:: POLYNOM3 semantic presentation
theorem Th1: :: POLYNOM3:1
theorem Th2: :: POLYNOM3:2
theorem Th3: :: POLYNOM3:3
theorem Th4: :: POLYNOM3:4
:: deftheorem Def1 defines < POLYNOM3:def 1 :
:: deftheorem Def2 defines <= POLYNOM3:def 2 :
theorem Th5: :: POLYNOM3:5
for b
1 being
Natfor b
2, b
3, b
4 being
Element of b
1 -tuples_on NAT holds
( ( b
2 < b
3 & b
3 < b
4 implies b
2 < b
4 ) & ( not ( not ( b
2 < b
3 & b
3 <= b
4 ) & not ( b
2 <= b
3 & b
3 < b
4 ) & not ( b
2 <= b
3 & b
3 <= b
4 ) ) implies b
2 <= b
4 ) )
theorem Th6: :: POLYNOM3:6
for b
1 being
Natfor b
2, b
3 being
Element of b
1 -tuples_on NAT holds
not ( b
2 <> b
3 & ( for b
4 being
Nat holds
not ( b
4 in Seg b
1 & b
2 . b
4 <> b
3 . b
4 & ( for b
5 being
Nat holds
( 1
<= b
5 & b
5 < b
4 implies b
2 . b
5 = b
3 . b
5 ) ) ) ) )
theorem Th7: :: POLYNOM3:7
:: deftheorem Def3 defines TuplesOrder POLYNOM3:def 3 :
:: deftheorem Def4 defines Decomp POLYNOM3:def 4 :
theorem Th8: :: POLYNOM3:8
theorem Th9: :: POLYNOM3:9
theorem Th10: :: POLYNOM3:10
theorem Th11: :: POLYNOM3:11
for b
1, b
2, b
3, b
4, b
5 being
Nat holds
(
(Decomp b3,2) . b
1 = <*b4,(b3 -' b4)*> &
(Decomp b3,2) . b
2 = <*b5,(b3 -' b5)*> implies ( not ( b
1 < b
2 & not b
4 < b
5 ) & not ( b
4 < b
5 & not b
1 < b
2 ) ) )
theorem Th12: :: POLYNOM3:12
theorem Th13: :: POLYNOM3:13
theorem Th14: :: POLYNOM3:14
:: deftheorem Def5 defines prodTuples POLYNOM3:def 5 :
theorem Th15: :: POLYNOM3:15
theorem Th16: :: POLYNOM3:16
theorem Th17: :: POLYNOM3:17
theorem Th18: :: POLYNOM3:18
theorem Th19: :: POLYNOM3:19
theorem Th20: :: POLYNOM3:20
theorem Th21: :: POLYNOM3:21
theorem Th22: :: POLYNOM3:22
theorem Th23: :: POLYNOM3:23
:: deftheorem Def6 defines + POLYNOM3:def 6 :
theorem Th24: :: POLYNOM3:24
theorem Th25: :: POLYNOM3:25
theorem Th26: :: POLYNOM3:26
:: deftheorem Def7 defines - POLYNOM3:def 7 :
:: deftheorem Def8 defines - POLYNOM3:def 8 :
theorem Th27: :: POLYNOM3:27
:: deftheorem Def9 defines 0_. POLYNOM3:def 9 :
theorem Th28: :: POLYNOM3:28
theorem Th29: :: POLYNOM3:29
theorem Th30: :: POLYNOM3:30
:: deftheorem Def10 defines 1_. POLYNOM3:def 10 :
theorem Th31: :: POLYNOM3:31
:: deftheorem Def11 defines *' POLYNOM3:def 11 :
theorem Th32: :: POLYNOM3:32
theorem Th33: :: POLYNOM3:33
theorem Th34: :: POLYNOM3:34
theorem Th35: :: POLYNOM3:35
theorem Th36: :: POLYNOM3:36
definition
let c
1 be non
empty add-associative right_zeroed right_complementable distributive doubleLoopStr ;
func Polynom-Ring c
1 -> non
empty strict doubleLoopStr means :
Def12:
:: POLYNOM3:def 12
( ( for b
1 being
set holds
( b
1 in the
carrier of a
2 iff b
1 is
Polynomial 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 ) ) &
0. a
2 = 0_. a
1 &
1_ a
2 = 1_. a
1 );
existence
ex b1 being non empty strict doubleLoopStr st
( ( for b2 being set holds
( b2 in the carrier of b1 iff b2 is Polynomial 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 ) ) & 0. b1 = 0_. c1 & 1_ b1 = 1_. c1 )
uniqueness
for b1, b2 being non empty strict doubleLoopStr holds
( ( for b3 being set holds
( b3 in the carrier of b1 iff b3 is Polynomial 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 ) ) & 0. b1 = 0_. c1 & 1_ b1 = 1_. c1 & ( for b3 being set holds
( b3 in the carrier of b2 iff b3 is Polynomial 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 ) ) & 0. b2 = 0_. c1 & 1_ b2 = 1_. c1 implies b1 = b2 )
end;
:: deftheorem Def12 defines Polynom-Ring POLYNOM3:def 12 :
theorem Th37: :: POLYNOM3:37