:: Fundamental Theorem of Algebra
:: by Robert Milewski
::
:: Received August 21, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users

theorem Th1: :: POLYNOM5:1

for n, m being Nat st n <> 0 & m <> 0 holds
(((n * m) - n) - m) + 1 >= 0

proof end;

theorem Th2: :: POLYNOM5:2

for x, y being Real st y > 0 holds
(min (x,y)) / (max (x,y)) <= 1

proof end;

theorem Th3: :: POLYNOM5:3

for x, y being Real st ( for c being Real st c > 0 & c < 1 holds
c * x >= y ) holds
y <= 0

proof end;

Lm1: for x, y being Real st ( for c being Real st c > 0 & c < 1 holds
c * x >= y ) holds
y <= 0
by Th3;

theorem Th4: :: POLYNOM5:4

for p being FinSequence of REAL st ( for n being Element of NAT st n in dom p holds
p . n >= 0 ) holds
for i being Element of NAT st i in dom p holds
Sum p >= p . i

proof end;

theorem Th5: :: POLYNOM5:5

for x, y being Real holds - [**x,y**] = [**(- x),(- y)**]

proof end;

theorem Th6: :: POLYNOM5:6

for x1, y1, x2, y2 being Real holds [**x1,y1**] - [**x2,y2**] = [**(x1 - x2),(y1 - y2)**]

proof end;

definition

let R be non empty multMagma ;
let z be Element of R;
let n be Nat;

func power (z,n) -> Element of R equals :: POLYNOM5:def 1
(power R) . (z,n);

coherence
(power R) . (z,n) is Element of R ;

end;

:: deftheorem defines power POLYNOM5:def 1 :
for R being non empty multMagma
for z being Element of R
for n being Nat holds power (z,n) = (power R) . (z,n);

theorem Th7: :: POLYNOM5:7

for z being Element of F_Complex st z <> 0. F_Complex holds
for n being Nat holds |.(power (z,n)).| = |.z.| to_power n

proof end;

definition

let p be complex-valued FinSequence;

:: original: |.
redefine func |.p.| -> FinSequence of REAL means :Def2: :: POLYNOM5:def 2
( len it = len p & ( for n being Nat st n in dom p holds
it . n = |.(p . n).| ) );

coherence
|.p.| is FinSequence of REAL by RVSUM_1:145;
compatibility
for b₁ being FinSequence of REAL holds
( b₁ = |.p.| iff ( len b₁ = len p & ( for n being Nat st n in dom p holds
b₁ . n = |.(p . n).| ) ) )

proof end;

end;

:: deftheorem Def2 defines |. POLYNOM5:def 2 :
for p being complex-valued FinSequence
for b₂ being FinSequence of REAL holds
( b₂ = |.p.| iff ( len b₂ = len p & ( for n being Nat st n in dom p holds
b₂ . n = |.(p . n).| ) ) );

theorem Th8: :: POLYNOM5:8

|.(<*> the carrier of F_Complex).| = <*> REAL

proof end;

theorem Th9: :: POLYNOM5:9

for x being Complex holds |.<*x*>.| = <*|.x.|*>

proof end;

theorem :: POLYNOM5:10

for x, y being Complex holds |.<*x,y*>.| = <*|.x.|,|.y.|*>

proof end;

theorem :: POLYNOM5:11

for x, y, z being Complex holds |.<*x,y,z*>.| = <*|.x.|,|.y.|,|.z.|*>

proof end;

theorem Th12: :: POLYNOM5:12

for p, q being complex-valued FinSequence holds |.(p ^ q).| = |.p.| ^ |.q.|

proof end;

theorem :: POLYNOM5:13

for p being complex-valued FinSequence
for x being Complex holds
( |.(p ^ <*x*>).| = |.p.| ^ <*|.x.|*> & |.(<*x*> ^ p).| = <*|.x.|*> ^ |.p.| )

proof end;

theorem Th14: :: POLYNOM5:14

for p being FinSequence of the carrier of F_Complex holds |.(Sum p).| <= Sum |.p.|

proof end;

definition

let L be non empty right_complementable Abelian add-associative right_zeroed right_unital distributive commutative doubleLoopStr ;
let p be Polynomial of L;
let n be Nat;

func p `^ n -> sequence of L equals :: POLYNOM5:def 3
(power (Polynom-Ring L)) . (p,n);

coherence
(power (Polynom-Ring L)) . (p,n) is sequence of L

proof end;

end;

:: deftheorem defines `^ POLYNOM5:def 3 :
for L being non empty right_complementable Abelian add-associative right_zeroed right_unital distributive commutative doubleLoopStr
for p being Polynomial of L
for n being Nat holds p `^ n = (power (Polynom-Ring L)) . (p,n);

registration

let L be non empty right_complementable Abelian add-associative right_zeroed right_unital distributive commutative doubleLoopStr ;
let p be Polynomial of L;
let n be Nat;

cluster p `^ n -> finite-Support ;

coherence
p `^ n is finite-Support

proof end;

end;

theorem Th15: :: POLYNOM5:15

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds p `^ 0 = 1_. L

proof end;

theorem :: POLYNOM5:16

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds p `^ 1 = p

proof end;

theorem :: POLYNOM5:17

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds p `^ 2 = p *' p

proof end;

theorem :: POLYNOM5:18

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds p `^ 3 = (p *' p) *' p

proof end;

theorem Th19: :: POLYNOM5:19

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L
for n being Nat holds p `^ (n + 1) = (p `^ n) *' p

proof end;

theorem Th20: :: POLYNOM5:20

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for n being Element of NAT holds (0_. L) `^ (n + 1) = 0_. L

proof end;

theorem :: POLYNOM5:21

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for n being Nat holds (1_. L) `^ n = 1_. L

proof end;

theorem Th22: :: POLYNOM5:22

for L being Field
for p being Polynomial of L
for x being Element of L
for n being Nat holds eval ((p `^ n),x) = (power L) . ((eval (p,x)),n)

proof end;

Lm2: for L being non empty ZeroStr
for p being AlgSequence of L st len p > 0 holds
p . ((len p) -' 1) <> 0. L

proof end;

theorem Th23: :: POLYNOM5:23

for L being domRing
for p being Polynomial of L st len p <> 0 holds
for n being Nat holds len (p `^ n) = ((n * (len p)) - n) + 1

proof end;

definition

let L be non empty multMagma ;
let p be sequence of L;
let v be Element of L;

func v * p -> sequence of L means :Def4: :: POLYNOM5:def 4
for n being Element of NAT holds it . n = v * (p . n);

existence
ex b₁ being sequence of L st
for n being Element of NAT holds b₁ . n = v * (p . n)

proof end;

uniqueness
for b₁, b₂ being sequence of L st ( for n being Element of NAT holds b₁ . n = v * (p . n) ) & ( for n being Element of NAT holds b₂ . n = v * (p . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines * POLYNOM5:def 4 :
for L being non empty multMagma
for p being sequence of L
for v being Element of L
for b₄ being sequence of L holds
( b₄ = v * p iff for n being Element of NAT holds b₄ . n = v * (p . n) );

registration

let L be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr ;
let p be Polynomial of L;
let v be Element of L;

cluster v * p -> finite-Support ;

coherence
v * p is finite-Support

proof end;

end;

theorem Th24: :: POLYNOM5:24

for L being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for p being Polynomial of L holds len ((0. L) * p) = 0

proof end;

theorem Th25: :: POLYNOM5:25

for L being non empty right_complementable almost_left_invertible add-associative right_zeroed well-unital distributive associative commutative doubleLoopStr
for p being Polynomial of L
for v being Element of L st v <> 0. L holds
len (v * p) = len p

proof end;

theorem Th26: :: POLYNOM5:26

for L being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for p being sequence of L holds (0. L) * p = 0_. L

proof end;

theorem Th27: :: POLYNOM5:27

for L being non empty well-unital multLoopStr
for p being sequence of L holds (1. L) * p = p

proof end;

theorem Th28: :: POLYNOM5:28

for L being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for v being Element of L holds v * (0_. L) = 0_. L

proof end;

theorem Th29: :: POLYNOM5:29

for L being non empty right_complementable add-associative right_zeroed right-distributive well-unital doubleLoopStr
for v being Element of L holds v * (1_. L) = <%v%>

proof end;

theorem Th30: :: POLYNOM5:30

for L being non empty right_complementable almost_left_invertible add-associative right_zeroed well-unital distributive associative commutative doubleLoopStr
for p being Polynomial of L
for v, x being Element of L holds eval ((v * p),x) = v * (eval (p,x))

proof end;

theorem Th31: :: POLYNOM5:31

for L being non empty right_complementable add-associative right_zeroed right-distributive unital doubleLoopStr
for p being Polynomial of L holds eval (p,(0. L)) = p . 0

proof end;

definition

let L be non empty ZeroStr ;
let z0, z1 be Element of L;

func <%z0,z1%> -> sequence of L equals :: POLYNOM5:def 5
((0_. L) +* (0,z0)) +* (1,z1);

coherence
((0_. L) +* (0,z0)) +* (1,z1) is sequence of L ;

end;

:: deftheorem defines <% POLYNOM5:def 5 :
for L being non empty ZeroStr
for z0, z1 being Element of L holds <%z0,z1%> = ((0_. L) +* (0,z0)) +* (1,z1);

theorem Th32: :: POLYNOM5:32

for L being non empty ZeroStr
for z0 being Element of L holds
( <%z0%> . 0 = z0 & ( for n being Element of NAT st n >= 1 holds
<%z0%> . n = 0. L ) )

proof end;

theorem Th33: :: POLYNOM5:33

for L being non empty ZeroStr
for z0 being Element of L st z0 <> 0. L holds
len <%z0%> = 1

proof end;

theorem Th34: :: POLYNOM5:34

for L being non empty ZeroStr holds <%(0. L)%> = 0_. L

proof end;

theorem Th35: :: POLYNOM5:35

for L being non empty right_complementable add-associative right_zeroed well-unital distributive associative commutative domRing-like doubleLoopStr
for x, y being Element of L holds <%x%> *' <%y%> = <%(x * y)%>

proof end;

theorem Th36: :: POLYNOM5:36

for L being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed well-unital distributive associative commutative doubleLoopStr
for x being Element of L
for n being Nat holds <%x%> `^ n = <%(power (x,n))%>

proof end;

theorem :: POLYNOM5:37

for L being non empty right_complementable add-associative right_zeroed unital doubleLoopStr
for z0, x being Element of L holds eval (<%z0%>,x) = z0

proof end;

theorem Th38: :: POLYNOM5:38

for L being non empty ZeroStr
for z0, z1 being Element of L holds
( <%z0,z1%> . 0 = z0 & <%z0,z1%> . 1 = z1 & ( for n being Nat st n >= 2 holds
<%z0,z1%> . n = 0. L ) )

proof end;

registration

let L be non empty ZeroStr ;
let z0, z1 be Element of L;

cluster <%z0,z1%> -> finite-Support ;

coherence
<%z0,z1%> is finite-Support

proof end;

end;

theorem Th39: :: POLYNOM5:39

for L being non empty ZeroStr
for z0, z1 being Element of L holds len <%z0,z1%> <= 2

proof end;

theorem Th40: :: POLYNOM5:40

for L being non empty ZeroStr
for z0, z1 being Element of L st z1 <> 0. L holds
len <%z0,z1%> = 2

proof end;

theorem Th41: :: POLYNOM5:41

for L being non empty ZeroStr
for z0 being Element of L st z0 <> 0. L holds
len <%z0,(0. L)%> = 1

proof end;

theorem Th42: :: POLYNOM5:42

for L being non empty ZeroStr holds <%(0. L),(0. L)%> = 0_. L

proof end;

theorem :: POLYNOM5:43

for L being non empty ZeroStr
for z0 being Element of L holds <%z0,(0. L)%> = <%z0%>

proof end;

theorem Th44: :: POLYNOM5:44

for L being non empty right_complementable add-associative right_zeroed left-distributive unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,z1%>,x) = z0 + (z1 * x)

proof end;

theorem :: POLYNOM5:45

for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,(0. L)%>,x) = z0

proof end;

theorem :: POLYNOM5:46

for L being non empty right_complementable add-associative right_zeroed left-distributive unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%(0. L),z1%>,x) = z1 * x

proof end;

theorem Th47: :: POLYNOM5:47

for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%z0,(1. L)%>,x) = z0 + x

proof end;

theorem :: POLYNOM5:48

for L being non empty right_complementable add-associative right_zeroed left-distributive well-unital doubleLoopStr
for z0, z1, x being Element of L holds eval (<%(0. L),(1. L)%>,x) = x

proof end;

definition

let L be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr ;
let p, q be Polynomial of L;

func Subst (p,q) -> Polynomial of L means :Def6: :: POLYNOM5:def 6
ex F being FinSequence of the carrier of (Polynom-Ring L) st
( it = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) );

existence
ex b₁ being Polynomial of L ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b₁ = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) )

proof end;

uniqueness
for b₁, b₂ being Polynomial of L st ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b₁ = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) & ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b<sub>2</sub> = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines Subst POLYNOM5:def 6 :
for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p, q, b₄ being Polynomial of L holds
( b₄ = Subst (p,q) iff ex F being FinSequence of the carrier of (Polynom-Ring L) st
( b₄ = Sum F & len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = (p . (n -' 1)) * (q `^ (n -' 1)) ) ) );

theorem Th49: :: POLYNOM5:49

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds Subst ((0_. L),p) = 0_. L

proof end;

theorem :: POLYNOM5:50

for L being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive commutative doubleLoopStr
for p being Polynomial of L holds Subst (p,(0_. L)) = <%(p . 0)%>

proof end;

theorem :: POLYNOM5:51

for L being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed well-unital distributive associative commutative doubleLoopStr
for p being Polynomial of L
for x being Element of L holds len (Subst (p,<%x%>)) <= 1

proof end;

theorem Th52: :: POLYNOM5:52

for L being Field
for p, q being Polynomial of L st len p <> 0 & len q > 1 holds
len (Subst (p,q)) = ((((len p) * (len q)) - (len p)) - (len q)) + 2

proof end;

theorem Th53: :: POLYNOM5:53

for L being Field
for p, q being Polynomial of L
for x being Element of L holds eval ((Subst (p,q)),x) = eval (p,(eval (q,x)))

proof end;

definition

let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;
let x be Element of L;

pred x is_a_root_of p means :: POLYNOM5:def 7
eval (p,x) = 0. L;

end;

:: deftheorem defines is_a_root_of POLYNOM5:def 7 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L
for x being Element of L holds
( x is_a_root_of p iff eval (p,x) = 0. L );

definition

let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;

attr p is with_roots means :: POLYNOM5:def 8
ex x being Element of L st x is_a_root_of p;

end;

:: deftheorem defines with_roots POLYNOM5:def 8 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L holds
( p is with_roots iff ex x being Element of L st x is_a_root_of p );

theorem Th54: :: POLYNOM5:54

for L being non empty unital doubleLoopStr holds 0_. L is with_roots

proof end;

registration

let L be non empty unital doubleLoopStr ;

cluster 0_. L -> with_roots ;

coherence
0_. L is with_roots by Th54;

end;

theorem :: POLYNOM5:55

for L being non empty unital doubleLoopStr
for x being Element of L holds x is_a_root_of 0_. L by POLYNOM4:17;

registration

let L be non empty unital doubleLoopStr ;

cluster Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots for Element of K1(K2(NAT, the carrier of L));

existence
ex b₁ being Polynomial of L st b₁ is with_roots

proof end;

end;

definition

let L be non empty unital doubleLoopStr ;

attr L is algebraic-closed means :: POLYNOM5:def 9
for p being Polynomial of L st len p > 1 holds
p is with_roots ;

end;

:: deftheorem defines algebraic-closed POLYNOM5:def 9 :
for L being non empty unital doubleLoopStr holds
( L is algebraic-closed iff for p being Polynomial of L st len p > 1 holds
p is with_roots );

definition

let L be non empty unital doubleLoopStr ;
let p be Polynomial of L;

func Roots p -> Subset of L means :Def10: :: POLYNOM5:def 10
for x being Element of L holds
( x in it iff x is_a_root_of p );

existence
ex b₁ being Subset of L st
for x being Element of L holds
( x in b₁ iff x is_a_root_of p )

proof end;

uniqueness
for b₁, b₂ being Subset of L st ( for x being Element of L holds
( x in b₁ iff x is_a_root_of p ) ) & ( for x being Element of L holds
( x in b₂ iff x is_a_root_of p ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines Roots POLYNOM5:def 10 :
for L being non empty unital doubleLoopStr
for p being Polynomial of L
for b₃ being Subset of L holds
( b₃ = Roots p iff for x being Element of L holds
( x in b₃ iff x is_a_root_of p ) );

definition

let L be non empty almost_left_invertible well-unital distributive associative commutative doubleLoopStr ;
let p be Polynomial of L;

func NormPolynomial p -> sequence of L means :Def11: :: POLYNOM5:def 11
for n being Element of NAT holds it . n = (p . n) / (p . ((len p) -' 1));

existence
ex b₁ being sequence of L st
for n being Element of NAT holds b₁ . n = (p . n) / (p . ((len p) -' 1))

proof end;

uniqueness
for b₁, b₂ being sequence of L st ( for n being Element of NAT holds b₁ . n = (p . n) / (p . ((len p) -' 1)) ) & ( for n being Element of NAT holds b₂ . n = (p . n) / (p . ((len p) -' 1)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines NormPolynomial POLYNOM5:def 11 :
for L being non empty almost_left_invertible well-unital distributive associative commutative doubleLoopStr
for p being Polynomial of L
for b₃ being sequence of L holds
( b₃ = NormPolynomial p iff for n being Element of NAT holds b₃ . n = (p . n) / (p . ((len p) -' 1)) );

registration

let L be non empty right_complementable almost_left_invertible add-associative right_zeroed well-unital distributive associative commutative doubleLoopStr ;
let p be Polynomial of L;

cluster NormPolynomial p -> finite-Support ;

coherence
NormPolynomial p is finite-Support

proof end;

end;

theorem Th56: :: POLYNOM5:56

for L being non empty almost_left_invertible well-unital distributive associative commutative doubleLoopStr
for p being Polynomial of L st len p <> 0 holds
(NormPolynomial p) . ((len p) -' 1) = 1. L

proof end;

theorem Th57: :: POLYNOM5:57

for L being Field
for p being Polynomial of L st len p <> 0 holds
len (NormPolynomial p) = len p

proof end;

theorem Th58: :: POLYNOM5:58

for L being Field
for p being Polynomial of L st len p <> 0 holds
for x being Element of L holds eval ((NormPolynomial p),x) = (eval (p,x)) / (p . ((len p) -' 1))

proof end;

theorem Th59: :: POLYNOM5:59

for L being Field
for p being Polynomial of L st len p <> 0 holds
for x being Element of L holds
( x is_a_root_of p iff x is_a_root_of NormPolynomial p )

proof end;

theorem Th60: :: POLYNOM5:60

for L being Field
for p being Polynomial of L st len p <> 0 holds
( p is with_roots iff NormPolynomial p is with_roots )

proof end;

theorem :: POLYNOM5:61

for L being Field
for p being Polynomial of L st len p <> 0 holds
Roots p = Roots (NormPolynomial p)

proof end;

theorem Th62: :: POLYNOM5:62

id COMPLEX is_continuous_on COMPLEX

proof end;

theorem Th63: :: POLYNOM5:63

for x being Element of COMPLEX holds COMPLEX --> x is_continuous_on COMPLEX

proof end;

definition

let L be non empty unital multMagma ;
let x be Element of L;
let n be Nat;

func FPower (x,n) -> Function of L,L means :Def12: :: POLYNOM5:def 12
for y being Element of L holds it . y = x * (power (y,n));

existence
ex b₁ being Function of L,L st
for y being Element of L holds b₁ . y = x * (power (y,n))

proof end;

uniqueness
for b₁, b₂ being Function of L,L st ( for y being Element of L holds b₁ . y = x * (power (y,n)) ) & ( for y being Element of L holds b₂ . y = x * (power (y,n)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines FPower POLYNOM5:def 12 :
for L being non empty unital multMagma
for x being Element of L
for n being Nat
for b₄ being Function of L,L holds
( b₄ = FPower (x,n) iff for y being Element of L holds b₄ . y = x * (power (y,n)) );

theorem :: POLYNOM5:64

for L being non empty unital multMagma holds FPower ((1_ L),1) = id the carrier of L

proof end;

theorem :: POLYNOM5:65

FPower ((1_ F_Complex),2) = (id COMPLEX) (#) (id COMPLEX)

proof end;

theorem Th66: :: POLYNOM5:66

for L being non empty unital multMagma
for x being Element of L holds FPower (x,0) = the carrier of L --> x

proof end;

theorem :: POLYNOM5:67

for x being Element of F_Complex ex x1 being Element of COMPLEX st
( x = x1 & FPower (x,1) = x1 (#) (id COMPLEX) )

proof end;

theorem :: POLYNOM5:68

for x being Element of F_Complex ex x1 being Element of COMPLEX st
( x = x1 & FPower (x,2) = x1 (#) ((id COMPLEX) (#) (id COMPLEX)) )

proof end;

theorem Th69: :: POLYNOM5:69

for x being Element of F_Complex
for n being Nat ex f being Function of COMPLEX,COMPLEX st
( f = FPower (x,n) & FPower (x,(n + 1)) = f (#) (id COMPLEX) )

proof end;

theorem Th70: :: POLYNOM5:70

for x being Element of F_Complex
for n being Nat ex f being Function of COMPLEX,COMPLEX st
( f = FPower (x,n) & f is_continuous_on COMPLEX )

proof end;

definition

let L be non empty well-unital doubleLoopStr ;
let p be Polynomial of L;

func Polynomial-Function (L,p) -> Function of L,L means :Def13: :: POLYNOM5:def 13
for x being Element of L holds it . x = eval (p,x);

existence
ex b₁ being Function of L,L st
for x being Element of L holds b₁ . x = eval (p,x)

proof end;

uniqueness
for b₁, b₂ being Function of L,L st ( for x being Element of L holds b₁ . x = eval (p,x) ) & ( for x being Element of L holds b₂ . x = eval (p,x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines Polynomial-Function POLYNOM5:def 13 :
for L being non empty well-unital doubleLoopStr
for p being Polynomial of L
for b₃ being Function of L,L holds
( b₃ = Polynomial-Function (L,p) iff for x being Element of L holds b₃ . x = eval (p,x) );

theorem Th71: :: POLYNOM5:71

for p being Polynomial of F_Complex ex f being Function of COMPLEX,COMPLEX st
( f = Polynomial-Function (F_Complex,p) & f is_continuous_on COMPLEX )

proof end;

theorem Th72: :: POLYNOM5:72

for p being Polynomial of F_Complex st len p > 2 & |.(p . ((len p) -' 1)).| = 1 holds
for F being FinSequence of REAL st len F = len p & ( for n being Element of NAT st n in dom F holds
F . n = |.(p . (n -' 1)).| ) holds
for z being Element of F_Complex st |.z.| > Sum F holds
|.(eval (p,z)).| > |.(p . 0).| + 1

proof end;

theorem Th73: :: POLYNOM5:73

for p being Polynomial of F_Complex st len p > 2 holds
ex z0 being Element of F_Complex st
for z being Element of F_Complex holds |.(eval (p,z)).| >= |.(eval (p,z0)).|

proof end;

:: Fundamental Theorem of Algebra

theorem Th74: :: POLYNOM5:74

for p being Polynomial of F_Complex st len p > 1 holds
p is with_roots

proof end;

registration

cluster F_Complex -> algebraic-closed ;

coherence
F_Complex is algebraic-closed by Th74;

end;

registration

cluster non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed right_unital well-unital distributive left_unital unital associative commutative algebraic-closed for doubleLoopStr ;

existence
ex b₁ being non empty well-unital doubleLoopStr st
( b₁ is algebraic-closed & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is Abelian & b₁ is commutative & b₁ is associative & b₁ is distributive & b₁ is almost_left_invertible & not b₁ is degenerated )

proof end;

end;