deffunc H1( non empty Element of NAT ) -> Polynomial of F_Complex = cyclotomic_poly $1;
set cPRFC = the carrier of (Polynom-Ring F_Complex );
set cFC = the carrier of F_Complex ;
defpred S1[ non empty Element of NAT ] means ( ( H1($1) . 0 = 1 or H1($1) . 0 = - 1 ) & ( for i being Element of NAT holds H1($1) . i is integer ) );
A1:
- (1_ F_Complex ) = - 1
by COMPLFLD:4, COMPLFLD:10;
A2:
now let k be non
empty Element of
NAT ;
( ( for n being non empty Element of NAT st n < k holds
S1[n] ) implies S1[b1] )assume A3:
for
n being non
empty Element of
NAT st
n < k holds
S1[
n]
;
S1[b1]A4:
1
<= k
by Lm1;
per cases
( k = 1 or k > 1 )
by A4, XXREAL_0:1;
suppose A6:
k > 1
;
S1[b1]consider f being
FinSequence of the
carrier of
(Polynom-Ring F_Complex ),
p being
Polynomial of
F_Complex such that A7:
p = Product f
and A8:
dom f = Seg k
and A9:
for
i being non
empty Element of
NAT st
i in Seg k holds
( ( ( not
i divides k or
i = k ) implies
f . i = <%(1_ F_Complex )%> ) & (
i divides k &
i <> k implies
f . i = H1(
i) ) )
and A10:
unital_poly F_Complex ,
k = (cyclotomic_poly k) *' p
by Th54;
defpred S2[
Nat,
set ]
means ex
g being
FinSequence of the
carrier of
(Polynom-Ring F_Complex ) ex
p being
Polynomial of
F_Complex st
(
g = f | (Seg $1) &
p = Product g & $2
= p & (
p . 0 = 1 or
p . 0 = - 1 ) & ( for
j being
Element of
NAT holds
p . j is
integer ) );
defpred S3[
Element of
NAT ]
means ( $1
in Seg (len f) implies ex
x being
set st
S2[$1,
x] );
A11:
k = len f
by A8, FINSEQ_1:def 3;
A12:
for
l being
Element of
NAT st
S3[
l] holds
S3[
l + 1]
proof
let l be
Element of
NAT ;
( S3[l] implies S3[l + 1] )
assume A13:
S3[
l]
;
S3[l + 1]
assume A14:
l + 1
in Seg (len f)
;
ex x being set st S2[l + 1,x]
per cases
( l = 0 or 0 < l )
;
suppose A15:
l = 0
;
ex x being set st S2[l + 1,x]reconsider fl1 =
f . (l + 1) as
Element of the
carrier of
(Polynom-Ring F_Complex ) by A8, A11, A14, FINSEQ_2:13;
reconsider g =
f | (Seg (l + 1)) as
FinSequence of the
carrier of
(Polynom-Ring F_Complex ) by FINSEQ_1:23;
reconsider p =
Product g as
Polynomial of
F_Complex by POLYNOM3:def 12;
<*> the
carrier of
(Polynom-Ring F_Complex ) = f | (Seg 0 )
;
then g =
(<*> the carrier of (Polynom-Ring F_Complex )) ^ <*(f . (l + 1))*>
by A8, A11, A14, A15, FINSEQ_5:11
.=
<*(f . (l + 1))*>
by FINSEQ_1:47
;
then A16:
p = fl1
by FINSOP_1:12;
take
p
;
S2[l + 1,p]take
g
;
ex p being Polynomial of F_Complex st
( g = f | (Seg (l + 1)) & p = Product g & p = p & ( p . 0 = 1 or p . 0 = - 1 ) & ( for j being Element of NAT holds p . j is integer ) )take
p
;
( g = f | (Seg (l + 1)) & p = Product g & p = p & ( p . 0 = 1 or p . 0 = - 1 ) & ( for j being Element of NAT holds p . j is integer ) )thus
(
g = f | (Seg (l + 1)) &
p = Product g &
p = p )
;
( ( p . 0 = 1 or p . 0 = - 1 ) & ( for j being Element of NAT holds p . j is integer ) )
1
divides k
by NAT_D:6;
then A17:
f . 1
= H1(1)
by A6, A9, A11, A14, A15;
hence
(
p . 0 = 1 or
p . 0 = - 1 )
by A1, A15, A16, Th52, POLYNOM5:39;
for j being Element of NAT holds p . j is integer let j be
Element of
NAT ;
p . j is integer thus
p . j is
integer
verum end; suppose A18:
0 < l
;
ex x being set st S2[l + 1,x]
(
l + 1
<= len f &
l <= l + 1 )
by A14, FINSEQ_1:3, NAT_1:12;
then A19:
l <= len f
by XXREAL_0:2;
0 + 1
<= l
by A18, NAT_1:13;
then consider x being
set such that A20:
S2[
l,
x]
by A13, A19, FINSEQ_1:3;
reconsider fl1 =
f . (l + 1) as
Element of the
carrier of
(Polynom-Ring F_Complex ) by A8, A11, A14, FINSEQ_2:13;
reconsider g1 =
f | (Seg (l + 1)) as
FinSequence of the
carrier of
(Polynom-Ring F_Complex ) by FINSEQ_1:23;
reconsider p1 =
Product g1 as
Polynomial of
F_Complex by POLYNOM3:def 12;
take
p1
;
S2[l + 1,p1]take
g1
;
ex p being Polynomial of F_Complex st
( g1 = f | (Seg (l + 1)) & p = Product g1 & p1 = p & ( p . 0 = 1 or p . 0 = - 1 ) & ( for j being Element of NAT holds p . j is integer ) )take
p1
;
( g1 = f | (Seg (l + 1)) & p1 = Product g1 & p1 = p1 & ( p1 . 0 = 1 or p1 . 0 = - 1 ) & ( for j being Element of NAT holds p1 . j is integer ) )thus
(
g1 = f | (Seg (l + 1)) &
p1 = Product g1 &
p1 = p1 )
;
( ( p1 . 0 = 1 or p1 . 0 = - 1 ) & ( for j being Element of NAT holds p1 . j is integer ) )reconsider fl1p =
fl1 as
Polynomial of
F_Complex by POLYNOM3:def 12;
reconsider m1 =
- 1 as
Element of
COMPLEX by XCMPLX_0:def 2;
consider g being
FinSequence of the
carrier of
(Polynom-Ring F_Complex ),
p being
Polynomial of
F_Complex such that A21:
g = f | (Seg l)
and A22:
p = Product g
and
x = p
and A23:
(
p . 0 = 1 or
p . 0 = - 1 )
and A24:
for
j being
Element of
NAT holds
p . j is
integer
by A20;
g1 = g ^ <*fl1*>
by A8, A11, A14, A21, FINSEQ_5:11;
then
Product g1 = (Product g) * fl1
by GROUP_4:9;
then A25:
p1 = p *' fl1p
by A22, POLYNOM3:def 12;
thus
( (
p1 . 0 = 1 or
p1 . 0 = - 1 ) & ( for
j being
Element of
NAT holds
p1 . j is
integer ) )
verumproof
per cases
( not l + 1 divides k or l + 1 = k or ( l + 1 divides k & l + 1 <> k ) )
;
suppose
( not
l + 1
divides k or
l + 1
= k )
;
( ( p1 . 0 = 1 or p1 . 0 = - 1 ) & ( for j being Element of NAT holds p1 . j is integer ) )then A26:
fl1p = <%(1_ F_Complex )%>
by A9, A11, A14;
consider r being
FinSequence of
F_Complex such that A27:
len r = 0 + 1
and A28:
p1 . 0 = Sum r
and A29:
for
m being
Element of
NAT st
m in dom r holds
r . m = (p . (m -' 1)) * (fl1p . ((0 + 1) -' m))
by A25, POLYNOM3:def 11;
1
in dom r
by A27, FINSEQ_3:27;
then reconsider r1 =
r . 1 as
Element of
F_Complex by FINSEQ_2:13;
r = <*r1*>
by A27, FINSEQ_1:57;
then A30:
p1 . 0 = r1
by A28, RLVECT_1:61;
1
in dom r
by A27, FINSEQ_3:27;
then A31:
p1 . 0 =
(p . (1 -' 1)) * (fl1p . ((0 + 1) -' 1))
by A29, A30
.=
(p . ((0 + 1) -' 1)) * (fl1p . 0 )
by NAT_D:34
.=
(p . 0 ) * (fl1p . 0 )
by NAT_D:34
.=
(p . 0 ) * (1_ F_Complex )
by A26, POLYNOM5:33
;
thus
(
p1 . 0 = 1 or
p1 . 0 = - 1 )
for j being Element of NAT holds p1 . j is integer let j be
Element of
NAT ;
p1 . j is integer consider r being
FinSequence of
F_Complex such that
len r = j + 1
and A32:
p1 . j = Sum r
and A33:
for
m being
Element of
NAT st
m in dom r holds
r . m = (p . (m -' 1)) * (fl1p . ((j + 1) -' m))
by A25, POLYNOM3:def 11;
for
i being
Element of
NAT st
i in dom r holds
r . i is
integer
hence
p1 . j is
integer
by A32, Th5;
verum end; suppose A35:
(
l + 1
divides k &
l + 1
<> k )
;
( ( p1 . 0 = 1 or p1 . 0 = - 1 ) & ( for j being Element of NAT holds p1 . j is integer ) )consider r being
FinSequence of
F_Complex such that A36:
len r = 0 + 1
and A37:
p1 . 0 = Sum r
and A38:
for
m being
Element of
NAT st
m in dom r holds
r . m = (p . (m -' 1)) * (fl1p . ((0 + 1) -' m))
by A25, POLYNOM3:def 11;
1
in dom r
by A36, FINSEQ_3:27;
then reconsider r1 =
r . 1 as
Element of
F_Complex by FINSEQ_2:13;
r = <*r1*>
by A36, FINSEQ_1:57;
then A39:
p1 . 0 = r1
by A37, RLVECT_1:61;
1
in dom r
by A36, FINSEQ_3:27;
then A40:
p1 . 0 =
(p . (1 -' 1)) * (fl1p . ((0 + 1) -' 1))
by A38, A39
.=
(p . ((0 + 1) -' 1)) * (fl1p . 0 )
by NAT_D:34
.=
(p . 0 ) * (fl1p . 0 )
by NAT_D:34
;
l + 1
<= k
by A35, NAT_D:7;
then A41:
l + 1
< k
by A35, XXREAL_0:1;
A42:
fl1p = H1(
l + 1)
by A9, A11, A14, A35;
then reconsider fl1p0 =
fl1p . 0 as
Integer by A3, A41;
A43:
(
fl1p0 = 1 or
fl1p0 = m1 )
by A3, A42, A41;
thus
(
p1 . 0 = 1 or
p1 . 0 = - 1 )
for j being Element of NAT holds p1 . j is integer let j be
Element of
NAT ;
p1 . j is integer consider r being
FinSequence of
F_Complex such that
len r = j + 1
and A44:
p1 . j = Sum r
and A45:
for
m being
Element of
NAT st
m in dom r holds
r . m = (p . (m -' 1)) * (fl1p . ((j + 1) -' m))
by A25, POLYNOM3:def 11;
for
i being
Element of
NAT st
i in dom r holds
r . i is
integer
hence
p1 . j is
integer
by A44, Th5;
verum end;
end;
end; defpred S4[
Nat]
means H1(
k)
. $1 is
integer ;
A46:
(0 + 1) -' 1
= 0
by NAT_D:34;
A47:
S3[
0 ]
by FINSEQ_1:3;
for
l being
Element of
NAT holds
S3[
l]
from NAT_1:sch 1(A47, A12);
then A48:
for
l being
Nat st
l in Seg (len f) holds
ex
x being
set st
S2[
l,
x]
;
consider F being
FinSequence such that
dom F = Seg (len f)
and A49:
for
i being
Nat st
i in Seg (len f) holds
S2[
i,
F . i]
from FINSEQ_1:sch 1(A48);
consider g being
FinSequence of the
carrier of
(Polynom-Ring F_Complex ),
p1 being
Polynomial of
F_Complex such that A50:
(
g = f | (Seg k) &
p1 = Product g )
and
F . k = p1
and A51:
(
p1 . 0 = 1 or
p1 . 0 = - 1 )
and A52:
for
j being
Element of
NAT holds
p1 . j is
integer
by A11, A49, FINSEQ_1:5;
A53:
p = p1
by A7, A11, A50, FINSEQ_3:55;
A54:
now let m be
Nat;
( ( for n being Nat st n < m holds
S4[n] ) implies S4[b1] )reconsider m1 =
m as
Element of
NAT by ORDINAL1:def 13;
consider r being
FinSequence of the
carrier of
F_Complex such that A55:
len r = m + 1
and A56:
(unital_poly F_Complex ,k) . m = Sum r
and A57:
for
l being
Element of
NAT st
l in dom r holds
r . l = (p . (l -' 1)) * (H1(k) . ((m1 + 1) -' l))
by A10, POLYNOM3:def 11;
reconsider Src =
Sum r as
Element of
COMPLEX by COMPLFLD:def 1;
then reconsider Sr =
Src as
Integer ;
A58:
(1,1 -cut r) ^ ((1 + 1),(len r) -cut r) = r
by A55, GRAPH_2:9, NAT_1:11;
set s =
(1 + 1),
(len r) -cut r;
reconsider Ssc =
Sum ((1 + 1),(len r) -cut r) as
Element of
COMPLEX by COMPLFLD:def 1;
assume A59:
for
n being
Nat st
n < m holds
S4[
n]
;
S4[b1]now let i be
Element of
NAT ;
( i in dom ((1 + 1),(len r) -cut r) implies ((1 + 1),(len r) -cut r) . b1 is integer )assume A60:
i in dom ((1 + 1),(len r) -cut r)
;
((1 + 1),(len r) -cut r) . b1 is integer per cases
( len r < 2 or 1 + 1 <= len r )
;
suppose A61:
1
+ 1
<= len r
;
((1 + 1),(len r) -cut r) . b1 is integer then A62:
(len ((1 + 1),(len r) -cut r)) + (1 + 1) = (len r) + 1
by GRAPH_2:def 1;
per cases
( m = 0 or m > 0 )
;
suppose A63:
m > 0
;
((1 + 1),(len r) -cut r) . b1 is integer
i <> 0
by A60, FINSEQ_3:27;
then reconsider cpkmi =
H1(
k)
. (m -' i) as
Integer by A59, A63, NAT_2:11;
reconsider ppi =
p . i as
Integer by A52, A53;
i <> 0
by A60, FINSEQ_3:27;
then consider i1 being
Nat such that A64:
i = i1 + 1
by NAT_1:6;
A65:
i <= len ((1 + 1),(len r) -cut r)
by A60, FINSEQ_3:27;
then
( 1
<= i + 1 &
i + 1
<= (len ((1 + 1),(len r) -cut r)) + 1 )
by NAT_1:11, XREAL_1:8;
then
1
+ i in dom r
by A62, FINSEQ_3:27;
then A66:
r . (1 + i) =
(p . ((1 + i) -' 1)) * (H1(k) . ((m + 1) -' (1 + i)))
by A57
.=
(p . ((i + 1) -' 1)) * (H1(k) . (((m + 1) -' 1) -' i))
by NAT_2:32
.=
(p . i) * (H1(k) . (((m + 1) -' 1) -' i))
by NAT_D:34
.=
ppi * cpkmi
by NAT_D:34
;
i1 < len ((1 + 1),(len r) -cut r)
by A65, A64, NAT_1:13;
then ((1 + 1),(len r) -cut r) . i =
r . ((1 + 1) + i1)
by A61, A64, GRAPH_2:def 1
.=
r . (1 + i)
by A64
;
hence
((1 + 1),(len r) -cut r) . i is
integer
by A66;
verum end; end; end; then reconsider Ss =
Ssc as
Integer by Th5;
A67:
1
<= len r
by A55, NAT_1:11;
then A68:
1
in dom r
by FINSEQ_3:27;
then reconsider r1 =
r . 1 as
Element of the
carrier of
F_Complex by FINSEQ_2:13;
reconsider r1c =
r1 as
Element of
COMPLEX by COMPLFLD:def 1;
1,1
-cut r = <*r1*>
by A67, GRAPH_2:6;
then
Sum r = r1 + (Sum ((1 + 1),(len r) -cut r))
by A58, FVSUM_1:89;
then
r1c = Sr - Ss
;
then reconsider r1i =
r1 as
Integer ;
A69:
r1i =
(p . (1 -' 1)) * (H1(k) . ((m + 1) -' 1))
by A57, A68
.=
(p . 0 ) * (H1(k) . m1)
by A46, NAT_D:34
;
end; A70:
for
i being
Nat holds
S4[
i]
from NAT_1:sch 4(A54);
consider r being
FinSequence of the
carrier of
F_Complex such that A71:
len r = 0 + 1
and A72:
(unital_poly F_Complex ,k) . 0 = Sum r
and A73:
for
l being
Element of
NAT st
l in dom r holds
r . l = (p . (l -' 1)) * (H1(k) . ((0 + 1) -' l))
by A10, POLYNOM3:def 11;
A74:
1
in dom r
by A71, FINSEQ_3:27;
then reconsider r1 =
r . 1 as
Element of the
carrier of
F_Complex by FINSEQ_2:13;
r = <*r1*>
by A71, FINSEQ_1:57;
then A75:
Sum r =
r . 1
by RLVECT_1:61
.=
(p . 0 ) * (H1(k) . 0 )
by A73, A46, A74
;
(
H1(
k)
. 0 = 1 or
H1(
k)
. 0 = - 1 )
hence
S1[
k]
by A70;
verum end; end;
for d being non empty Element of NAT holds S1[d]
from UNIROOTS:sch 1(A2);
hence
for d being non empty Element of NAT
for i being Element of NAT holds
( ( (cyclotomic_poly d) . 0 = 1 or (cyclotomic_poly d) . 0 = - 1 ) & (cyclotomic_poly d) . i is integer )
; verum