Article HILBASIS, MML version 4.99.1005

:: HILBASIS:th 1
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being Relation-like Function-like set
      st (proj2 b1) \/ proj2 b2 c= proj1 b3
   holds ex b4, b5 being Relation-like Function-like FinSequence-like set st
      b4 = b1 * b3 & b5 = b2 * b3 & (b1 ^ b2) * b3 = b4 ^ b5;

:: HILBASIS:th 2
theorem
for b1 being natural-valued finite-support ManySortedSet of 0 holds
   decomp b1 = <*<*{},{}*>*>;

:: HILBASIS:th 3
theorem
for b1, b2 being Element of NAT
for b3 being natural-valued finite-support ManySortedSet of b2
      st b1 <= b2
   holds b3 | b1 is Element of Bags b1;

:: HILBASIS:th 4
theorem
for b1, b2 being set
for b3, b4 being natural-valued finite-support ManySortedSet of b2
for b5, b6 being natural-valued finite-support ManySortedSet of b1
      st b5 = b3 | b1 & b6 = b4 | b1 & b3 divides b4
   holds b5 divides b6;

:: HILBASIS:th 5
theorem
for b1, b2 being set
for b3, b4 being natural-valued finite-support ManySortedSet of b2
for b5, b6 being natural-valued finite-support ManySortedSet of b1
      st b5 = b3 | b1 & b6 = b4 | b1
   holds (b3 -' b4) | b1 = b5 -' b6 & (b3 + b4) | b1 = b5 + b6;

:: HILBASIS:funcnot 1 => HILBASIS:func 1
definition
  let a1, a2 be natural set;
  let a3 be natural-valued finite-support ManySortedSet of a1;
  func A3 bag_extend A2 -> Element of Bags (a1 + 1) means
    it | a1 = a3 & it . a1 = a2;
end;

:: HILBASIS:def 1
theorem
for b1, b2 being natural set
for b3 being natural-valued finite-support ManySortedSet of b1
for b4 being Element of Bags (b1 + 1) holds
      b4 = b3 bag_extend b2
   iff
      b4 | b1 = b3 & b4 . b1 = b2;

:: HILBASIS:th 6
theorem
for b1 being Element of NAT holds
   EmptyBag (b1 + 1) = (EmptyBag b1) bag_extend 0;

:: HILBASIS:th 7
theorem
for b1 being ordinal set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
   b3 in rng divisors b2
iff
   b3 divides b2;

:: HILBASIS:funcnot 2 => HILBASIS:func 2
definition
  let a1 be set;
  let a2 be Element of a1;
  func UnitBag A2 -> Element of Bags a1 equals
    (EmptyBag a1) +*(a2,1);
end;

:: HILBASIS:def 2
theorem
for b1 being set
for b2 being Element of b1 holds
   UnitBag b2 = (EmptyBag b1) +*(b2,1);

:: HILBASIS:th 8
theorem
for b1 being non empty set
for b2 being Element of b1 holds
   support UnitBag b2 = {b2};

:: HILBASIS:th 9
theorem
for b1 being non empty set
for b2 being Element of b1 holds
   (UnitBag b2) . b2 = 1 &
    (for b3 being Element of b1
          st b2 <> b3
       holds (UnitBag b2) . b3 = 0);

:: HILBASIS:th 10
theorem
for b1 being non empty set
for b2, b3 being Element of b1
      st UnitBag b2 = UnitBag b3
   holds b2 = b3;

:: HILBASIS:th 11
theorem
for b1 being non empty ordinal set
for b2 being Element of b1
for b3 being non empty non trivial well-unital doubleLoopStr
for b4 being Function-like quasi_total Relation of b1,the carrier of b3 holds
   eval(UnitBag b2,b4) = b4 . b2;

:: HILBASIS:funcnot 3 => HILBASIS:func 3
definition
  let a1 be set;
  let a2 be Element of a1;
  let a3 be non empty unital multLoopStr_0;
  func 1_1(A2,A3) -> Function-like quasi_total Relation of Bags a1,the carrier of a3 equals
    (0_(a1,a3)) +*(UnitBag a2,1_ a3);
end;

:: HILBASIS:def 3
theorem
for b1 being set
for b2 being Element of b1
for b3 being non empty unital multLoopStr_0 holds
   1_1(b2,b3) = (0_(b1,b3)) +*(UnitBag b2,1_ b3);

:: HILBASIS:th 12
theorem
for b1 being set
for b2 being non empty non trivial unital doubleLoopStr
for b3 being Element of b1 holds
   (1_1(b3,b2)) . UnitBag b3 = 1_ b2 &
    (for b4 being natural-valued finite-support ManySortedSet of b1
          st b4 <> UnitBag b3
       holds (1_1(b3,b2)) . b4 = 0. b2);

:: HILBASIS:th 13
theorem
for b1 being set
for b2 being Element of b1
for b3 being non empty non trivial right_complementable add-associative right_zeroed right-distributive well-unital doubleLoopStr holds
   Support 1_1(b2,b3) = {UnitBag b2};

:: HILBASIS:funcreg 1
registration
  let a1 be ordinal set;
  let a2 be Element of a1;
  let a3 be non empty non trivial right_complementable add-associative right_zeroed right-distributive well-unital doubleLoopStr;
  cluster 1_1(a2,a3) -> Function-like quasi_total finite-Support;
end;

:: HILBASIS:th 14
theorem
for b1 being non empty non trivial right_complementable add-associative right_zeroed right-distributive well-unital doubleLoopStr
for b2 being non empty set
for b3, b4 being Element of b2
      st 1_1(b3,b1) = 1_1(b4,b1)
   holds b3 = b4;

:: HILBASIS:th 15
theorem
for b1 being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for b2 being Element of the carrier of Polynom-Ring b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 = b3
   holds - b2 = - b3;

:: HILBASIS:th 16
theorem
for b1 being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for b2, b3 being Element of the carrier of Polynom-Ring b1
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 = b4 & b3 = b5
   holds b2 - b3 = b4 - b5;

:: HILBASIS:funcnot 4 => HILBASIS:func 4
definition
  let a1 be non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a2 be non empty Element of bool the carrier of Polynom-Ring a1;
  func minlen A2 -> non empty Element of bool a2 equals
    {b1 where b1 is Element of a2: for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of a1
          st b2 = b1 & b3 in a2
       holds len b2 <= len b3};
end;

:: HILBASIS:def 4
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being non empty Element of bool the carrier of Polynom-Ring b1 holds
   minlen b2 = {b3 where b3 is Element of b2: for b4, b5 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
         st b4 = b3 & b5 in b2
      holds len b4 <= len b5};

:: HILBASIS:th 17
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being non empty Element of bool the carrier of Polynom-Ring b1
for b3, b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st b3 in minlen b2 & b4 in b2
   holds b3 in b2 & len b3 <= len b4;

:: HILBASIS:funcnot 5 => HILBASIS:func 5
definition
  let a1 be non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a2 be natural set;
  let a3 be Element of the carrier of a1;
  func monomial(A3,A2) -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 means
    for b1 being natural set holds
       (b1 = a2 implies it . b1 = a3) & (b1 = a2 or it . b1 = 0. a1);
end;

:: HILBASIS:def 5
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being natural set
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b4 = monomial(b3,b2)
   iff
      for b5 being natural set holds
         (b5 = b2 implies b4 . b5 = b3) & (b5 = b2 or b4 . b5 = 0. b1);

:: HILBASIS:th 18
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being Element of NAT
for b3 being Element of the carrier of b1 holds
   (b3 = 0. b1 or len monomial(b3,b2) = b2 + 1) &
    (b3 = 0. b1 implies len monomial(b3,b2) = 0) &
    len monomial(b3,b2) <= b2 + 1;

:: HILBASIS:th 19
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being Element of NAT
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   ((monomial(b4,b2)) *' b5) . (b3 + b2) = b4 * (b5 . b3);

:: HILBASIS:th 20
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being Element of NAT
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   (b5 *' monomial(b4,b2)) . (b3 + b2) = (b5 . b3) * b4;

:: HILBASIS:th 21
theorem
for b1 being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
len (b2 *' b3) <= ((len b2) + len b3) -' 1;

:: HILBASIS:th 22
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being non empty add-closed left-ideal right-ideal Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is RingIsomorphism(b1, b2)
   holds b4 .: b3 is non empty add-closed left-ideal right-ideal Element of bool the carrier of b2;

:: HILBASIS:th 23
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is RingHomomorphism(b1, b2)
   holds b3 . 0. b1 = 0. b2;

:: HILBASIS:th 24
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being non empty Element of bool the carrier of b1
for b4 being non empty Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being LinearCombination of b3
for b7 being LinearCombination of b4
for b8 being FinSequence of [:the carrier of b1,the carrier of b1,the carrier of b1:]
      st b5 is RingHomomorphism(b1, b2) &
         len b6 = len b7 &
         b8 represents b6 &
         (for b9 being set
               st b9 in dom b7
            holds b7 . b9 = ((b5 . ((b8 /. b9) `1)) * (b5 . ((b8 /. b9) `2))) * (b5 . ((b8 /. b9) `3)))
   holds b5 . Sum b6 = Sum b7;

:: HILBASIS:th 25
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is RingIsomorphism(b1, b2)
   holds b3 /" is RingIsomorphism(b2, b1);

:: HILBASIS:th 26
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being non empty Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is RingIsomorphism(b1, b2)
   holds b4 .: (b3 -Ideal) = (b4 .: b3) -Ideal;

:: HILBASIS:th 27
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is RingIsomorphism(b1, b2) & b1 is Noetherian
   holds b2 is Noetherian;

:: HILBASIS:th 28
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr holds
   ex b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of Polynom-Ring(0,b1) st
      b2 is RingIsomorphism(b1, Polynom-Ring(0,b1));

:: HILBASIS:th 29
theorem
for b1 being non empty non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being Element of NAT
for b3 being natural-valued finite-support ManySortedSet of b2
for b4 being Function-like quasi_total finite-Support Relation of Bags b2,the carrier of b1
for b5 being FinSequence of the carrier of Polynom-Ring(b2,b1)
      st b4 = Sum b5
   holds ex b6 being Function-like quasi_total Relation of the carrier of Polynom-Ring(b2,b1),the carrier of b1 st
      (for b7 being Function-like quasi_total finite-Support Relation of Bags b2,the carrier of b1 holds
          b6 . b7 = b7 . b3) &
       b4 . b3 = Sum (b6 * b5);

:: HILBASIS:funcnot 6 => HILBASIS:func 6
definition
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  func upm(A2,A1) -> Function-like quasi_total Relation of the carrier of Polynom-Ring Polynom-Ring(a2,a1),the carrier of Polynom-Ring(a2 + 1,a1) means
    for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of Polynom-Ring(a2,a1)
    for b2 being Function-like quasi_total finite-Support Relation of Bags a2,the carrier of a1
    for b3 being Function-like quasi_total finite-Support Relation of Bags (a2 + 1),the carrier of a1
    for b4 being natural-valued finite-support ManySortedSet of a2 + 1
          st b3 = it . b1 & b2 = b1 . (b4 . a2)
       holds b3 . b4 = b2 . (b4 | a2);
end;

:: HILBASIS:def 6
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of the carrier of Polynom-Ring Polynom-Ring(b2,b1),the carrier of Polynom-Ring(b2 + 1,b1) holds
      b3 = upm(b2,b1)
   iff
      for b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of Polynom-Ring(b2,b1)
      for b5 being Function-like quasi_total finite-Support Relation of Bags b2,the carrier of b1
      for b6 being Function-like quasi_total finite-Support Relation of Bags (b2 + 1),the carrier of b1
      for b7 being natural-valued finite-support ManySortedSet of b2 + 1
            st b6 = b3 . b4 & b5 = b4 . (b7 . b2)
         holds b6 . b7 = b5 . (b7 | b2);

:: HILBASIS:funcreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  cluster upm(a2,a1) -> Function-like quasi_total additive;
end;

:: HILBASIS:funcreg 3
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  cluster upm(a2,a1) -> Function-like quasi_total multiplicative;
end;

:: HILBASIS:funcreg 4
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  cluster upm(a2,a1) -> Function-like quasi_total unity-preserving;
end;

:: HILBASIS:funcreg 5
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  cluster upm(a2,a1) -> Function-like one-to-one quasi_total;
end;

:: HILBASIS:funcnot 7 => HILBASIS:func 7
definition
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be Element of NAT;
  func mpu(A2,A1) -> Function-like quasi_total Relation of the carrier of Polynom-Ring(a2 + 1,a1),the carrier of Polynom-Ring Polynom-Ring(a2,a1) means
    for b1 being Function-like quasi_total finite-Support Relation of Bags (a2 + 1),the carrier of a1
    for b2 being Function-like quasi_total finite-Support Relation of Bags a2,the carrier of a1
    for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of Polynom-Ring(a2,a1)
    for b4 being Element of NAT
    for b5 being natural-valued finite-support ManySortedSet of a2
          st b3 = it . b1 & b2 = b3 . b4
       holds b2 . b5 = b1 . (b5 bag_extend b4);
end;

:: HILBASIS:def 7
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of the carrier of Polynom-Ring(b2 + 1,b1),the carrier of Polynom-Ring Polynom-Ring(b2,b1) holds
      b3 = mpu(b2,b1)
   iff
      for b4 being Function-like quasi_total finite-Support Relation of Bags (b2 + 1),the carrier of b1
      for b5 being Function-like quasi_total finite-Support Relation of Bags b2,the carrier of b1
      for b6 being Function-like quasi_total finite-Support Relation of NAT,the carrier of Polynom-Ring(b2,b1)
      for b7 being Element of NAT
      for b8 being natural-valued finite-support ManySortedSet of b2
            st b6 = b3 . b4 & b5 = b6 . b7
         holds b5 . b8 = b4 . (b8 bag_extend b7);

:: HILBASIS:th 30
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of NAT
for b3 being Element of the carrier of Polynom-Ring(b2 + 1,b1) holds
   (upm(b2,b1)) . ((mpu(b2,b1)) . b3) = b3;

:: HILBASIS:th 31
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of NAT holds
   ex b3 being Function-like quasi_total Relation of the carrier of Polynom-Ring Polynom-Ring(b2,b1),the carrier of Polynom-Ring(b2 + 1,b1) st
      b3 is RingIsomorphism(Polynom-Ring Polynom-Ring(b2,b1), Polynom-Ring(b2 + 1,b1));

:: HILBASIS:funcreg 6
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive Noetherian doubleLoopStr;
  cluster Polynom-Ring a1 -> non empty strict Noetherian;
end;

:: HILBASIS:th 33
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
   st b1 is Noetherian
for b2 being Element of NAT holds
   Polynom-Ring(b2,b1) is Noetherian;

:: HILBASIS:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
   b1 is Noetherian;

:: HILBASIS:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of NAT holds
   Polynom-Ring(b2,b1) is Noetherian;

:: HILBASIS:th 36
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being ordinal infinite set holds
   Polynom-Ring(b2,b1) is not Noetherian;