:: POLYEQ_3 semantic presentation
Lemma19:
0 = [*0,0*]
by ARYTM_0:def 7;
:: deftheorem Def1 defines ^2 POLYEQ_3:def 1 :
theorem Th1: :: POLYEQ_3:1
theorem Th2: :: POLYEQ_3:2
theorem Th3: :: POLYEQ_3:3
canceled;
theorem Th4: :: POLYEQ_3:4
for
a,
b,
c being
Real for
z being
Element of
COMPLEX st
a <> 0 &
delta a,
b,
c >= 0 &
Polynom a,
b,
c,
z = 0 & not
z = ((- b) + (sqrt (delta a,b,c))) / (2 * a) & not
z = ((- b) - (sqrt (delta a,b,c))) / (2 * a) holds
z = - (b / (2 * a))
theorem Th5: :: POLYEQ_3:5
for
a,
b,
c being
Real for
z being
Element of
COMPLEX st
a <> 0 &
delta a,
b,
c < 0 &
Polynom a,
b,
c,
z = 0 & not
z = (- (b / (2 * a))) + (((sqrt (- (delta a,b,c))) / (2 * a)) * <i> ) holds
z = (- (b / (2 * a))) + ((- ((sqrt (- (delta a,b,c))) / (2 * a))) * <i> )
theorem Th6: :: POLYEQ_3:6
theorem Th7: :: POLYEQ_3:7
for
a,
b,
c being
Real for
z,
z1,
z2 being
complex number st
a <> 0 & ( for
z being
complex number holds
Polynom a,
b,
c,
z = Quard a,
z1,
z2,
z ) holds
(
b / a = - (z1 + z2) &
c / a = z1 * z2 )
:: deftheorem Def2 defines ^3 POLYEQ_3:def 2 :
Lemma66:
for z being complex number holds z |^ 2 = z * z
definition
let a be
complex number ,
b be
complex number ,
c be
complex number ,
d be
complex number ,
z be
complex number ;
redefine func Polynom c1,
c2,
c3,
c4,
c5 -> set equals :: POLYEQ_3:def 3
(((a * (z ^3 )) + (b * (z ^2 ))) + (c * z)) + d;
compatibility
for b1 being set holds
( b1 = Polynom a,b,c,d,z iff b1 = (((a * (z ^3 )) + (b * (z ^2 ))) + (c * z)) + d )
end;
:: deftheorem Def3 defines Polynom POLYEQ_3:def 3 :
theorem Th8: :: POLYEQ_3:8
theorem Th9: :: POLYEQ_3:9
theorem Th10: :: POLYEQ_3:10
theorem Th11: :: POLYEQ_3:11
theorem Th12: :: POLYEQ_3:12
for
a,
b,
c,
d,
a',
b',
c',
d' being
Real st ( for
z being
complex number holds
Polynom a,
b,
c,
d,
z = Polynom a',
b',
c',
d',
z ) holds
(
a = a' &
b = b' &
c = c' &
d = d' )
theorem Th13: :: POLYEQ_3:13
theorem Th14: :: POLYEQ_3:14
theorem Th15: :: POLYEQ_3:15
for
b,
c,
d being
Real for
z being
Element of
COMPLEX st
b <> 0 &
delta b,
c,
d >= 0 &
Polynom 0,
b,
c,
d,
z = 0 & not
z = ((- c) + (sqrt (delta b,c,d))) / (2 * b) & not
z = ((- c) - (sqrt (delta b,c,d))) / (2 * b) holds
z = - (c / (2 * b))
theorem Th16: :: POLYEQ_3:16
for
b,
c,
d being
Real for
z being
Element of
COMPLEX st
b <> 0 &
delta b,
c,
d < 0 &
Polynom 0,
b,
c,
d,
z = 0 & not
z = (- (c / (2 * b))) + (((sqrt (- (delta b,c,d))) / (2 * b)) * <i> ) holds
z = (- (c / (2 * b))) + ((- ((sqrt (- (delta b,c,d))) / (2 * b))) * <i> )
theorem Th17: :: POLYEQ_3:17
theorem Th18: :: POLYEQ_3:18
for
a,
b,
c being
Real for
z being
Element of
COMPLEX st
a <> 0 &
delta a,
b,
c >= 0 &
Polynom a,
b,
c,0,
z = 0 & not
z = ((- b) + (sqrt (delta a,b,c))) / (2 * a) & not
z = ((- b) - (sqrt (delta a,b,c))) / (2 * a) & not
z = - (b / (2 * a)) holds
z = 0
theorem Th19: :: POLYEQ_3:19
for
a,
b,
c being
Real for
z being
Element of
COMPLEX st
a <> 0 &
delta a,
b,
c < 0 &
Polynom a,
b,
c,0,
z = 0 & not
z = (- (b / (2 * a))) + (((sqrt (- (delta a,b,c))) / (2 * a)) * <i> ) & not
z = (- (b / (2 * a))) + ((- ((sqrt (- (delta a,b,c))) / (2 * a))) * <i> ) holds
z = 0
theorem Th20: :: POLYEQ_3:20
theorem Th21: :: POLYEQ_3:21
theorem Th22: :: POLYEQ_3:22
theorem Th23: :: POLYEQ_3:23
theorem Th24: :: POLYEQ_3:24
theorem Th25: :: POLYEQ_3:25
theorem Th26: :: POLYEQ_3:26
for
z1,
z2,
z3,
s1,
s2,
s3 being
Element of
COMPLEX st ( for
z being
Element of
COMPLEX holds
Polynom z1,
z2,
z3,
z = Polynom s1,
s2,
s3,
z ) holds
(
z1 = s1 &
z2 = s2 &
z3 = s3 )
theorem Th27: :: POLYEQ_3:27
theorem Th28: :: POLYEQ_3:28
theorem Th29: :: POLYEQ_3:29
theorem Th30: :: POLYEQ_3:30
theorem Th31: :: POLYEQ_3:31
theorem Th32: :: POLYEQ_3:32
theorem Th33: :: POLYEQ_3:33
theorem Th34: :: POLYEQ_3:34
theorem Th35: :: POLYEQ_3:35
theorem Th36: :: POLYEQ_3:36
theorem Th37: :: POLYEQ_3:37
definition
let z1 be
complex number ,
z2 be
complex number ,
z3 be
complex number ,
z4 be
complex number ,
z be
complex number ;
canceled;redefine func Polynom c1,
c2,
c3,
c4,
c5 -> set equals :: POLYEQ_3:def 5
(((z1 * (z ^3 )) + (z2 * (z ^2 ))) + (z3 * z)) + z4;
compatibility
for b1 being set holds
( b1 = Polynom z1,z2,z3,z4,z iff b1 = (((z1 * (z ^3 )) + (z2 * (z ^2 ))) + (z3 * z)) + z4 )
;
end;
:: deftheorem Def4 POLYEQ_3:def 4 :
canceled;
:: deftheorem Def5 defines Polynom POLYEQ_3:def 5 :
theorem Th38: :: POLYEQ_3:38
theorem Th39: :: POLYEQ_3:39
theorem Th40: :: POLYEQ_3:40
theorem Th41: :: POLYEQ_3:41
theorem Th42: :: POLYEQ_3:42
theorem Th43: :: POLYEQ_3:43
theorem Th44: :: POLYEQ_3:44
theorem Th45: :: POLYEQ_3:45
for
z1,
z2,
z3,
z being
Element of
COMPLEX st
Polynom 1r ,
z1,
z2,
z3,
z = 0 holds
for
p,
q,
s being
Element of
COMPLEX st
z = s - ((1 / 3) * z1) &
p = (- ((1 / 3) * (z1 ^2 ))) + z2 &
q = (((2 / 27) * (z1 ^3 )) - (((1 / 3) * z1) * z2)) + z3 holds
Polynom 1r ,0,
p,
q,
s = 0
by COMPLEX1:def 7;
theorem Th46: :: POLYEQ_3:46
theorem Th47: :: POLYEQ_3:47
theorem Th48: :: POLYEQ_3:48
theorem Th49: :: POLYEQ_3:49
theorem Th50: :: POLYEQ_3:50
for
n being
Nat st
n > 0 holds
0
|^ n = 0
theorem Th51: :: POLYEQ_3:51
theorem Th52: :: POLYEQ_3:52
theorem Th53: :: POLYEQ_3:53
theorem Th54: :: POLYEQ_3:54
theorem Th55: :: POLYEQ_3:55
theorem Th56: :: POLYEQ_3:56
definition
canceled;
end;
:: deftheorem Def6 POLYEQ_3:def 6 :
canceled;
theorem Th57: :: POLYEQ_3:57
theorem Th58: :: POLYEQ_3:58
theorem Th59: :: POLYEQ_3:59
theorem Th60: :: POLYEQ_3:60
theorem Th61: :: POLYEQ_3:61
theorem Th62: :: POLYEQ_3:62
canceled;
theorem Th63: :: POLYEQ_3:63