:: POLYEQ_3 semantic presentation
Lemma1:
0 = [*0,0*]
by ARYTM_0:def 7;
:: deftheorem Def1 defines ^2 POLYEQ_3:def 1 :
theorem Th1: :: POLYEQ_3:1
for
b1,
b2,
b3,
b4 being
Real holds
(b1 + (b2 * <i> )) * (b3 + (b4 * <i> )) = ((b1 * b3) - (b2 * b4)) + (((b1 * b4) + (b3 * b2)) * <i> ) ;
theorem Th2: :: POLYEQ_3:2
theorem Th3: :: POLYEQ_3:3
canceled;
theorem Th4: :: POLYEQ_3:4
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 >= 0 &
Polynom b1,
b2,
b3,
b4 = 0 & not
b4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not
b4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) holds
b4 = - (b2 / (2 * b1))
theorem Th5: :: POLYEQ_3:5
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 < 0 &
Polynom b1,
b2,
b3,
b4 = 0 & not
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> )
theorem Th6: :: POLYEQ_3:6
theorem Th7: :: POLYEQ_3:7
for
b1,
b2,
b3 being
Realfor
b4,
b5,
b6 being
complex number st
b1 <> 0 & ( for
b7 being
complex number holds
Polynom b1,
b2,
b3,
b7 = Quard b1,
b5,
b6,
b7 ) holds
(
b2 / b1 = - (b5 + b6) &
b3 / b1 = b5 * b6 )
:: deftheorem Def2 defines ^3 POLYEQ_3:def 2 :
Lemma4:
for b1 being complex number holds b1 |^ 2 = b1 * b1
definition
let c1,
c2,
c3,
c4,
c5 be
complex number ;
redefine func Polynom c1,
c2,
c3,
c4,
c5 -> set equals :: POLYEQ_3:def 3
(((a1 * (a5 ^3 )) + (a2 * (a5 ^2 ))) + (a3 * a5)) + a4;
compatibility
for b1 being set holds
( b1 = Polynom c1,c2,c3,c4,c5 iff b1 = (((c1 * (c5 ^3 )) + (c2 * (c5 ^2 ))) + (c3 * c5)) + c4 )
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
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
Real st ( for
b9 being
complex number holds
Polynom b1,
b2,
b3,
b4,
b9 = Polynom b5,
b6,
b7,
b8,
b9 ) holds
(
b1 = b5 &
b2 = b6 &
b3 = b7 &
b4 = b8 )
theorem Th13: :: POLYEQ_3:13
theorem Th14: :: POLYEQ_3:14
theorem Th15: :: POLYEQ_3:15
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 >= 0 &
Polynom 0,
b1,
b2,
b3,
b4 = 0 & not
b4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not
b4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) holds
b4 = - (b2 / (2 * b1))
theorem Th16: :: POLYEQ_3:16
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 < 0 &
Polynom 0,
b1,
b2,
b3,
b4 = 0 & not
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> )
theorem Th17: :: POLYEQ_3:17
theorem Th18: :: POLYEQ_3:18
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 >= 0 &
Polynom b1,
b2,
b3,0,
b4 = 0 & not
b4 = ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not
b4 = ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) & not
b4 = - (b2 / (2 * b1)) holds
b4 = 0
theorem Th19: :: POLYEQ_3:19
for
b1,
b2,
b3 being
Realfor
b4 being
Element of
COMPLEX st
b1 <> 0 &
delta b1,
b2,
b3 < 0 &
Polynom b1,
b2,
b3,0,
b4 = 0 & not
b4 = (- (b2 / (2 * b1))) + (((sqrt (- (delta b1,b2,b3))) / (2 * b1)) * <i> ) & not
b4 = (- (b2 / (2 * b1))) + ((- ((sqrt (- (delta b1,b2,b3))) / (2 * b1))) * <i> ) holds
b4 = 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
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 )
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 c1,
c2,
c3,
c4,
c5 be
complex number ;
canceled;redefine func Polynom c1,
c2,
c3,
c4,
c5 -> set equals :: POLYEQ_3:def 5
(((a1 * (a5 ^3 )) + (a2 * (a5 ^2 ))) + (a3 * a5)) + a4;
compatibility
for b1 being set holds
( b1 = Polynom c1,c2,c3,c4,c5 iff b1 = (((c1 * (c5 ^3 )) + (c2 * (c5 ^2 ))) + (c3 * c5)) + c4 )
;
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
b1,
b2,
b3,
b4 being
Element of
COMPLEX st
Polynom 1r ,
b1,
b2,
b3,
b4 = 0 holds
for
b5,
b6,
b7 being
Element of
COMPLEX st
b4 = b7 - ((1 / 3) * b1) &
b5 = (- ((1 / 3) * (b1 ^2 ))) + b2 &
b6 = (((2 / 27) * (b1 ^3 )) - (((1 / 3) * b1) * b2)) + b3 holds
Polynom 1r ,0,
b5,
b6,
b7 = 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
b1 being
Nat st
b1 > 0 holds
0
|^ b1 = 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
:: deftheorem Def6 defines CRoot POLYEQ_3:def 6 :
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