:: Multivariate polynomials with arbitrary number of variables
:: by Piotr Rudnicki and Andrzej Trybulec
::
:: Received September 22, 1999
:: Copyright (c) 1999 Association of Mizar Users
theorem :: POLYNOM1:1
canceled;
theorem :: POLYNOM1:2
canceled;
theorem :: POLYNOM1:3
canceled;
theorem :: POLYNOM1:4
canceled;
theorem Th5: :: POLYNOM1:5
theorem :: POLYNOM1:6
canceled;
theorem :: POLYNOM1:7
canceled;
theorem :: POLYNOM1:8
canceled;
theorem :: POLYNOM1:9
canceled;
theorem :: POLYNOM1:10
canceled;
theorem :: POLYNOM1:11
canceled;
theorem :: POLYNOM1:12
canceled;
theorem :: POLYNOM1:13
canceled;
theorem :: POLYNOM1:14
canceled;
theorem Th15: :: POLYNOM1:15
theorem Th16: :: POLYNOM1:16
theorem Th17: :: POLYNOM1:17
theorem Th18: :: POLYNOM1:18
:: deftheorem POLYNOM1:def 1 :
canceled;
:: deftheorem Def2 defines * POLYNOM1:def 2 :
:: deftheorem Def3 defines * POLYNOM1:def 3 :
theorem Th19: :: POLYNOM1:19
theorem Th20: :: POLYNOM1:20
theorem Th21: :: POLYNOM1:21
theorem Th22: :: POLYNOM1:22
theorem Th23: :: POLYNOM1:23
theorem Th24: :: POLYNOM1:24
theorem :: POLYNOM1:25
canceled;
theorem :: POLYNOM1:26
canceled;
theorem :: POLYNOM1:27
canceled;
theorem Th28: :: POLYNOM1:28
theorem Th29: :: POLYNOM1:29
theorem :: POLYNOM1:30
canceled;
theorem :: POLYNOM1:31
canceled;
theorem :: POLYNOM1:32
canceled;
theorem :: POLYNOM1:33
canceled;
theorem Th34: :: POLYNOM1:34
:: deftheorem POLYNOM1:def 4 :
canceled;
:: deftheorem POLYNOM1:def 5 :
canceled;
:: deftheorem POLYNOM1:def 6 :
canceled;
:: deftheorem POLYNOM1:def 7 :
canceled;
:: deftheorem POLYNOM1:def 8 :
canceled;
:: deftheorem Def9 defines Support POLYNOM1:def 9 :
:: deftheorem Def10 defines finite-Support POLYNOM1:def 10 :
:: deftheorem POLYNOM1:def 11 :
canceled;
:: deftheorem POLYNOM1:def 12 :
canceled;
:: deftheorem POLYNOM1:def 13 :
canceled;
:: deftheorem POLYNOM1:def 14 :
canceled;
:: deftheorem POLYNOM1:def 15 :
canceled;
:: deftheorem POLYNOM1:def 16 :
canceled;
:: deftheorem POLYNOM1:def 17 :
canceled;
:: deftheorem POLYNOM1:def 18 :
canceled;
:: deftheorem POLYNOM1:def 19 :
canceled;
:: deftheorem POLYNOM1:def 20 :
canceled;
:: deftheorem Def21 defines + POLYNOM1:def 21 :
theorem :: POLYNOM1:35
canceled;
theorem :: POLYNOM1:36
canceled;
theorem :: POLYNOM1:37
canceled;
theorem :: POLYNOM1:38
canceled;
theorem :: POLYNOM1:39
canceled;
theorem :: POLYNOM1:40
canceled;
theorem :: POLYNOM1:41
canceled;
theorem :: POLYNOM1:42
canceled;
theorem :: POLYNOM1:43
canceled;
theorem :: POLYNOM1:44
canceled;
theorem :: POLYNOM1:45
canceled;
theorem :: POLYNOM1:46
canceled;
theorem :: POLYNOM1:47
canceled;
theorem :: POLYNOM1:48
canceled;
theorem :: POLYNOM1:49
canceled;
theorem :: POLYNOM1:50
canceled;
theorem :: POLYNOM1:51
canceled;
theorem :: POLYNOM1:52
canceled;
theorem :: POLYNOM1:53
canceled;
theorem :: POLYNOM1:54
canceled;
theorem :: POLYNOM1:55
canceled;
theorem :: POLYNOM1:56
canceled;
theorem :: POLYNOM1:57
canceled;
theorem :: POLYNOM1:58
canceled;
theorem :: POLYNOM1:59
canceled;
theorem :: POLYNOM1:60
canceled;
theorem :: POLYNOM1:61
canceled;
theorem :: POLYNOM1:62
canceled;
theorem :: POLYNOM1:63
canceled;
theorem :: POLYNOM1:64
canceled;
theorem :: POLYNOM1:65
canceled;
theorem :: POLYNOM1:66
canceled;
theorem :: POLYNOM1:67
canceled;
theorem :: POLYNOM1:68
canceled;
theorem :: POLYNOM1:69
canceled;
theorem :: POLYNOM1:70
canceled;
theorem :: POLYNOM1:71
canceled;
theorem :: POLYNOM1:72
canceled;
theorem :: POLYNOM1:73
canceled;
theorem :: POLYNOM1:74
canceled;
theorem :: POLYNOM1:75
canceled;
theorem :: POLYNOM1:76
canceled;
theorem :: POLYNOM1:77
canceled;
theorem :: POLYNOM1:78
canceled;
theorem Th79: :: POLYNOM1:79
theorem Th80: :: POLYNOM1:80
:: deftheorem Def22 defines - POLYNOM1:def 22 :
:: deftheorem defines - POLYNOM1:def 23 :
:: deftheorem defines 0_ POLYNOM1:def 24 :
theorem Th81: :: POLYNOM1:81
theorem Th82: :: POLYNOM1:82
:: deftheorem defines 1_ POLYNOM1:def 25 :
theorem Th83: :: POLYNOM1:83
theorem Th84: :: POLYNOM1:84
definition
let n be
Ordinal;
let L be non
empty right_complementable add-associative right_zeroed doubleLoopStr ;
let p,
q be
Series of
n,
L;
func p *' q -> Series of
n,
L means :
Def26:
:: POLYNOM1:def 26
for
b being
bag of ex
s being
FinSequence of the
carrier of
L st
(
it . b = Sum s &
len s = len (decomp b) & ( for
k being
Element of
NAT st
k in dom s holds
ex
b1,
b2 being
bag of st
(
(decomp b) /. k = <*b1,b2*> &
s /. k = (p . b1) * (q . b2) ) ) );
existence
ex b1 being Series of n,L st
for b being bag of ex s being FinSequence of the carrier of L st
( b1 . b = Sum s & len s = len (decomp b) & ( for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of st
( (decomp b) /. k = <*b1,b2*> & s /. k = (p . b1) * (q . b2) ) ) )
uniqueness
for b1, b2 being Series of n,L st ( for b being bag of ex s being FinSequence of the carrier of L st
( b1 . b = Sum s & len s = len (decomp b) & ( for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of st
( (decomp b) /. k = <*b1,b2*> & s /. k = (p . b1) * (q . b2) ) ) ) ) & ( for b being bag of ex s being FinSequence of the carrier of L st
( b2 . b = Sum s & len s = len (decomp b) & ( for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of st
( (decomp b) /. k = <*b1,b2*> & s /. k = (p . b1) * (q . b2) ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def26 defines *' POLYNOM1:def 26 :
theorem Th85: :: POLYNOM1:85
theorem Th86: :: POLYNOM1:86
theorem :: POLYNOM1:87
theorem Th88: :: POLYNOM1:88
theorem Th89: :: POLYNOM1:89
definition
let n be
Ordinal;
let L be non
empty non
trivial right_complementable right_unital distributive add-associative right_zeroed doubleLoopStr ;
func Polynom-Ring n,
L -> non
empty strict doubleLoopStr means :
Def27:
:: POLYNOM1:def 27
( ( for
x being
set holds
(
x in the
carrier of
it iff
x is
Polynomial of
n,
L ) ) & ( for
x,
y being
Element of
it for
p,
q being
Polynomial of
n,
L st
x = p &
y = q holds
x + y = p + q ) & ( for
x,
y being
Element of
it for
p,
q being
Polynomial of
n,
L st
x = p &
y = q holds
x * y = p *' q ) &
0. it = 0_ n,
L &
1. it = 1_ n,
L );
existence
ex b1 being non empty strict doubleLoopStr st
( ( for x being set holds
( x in the carrier of b1 iff x is Polynomial of n,L ) ) & ( for x, y being Element of b1
for p, q being Polynomial of n,L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b1
for p, q being Polynomial of n,L st x = p & y = q holds
x * y = p *' q ) & 0. b1 = 0_ n,L & 1. b1 = 1_ n,L )
uniqueness
for b1, b2 being non empty strict doubleLoopStr st ( for x being set holds
( x in the carrier of b1 iff x is Polynomial of n,L ) ) & ( for x, y being Element of b1
for p, q being Polynomial of n,L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b1
for p, q being Polynomial of n,L st x = p & y = q holds
x * y = p *' q ) & 0. b1 = 0_ n,L & 1. b1 = 1_ n,L & ( for x being set holds
( x in the carrier of b2 iff x is Polynomial of n,L ) ) & ( for x, y being Element of b2
for p, q being Polynomial of n,L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b2
for p, q being Polynomial of n,L st x = p & y = q holds
x * y = p *' q ) & 0. b2 = 0_ n,L & 1. b2 = 1_ n,L holds
b1 = b2
end;
:: deftheorem Def27 defines Polynom-Ring POLYNOM1:def 27 :
for
n being
Ordinal for
L being non
empty non
trivial right_complementable right_unital distributive add-associative right_zeroed doubleLoopStr for
b3 being non
empty strict doubleLoopStr holds
(
b3 = Polynom-Ring n,
L iff ( ( for
x being
set holds
(
x in the
carrier of
b3 iff
x is
Polynomial of
n,
L ) ) & ( for
x,
y being
Element of
b3 for
p,
q being
Polynomial of
n,
L st
x = p &
y = q holds
x + y = p + q ) & ( for
x,
y being
Element of
b3 for
p,
q being
Polynomial of
n,
L st
x = p &
y = q holds
x * y = p *' q ) &
0. b3 = 0_ n,
L &
1. b3 = 1_ n,
L ) );
Lm5:
now
let n be
Ordinal;
:: thesis: for L being non empty non trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for x, e being Element of (Polynom-Ring n,L) st e = 1. (Polynom-Ring n,L) holds
( x * e = x & e * x = x )let L be non
empty non
trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ;
:: thesis: for x, e being Element of (Polynom-Ring n,L) st e = 1. (Polynom-Ring n,L) holds
( x * e = x & e * x = x )set Pm =
Polynom-Ring n,
L;
let x,
e be
Element of
(Polynom-Ring n,L);
:: thesis: ( e = 1. (Polynom-Ring n,L) implies ( x * e = x & e * x = x ) )reconsider p =
x as
Polynomial of
n,
L by Def27;
assume A1:
e = 1. (Polynom-Ring n,L)
;
:: thesis: ( x * e = x & e * x = x )A2:
1. (Polynom-Ring n,L) = 1_ n,
L
by Def27;
hence x * e =
p *' (1_ n,L)
by A1, Def27
.=
x
by Th88
;
:: thesis: e * x = xthus e * x =
(1_ n,L) *' p
by A1, A2, Def27
.=
x
by Th89
;
:: thesis: verum
end;
theorem :: POLYNOM1:90