 / (4 * (b1 ^2))) + ((b2 / (3 * b1)) |^ 3))));
:: POLYEQ_3:th 22
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of COMPLEX
st b1 <> 0 & Im b4 <> 0 & Polynom(b1,0,b2,b3,b4) = 0
for b5, b6 being Element of REAL
st Re b4 = b5 + b6 &
((3 * b6) * b5) + (b2 / (4 * b1)) = 0 &
0 <= b2 / b1 &
b4 <> ((3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3)))) + (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) + ((sqrt ((3 * (((3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3)))) + (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>) &
b4 <> ((3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3)))) + (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) - ((sqrt ((3 * (((3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3)))) + (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>) &
b4 <> (2 * (3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) + ((sqrt ((3 * ((2 * (3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>) &
b4 <> (2 * (3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) - ((sqrt ((3 * ((2 * (3 -root ((b3 / (16 * b1)) + sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>) &
b4 <> (2 * (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) + ((sqrt ((3 * ((2 * (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>)
holds b4 = (2 * (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) - ((sqrt ((3 * ((2 * (3 -root ((b3 / (16 * b1)) - sqrt ((b3 / (16 * b1)) ^2 + ((b2 / (12 * b1)) |^ 3))))) ^2)) + (b2 / b1))) * <i>);
:: POLYEQ_3:th 23
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,b3,b4,b5) = 0 & Im b5 = 0
for b6, b7, b8 being Element of REAL
st b8 = (Re b5) + (b2 / (3 * b1)) &
Re b5 = (b6 + b7) - (b2 / (3 * b1)) &
((3 * b6) * b7) + ((((3 * b1) * b3) - (b2 ^2)) / (3 * (b1 ^2))) = 0 &
b5 <> (((3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) + sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3)))) + (3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) - sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3))))) - (b2 / (3 * b1))) + (0 * <i>) &
b5 <> (((3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) + sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3)))) + (3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) + sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3))))) - (b2 / (3 * b1))) + (0 * <i>)
holds b5 = (((3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) - sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3)))) + (3 -root (((- ((b2 / (3 * b1)) |^ 3)) - ((((3 * b1) * b4) - (b2 * b3)) / (6 * (b1 ^2)))) - sqrt ((((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) ^2 / 4) + (((((3 * b1) * b3) - (b2 ^2)) / (9 * (b1 ^2))) |^ 3))))) - (b2 / (3 * b1))) + (0 * <i>);
:: POLYEQ_3:th 24
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,b3) = 0
holds b3 = - (b2 / b1);
:: POLYEQ_3:th 25
theorem
for b1 being Element of COMPLEX
st b1 <> 0
for b2 being Element of COMPLEX holds
Polynom(0,b1,b2) <> 0;
:: POLYEQ_3:th 26
theorem
for b1, b2, b3, b4, b5, b6 being Element of COMPLEX
st for b7 being Element of COMPLEX holds
Polynom(b1,b2,b3,b7) = Polynom(b4,b5,b6,b7)
holds b1 = b4 & b2 = b5 & b3 = b6;
:: POLYEQ_3:th 27
theorem
for b1, b2 being Element of REAL holds
0 <= ((- b1) + sqrt (b1 ^2 + (b2 ^2))) / 2 &
0 <= (b1 + sqrt (b1 ^2 + (b2 ^2))) / 2;
:: POLYEQ_3:th 28
theorem
for b1, b2 being Element of COMPLEX
st b1 ^2 = b2 &
0 <= Im b2 &
b1 <> (sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>)
holds b1 = (- sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((- sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>);
:: POLYEQ_3:th 29
theorem
for b1, b2 being Element of COMPLEX
st b1 ^2 = b2 & Im b2 = 0 & 0 < Re b2 & b1 <> sqrt Re b2
holds b1 = - sqrt Re b2;
:: POLYEQ_3:th 30
theorem
for b1, b2 being Element of COMPLEX
st b1 ^2 = b2 &
Im b2 = 0 &
Re b2 < 0 &
b1 <> (sqrt - Re b2) * <i>
holds b1 = - ((sqrt - Re b2) * <i>);
:: POLYEQ_3:th 31
theorem
for b1, b2 being Element of COMPLEX
st b1 ^2 = b2 &
Im b2 < 0 &
b1 <> (sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((- sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>)
holds b1 = (- sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>);
:: POLYEQ_3:th 32
theorem
for b1, b2 being Element of COMPLEX
st b1 ^2 = b2 &
b1 <> (sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>) &
b1 <> (- sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((- sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>) &
b1 <> (sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((- sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>)
holds b1 = (- sqrt (((Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) + ((sqrt (((- Re b2) + sqrt ((Re b2) ^2 + ((Im b2) ^2))) / 2)) * <i>);
:: POLYEQ_3:th 33
theorem
for b1 being Element of COMPLEX holds
Polynom(0,0,0,b1) = 0;
:: POLYEQ_3:th 34
theorem
for b1, b2 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,0,0,b2) = 0
holds b2 = 0;
:: POLYEQ_3:th 35
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,0,b3) = 0 & b3 <> - (b2 / b1)
holds b3 = 0;
:: POLYEQ_3:th 36
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,0,b2,b3) = 0
for b4 being Element of COMPLEX
st b4 = - (b2 / b1) &
b3 <> (sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>) &
b3 <> (- sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((- sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>) &
b3 <> (sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((- sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>)
holds b3 = (- sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>);
:: POLYEQ_3:th 37
theorem
for b1, b2, b3, b4 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,b3,b4) = 0
for b5, b6 being Element of COMPLEX
st b5 = (b2 / (2 * b1)) ^2 - (b3 / b1) &
b6 = b2 / (2 * b1) &
b4 <> ((sqrt (((Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) + ((sqrt (((- Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) * <i>)) - b6 &
b4 <> ((- sqrt (((Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) + ((- sqrt (((- Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) * <i>)) - b6 &
b4 <> ((sqrt (((Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) + ((- sqrt (((- Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) * <i>)) - b6
holds b4 = ((- sqrt (((Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) + ((sqrt (((- Re b5) + sqrt ((Re b5) ^2 + ((Im b5) ^2))) / 2)) * <i>)) - b6;
:: POLYEQ_3:funcnot 7 => POLYEQ_1:func 7
definition
let a1, a2, a3, a4, a5 be complex set;
func Polynom(A1,A2,A3,A4,A5) -> set equals
(((a1 * (a5 ^3)) + (a2 * (a5 ^2))) + (a3 * a5)) + a4;
end;
:: POLYEQ_3:def 5
theorem
for b1, b2, b3, b4, b5 being complex set holds
Polynom(b1,b2,b3,b4,b5) = (((b1 * (b5 ^3)) + (b2 * (b5 ^2))) + (b3 * b5)) + b4;
:: POLYEQ_3:th 39
theorem
for b1 being Element of COMPLEX holds
b1 |^ 3 = (b1 * b1) * b1 & b1 |^ 3 = b1 ^2 * b1 & b1 |^ 3 = b1 ^3;
:: POLYEQ_3:th 40
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,0,0,b3) = 0 & b3 <> - (b2 / b1)
holds b3 = 0;
:: POLYEQ_3:th 41
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,0,b2,0,b3) = 0
for b4 being Element of COMPLEX
st b4 = - (b2 / b1) &
b3 <> 0 &
b3 <> (sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>) &
b3 <> (- sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((- sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>) &
b3 <> (sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((- sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>)
holds b3 = (- sqrt (((Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) + ((sqrt (((- Re b4) + sqrt ((Re b4) ^2 + ((Im b4) ^2))) / 2)) * <i>);
:: POLYEQ_3:th 42
theorem
for b1, b2, b3, b4 being Element of COMPLEX
st b1 <> 0 & Polynom(b1,b2,b3,0,b4) = 0
for b5, b6, b7 being Element of COMPLEX
st b5 = - (b3 / b1) &
b6 = (b2 / (2 * b1)) ^2 - (b3 / b1) &
b7 = b2 / (2 * b1) &
b4 <> 0 &
b4 <> ((sqrt (((Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) + ((sqrt (((- Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) * <i>)) - b7 &
b4 <> ((- sqrt (((Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) + ((- sqrt (((- Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) * <i>)) - b7 &
b4 <> ((sqrt (((Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) + ((- sqrt (((- Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) * <i>)) - b7
holds b4 = ((- sqrt (((Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) + ((sqrt (((- Re b6) + sqrt ((Re b6) ^2 + ((Im b6) ^2))) / 2)) * <i>)) - b7;
:: POLYEQ_3:th 43
theorem
for b1, b2 being Element of COMPLEX holds
(b1 - ((1 / 3) * b2)) ^2 = (b1 ^2 + (((- (2 / 3)) * b2) * b1)) + ((1 / 9) * (b2 ^2));
:: POLYEQ_3:th 44
theorem
for b1, b2, b3 being Element of COMPLEX
st b1 = b2 - ((1 / 3) * b3)
holds b1 ^3 = ((b2 ^3 - (b3 * (b2 ^2))) + (((1 / 3) * (b3 ^2)) * b2)) - ((1 / 27) * (b3 ^3));
:: POLYEQ_3:th 46
theorem
for b1 being Element of COMPLEX holds
[*|.b1.| * cos Arg b1,|.b1.| * sin Arg b1*] = [*|.b1.|,0*] * [*cos Arg b1,sin Arg b1*];
:: POLYEQ_3:th 49
theorem
for b1 being complex set holds
b1 |^ 2 = b1 * b1;
:: POLYEQ_3:th 50
theorem
for b1 being natural set
st 0 < b1
holds 0 |^ b1 = 0;
:: POLYEQ_3:th 51
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of NAT holds
(b1 * b2) |^ b3 = (b1 |^ b3) * (b2 |^ b3);
:: POLYEQ_3:th 52
theorem
for b1 being Element of REAL
st 0 < b1
for b2 being Element of NAT holds
[*b1,0*] |^ b2 = b1 to_power b2;
:: POLYEQ_3:th 53
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
[*cos b1,sin b1*] |^ b2 = (cos (b2 * b1)) + ((sin (b2 * b1)) * <i>);
:: POLYEQ_3:th 54
theorem
for b1 being Element of COMPLEX
for b2 being Element of NAT
st (b1 = 0 implies 0 < b2)
holds b1 |^ b2 = ((|.b1.| to_power b2) * cos (b2 * Arg b1)) + (((|.b1.| to_power b2) * sin (b2 * Arg b1)) * <i>);
:: POLYEQ_3:th 55
theorem
for b1, b2 being Element of NAT
for b3 being Element of REAL
st b1 <> 0
holds [*cos ((b3 + ((2 * PI) * b2)) / b1),sin ((b3 + ((2 * PI) * b2)) / b1)*] |^ b1 = (cos b3) + ((sin b3) * <i>);
:: POLYEQ_3:th 56
theorem
for b1 being complex set
for b2, b3 being Element of NAT
st b2 <> 0
holds b1 = [*(b2 -root |.b1.|) * cos (((Arg b1) + ((2 * PI) * b3)) / b2),(b2 -root |.b1.|) * sin (((Arg b1) + ((2 * PI) * b3)) / b2)*] |^ b2;
:: POLYEQ_3:th 57
theorem
for b1 being Element of COMPLEX
for b2 being non empty Element of NAT
for b3 being Element of NAT holds
[*(b2 -root |.b1.|) * cos (((Arg b1) + ((2 * PI) * b3)) / b2),(b2 -root |.b1.|) * sin (((Arg b1) + ((2 * PI) * b3)) / b2)*] is CRoot of b2,b1;
:: POLYEQ_3:th 58
theorem
for b1 being complex set
for b2 being CRoot of 1,b1 holds
b2 = b1;
:: POLYEQ_3:th 59
theorem
for b1 being non empty natural set
for b2 being CRoot of b1,0 holds
b2 = 0;
:: POLYEQ_3:th 60
theorem
for b1 being non empty Element of NAT
for b2 being complex set
for b3 being CRoot of b1,b2
st b3 = 0
holds b2 = 0;
:: POLYEQ_3:th 61
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT holds
(cos (((2 * PI) * b2) / b1)) + ((sin (((2 * PI) * b2) / b1)) * <i>) is CRoot of b1,1;
:: POLYEQ_3:th 63
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of NAT
st b2 <> 0 & b1 <> 0 & 1 <= b3 & b2 |^ b3 = b1 |^ b3
holds |.b2.| = |.b1.|;