:: The Algebra of Polynomials
:: by Ewa Gr\c{a}dzka
::
:: Received February 24, 2001
:: Copyright (c) 2001 Association of Mizar Users
:: deftheorem defines mix-associative POLYALG1:def 1 :
theorem Th1: :: POLYALG1:1
theorem Th2: :: POLYALG1:2
definition
let L be non
empty doubleLoopStr ;
func Formal-Series L -> non
empty strict AlgebraStr of
L means :
Def2:
:: POLYALG1:def 2
( ( for
x being
set holds
(
x in the
carrier of
it iff
x is
sequence of
L ) ) & ( for
x,
y being
Element of
it for
p,
q being
sequence of
L st
x = p &
y = q holds
x + y = p + q ) & ( for
x,
y being
Element of
it for
p,
q being
sequence of
L st
x = p &
y = q holds
x * y = p *' q ) & ( for
a being
Element of
L for
x being
Element of
it for
p being
sequence of
L st
x = p holds
a * x = a * p ) &
0. it = 0_. L &
1. it = 1_. L );
existence
ex b1 being non empty strict AlgebraStr of L st
( ( for x being set holds
( x in the carrier of b1 iff x is sequence of L ) ) & ( for x, y being Element of b1
for p, q being sequence of L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b1
for p, q being sequence of L st x = p & y = q holds
x * y = p *' q ) & ( for a being Element of L
for x being Element of b1
for p being sequence of L st x = p holds
a * x = a * p ) & 0. b1 = 0_. L & 1. b1 = 1_. L )
uniqueness
for b1, b2 being non empty strict AlgebraStr of L st ( for x being set holds
( x in the carrier of b1 iff x is sequence of L ) ) & ( for x, y being Element of b1
for p, q being sequence of L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b1
for p, q being sequence of L st x = p & y = q holds
x * y = p *' q ) & ( for a being Element of L
for x being Element of b1
for p being sequence of L st x = p holds
a * x = a * p ) & 0. b1 = 0_. L & 1. b1 = 1_. L & ( for x being set holds
( x in the carrier of b2 iff x is sequence of L ) ) & ( for x, y being Element of b2
for p, q being sequence of L st x = p & y = q holds
x + y = p + q ) & ( for x, y being Element of b2
for p, q being sequence of L st x = p & y = q holds
x * y = p *' q ) & ( for a being Element of L
for x being Element of b2
for p being sequence of L st x = p holds
a * x = a * p ) & 0. b2 = 0_. L & 1. b2 = 1_. L holds
b1 = b2
end;
:: deftheorem Def2 defines Formal-Series POLYALG1:def 2 :
theorem Th3: :: POLYALG1:3
theorem Th4: :: POLYALG1:4
theorem Th5: :: POLYALG1:5
theorem Th6: :: POLYALG1:6
theorem Th7: :: POLYALG1:7
theorem Th8: :: POLYALG1:8
theorem Th9: :: POLYALG1:9
consider D00 being non empty right_complementable associative distributive add-associative right_zeroed doubleLoopStr ;
theorem Th10: :: POLYALG1:10
:: deftheorem Def3 defines Subalgebra POLYALG1:def 3 :
theorem Th11: :: POLYALG1:11
theorem :: POLYALG1:12
theorem :: POLYALG1:13
for
L being
1-sorted for
A,
B being
AlgebraStr of
L st
A is
Subalgebra of
B &
B is
Subalgebra of
A holds
AlgebraStr(# the
carrier of
A,the
addF of
A,the
multF of
A,the
ZeroF of
A,the
OneF of
A,the
lmult of
A #)
= AlgebraStr(# the
carrier of
B,the
addF of
B,the
multF of
B,the
ZeroF of
B,the
OneF of
B,the
lmult of
B #)
theorem Th14: :: POLYALG1:14
for
L being
1-sorted for
A,
B being
AlgebraStr of
L st
AlgebraStr(# the
carrier of
A,the
addF of
A,the
multF of
A,the
ZeroF of
A,the
OneF of
A,the
lmult of
A #)
= AlgebraStr(# the
carrier of
B,the
addF of
B,the
multF of
B,the
ZeroF of
B,the
OneF of
B,the
lmult of
B #) holds
A is
Subalgebra of
B
:: deftheorem Def4 defines opers_closed POLYALG1:def 4 :
theorem Th15: :: POLYALG1:15
theorem Th16: :: POLYALG1:16
theorem Th17: :: POLYALG1:17
theorem :: POLYALG1:18
canceled;
theorem :: POLYALG1:19
theorem Th20: :: POLYALG1:20
theorem Th21: :: POLYALG1:21
:: deftheorem Def5 defines GenAlg POLYALG1:def 5 :
theorem Th22: :: POLYALG1:22
:: deftheorem Def6 defines Polynom-Algebra POLYALG1:def 6 :
theorem Th23: :: POLYALG1:23
theorem :: POLYALG1:24
theorem :: POLYALG1:25