:: POLYALG1 semantic presentation
:: deftheorem Def1 defines mix-associative POLYALG1:def 1 :
theorem Th1: :: POLYALG1:1
theorem Th2: :: POLYALG1:2
definition
let c
1 be non
empty doubleLoopStr ;
func Formal-Series c
1 -> non
empty strict AlgebraStr of a
1 means :
Def2:
:: POLYALG1:def 2
( ( for b
1 being
set holds
( b
1 in the
carrier of a
2 iff b
1 is
sequence of a
1 ) ) & ( for b
1, b
2 being
Element of a
2for b
3, b
4 being
sequence of a
1 holds
( b
1 = b
3 & b
2 = b
4 implies b
1 + b
2 = b
3 + b
4 ) ) & ( for b
1, b
2 being
Element of a
2for b
3, b
4 being
sequence of a
1 holds
( b
1 = b
3 & b
2 = b
4 implies b
1 * b
2 = b
3 *' b
4 ) ) & ( for b
1 being
Element of a
1for b
2 being
Element of a
2for b
3 being
sequence of a
1 holds
( b
2 = b
3 implies b
1 * b
2 = b
1 * b
3 ) ) &
0. a
2 = 0_. a
1 &
1_ a
2 = 1_. a
1 );
existence
ex b1 being non empty strict AlgebraStr of c1 st
( ( for b2 being set holds
( b2 in the carrier of b1 iff b2 is sequence of c1 ) ) & ( for b2, b3 being Element of b1
for b4, b5 being sequence of c1 holds
( b2 = b4 & b3 = b5 implies b2 + b3 = b4 + b5 ) ) & ( for b2, b3 being Element of b1
for b4, b5 being sequence of c1 holds
( b2 = b4 & b3 = b5 implies b2 * b3 = b4 *' b5 ) ) & ( for b2 being Element of c1
for b3 being Element of b1
for b4 being sequence of c1 holds
( b3 = b4 implies b2 * b3 = b2 * b4 ) ) & 0. b1 = 0_. c1 & 1_ b1 = 1_. c1 )
uniqueness
for b1, b2 being non empty strict AlgebraStr of c1 holds
( ( for b3 being set holds
( b3 in the carrier of b1 iff b3 is sequence of c1 ) ) & ( for b3, b4 being Element of b1
for b5, b6 being sequence of c1 holds
( b3 = b5 & b4 = b6 implies b3 + b4 = b5 + b6 ) ) & ( for b3, b4 being Element of b1
for b5, b6 being sequence of c1 holds
( b3 = b5 & b4 = b6 implies b3 * b4 = b5 *' b6 ) ) & ( for b3 being Element of c1
for b4 being Element of b1
for b5 being sequence of c1 holds
( b4 = b5 implies b3 * b4 = b3 * b5 ) ) & 0. b1 = 0_. c1 & 1_ b1 = 1_. c1 & ( for b3 being set holds
( b3 in the carrier of b2 iff b3 is sequence of c1 ) ) & ( for b3, b4 being Element of b2
for b5, b6 being sequence of c1 holds
( b3 = b5 & b4 = b6 implies b3 + b4 = b5 + b6 ) ) & ( for b3, b4 being Element of b2
for b5, b6 being sequence of c1 holds
( b3 = b5 & b4 = b6 implies b3 * b4 = b5 *' b6 ) ) & ( for b3 being Element of c1
for b4 being Element of b2
for b5 being sequence of c1 holds
( b4 = b5 implies b3 * b4 = b3 * b5 ) ) & 0. b2 = 0_. c1 & 1_ b2 = 1_. c1 implies 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
theorem Th10: :: POLYALG1:10
:: deftheorem Def3 defines Subalgebra POLYALG1:def 3 :
theorem Th11: :: POLYALG1:11
theorem Th12: :: POLYALG1:12
theorem Th13: :: POLYALG1:13
for b
1 being
1-sorted for b
2, b
3 being
AlgebraStr of b
1 holds
( b
2 is
Subalgebra of b
3 & b
3 is
Subalgebra of b
2 implies
AlgebraStr(# the
carrier of b
2,the
add of b
2,the
mult of b
2,the
Zero of b
2,the
unity of b
2,the
lmult of b
2 #)
= AlgebraStr(# the
carrier of b
3,the
add of b
3,the
mult of b
3,the
Zero of b
3,the
unity of b
3,the
lmult of b
3 #) )
theorem Th14: :: POLYALG1:14
for b
1 being
1-sorted for b
2, b
3 being
AlgebraStr of b
1 holds
(
AlgebraStr(# the
carrier of b
2,the
add of b
2,the
mult of b
2,the
Zero of b
2,the
unity of b
2,the
lmult of b
2 #)
= AlgebraStr(# the
carrier of b
3,the
add of b
3,the
mult of b
3,the
Zero of b
3,the
unity of b
3,the
lmult of b
3 #) implies b
2 is
Subalgebra of b
3 )
:: deftheorem Def4 defines opers_closed POLYALG1:def 4 :
theorem Th15: :: POLYALG1:15
theorem Th16: :: POLYALG1:16
theorem Th17: :: POLYALG1:17
theorem Th18: :: POLYALG1:18
canceled;
theorem Th19: :: 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 Th24: :: POLYALG1:24
theorem Th25: :: POLYALG1:25