Article INT_7, MML version 4.99.1005

:: INT_7:th 1
theorem
for b1, b2 being set
for b3 being ManySortedSet of b1
      st support b3 = {b2}
   holds b3 = (b1 --> 0) +*(b2,b3 . b2);

:: INT_7:th 2
theorem
for b1 being set
for b2, b3, b4 being real-valued ManySortedSet of b1
      st (support b2) /\ support b3 = {} & (support b2) \/ support b3 = support b4 & b2 | support b2 = b4 | support b2 & b3 | support b3 = b4 | support b3
   holds b2 + b3 = b4;

:: INT_7:th 3
theorem
for b1 being set
for b2, b3 being ManySortedSet of b1
      st b2 | support b2 = b3 | support b3
   holds b2 = b3;

:: INT_7:th 4
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1
      st support b2 = {} & support b3 = {}
   holds b2 = b3;

:: INT_7:attrnot 1 => INT_7:attr 1
definition
  let a1 be natural-valued finite-support ManySortedSet of SetPrimes;
  attr a1 is prime-factorization-like means
    for b1 being natural prime set
          st b1 in support a1
       holds ex b2 being natural set st
          0 < b2 & a1 . b1 = b1 |^ b2;
end;

:: INT_7:dfs 1
definiens
  let a1 be natural-valued finite-support ManySortedSet of SetPrimes;
To prove
     a1 is prime-factorization-like
it is sufficient to prove
  thus for b1 being natural prime set
          st b1 in support a1
       holds ex b2 being natural set st
          0 < b2 & a1 . b1 = b1 |^ b2;

:: INT_7:def 1
theorem
for b1 being natural-valued finite-support ManySortedSet of SetPrimes holds
      b1 is prime-factorization-like
   iff
      for b2 being natural prime set
            st b2 in support b1
         holds ex b3 being natural set st
            0 < b3 & b1 . b2 = b2 |^ b3;

:: INT_7:funcreg 1
registration
  let a1 be non empty natural set;
  cluster prime_factorization a1 -> prime-factorization-like;
end;

:: INT_7:th 5
theorem
for b1, b2 being natural prime set
for b3, b4 being natural set
      st b1 divides b4 * (b2 |^ b3) & b1 <> b2
   holds b1 divides b4;

:: INT_7:th 6
theorem
for b1 being FinSequence of NAT
for b2 being natural-valued finite-support ManySortedSet of SetPrimes
for b3 being natural prime set
      st b2 is prime-factorization-like & Product b2 <> 1 & b3 divides Product b2 & Product b2 = Product b1 & b1 = (canFS support b2) * b2
   holds b3 in support b2;

:: INT_7:th 7
theorem
for b1, b2 being natural-valued finite-support ManySortedSet of SetPrimes
      st support b1 c= support b2 & b1 | support b1 = b2 | support b1
   holds Product b1 divides Product b2;

:: INT_7:th 8
theorem
for b1 being natural-valued finite-support ManySortedSet of SetPrimes
for b2 being natural prime set
      st b1 is prime-factorization-like
   holds    b2 divides Product b1
   iff
      b2 in support b1;

:: INT_7:th 9
theorem
for b1, b2, b3 being non empty natural set
      st b3 = b1 lcm b2
   holds support prime_factorization b3 = (support prime_factorization b1) \/ support prime_factorization b2;

:: INT_7:th 10
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
support min(b2,b3) = (support b2) /\ support b3;

:: INT_7:th 11
theorem
for b1, b2, b3 being non empty natural set
      st b3 = b1 gcd b2
   holds support prime_factorization b3 = (support prime_factorization b1) /\ support prime_factorization b2;

:: INT_7:th 12
theorem
for b1, b2 being natural-valued finite-support ManySortedSet of SetPrimes
      st b1 is prime-factorization-like & b2 is prime-factorization-like & support b1 misses support b2
   holds Product b1,Product b2 are_relative_prime;

:: INT_7:th 13
theorem
for b1 being natural-valued finite-support ManySortedSet of SetPrimes
      st b1 is prime-factorization-like
   holds Product b1 <> 0;

:: INT_7:th 14
theorem
for b1 being natural-valued finite-support ManySortedSet of SetPrimes
      st b1 is prime-factorization-like
   holds    Product b1 = 1
   iff
      support b1 = {};

:: INT_7:th 15
theorem
for b1, b2 being natural-valued finite-support ManySortedSet of SetPrimes
      st b1 is prime-factorization-like & b2 is prime-factorization-like & Product b1 = Product b2
   holds b1 = b2;

:: INT_7:th 16
theorem
for b1 being natural-valued finite-support ManySortedSet of SetPrimes
for b2 being non empty natural set
      st b1 is prime-factorization-like & b2 = Product b1
   holds prime_factorization b2 = b1;

:: INT_7:th 17
theorem
for b1, b2 being Element of NAT
      st 1 <= b1 & 1 <= b2
   holds ex b3, b4 being Element of NAT st
      b1 lcm b2 = b4 * b3 & b4 gcd b3 = 1 & b4 divides b1 & b3 divides b2 & b4 <> 0 & b3 <> 0;

:: INT_7:funcnot 1 => INT_7:func 1
definition
  let a1 be natural set;
  assume 1 < a1;
  func Segm0 A1 -> non empty finite Element of bool NAT equals
    (Segm a1) \ {0};
end;

:: INT_7:def 2
theorem
for b1 being natural set
      st 1 < b1
   holds Segm0 b1 = (Segm b1) \ {0};

:: INT_7:th 18
theorem
for b1 being natural set
      st 1 < b1
   holds Card Segm0 b1 = b1 - 1;

:: INT_7:funcnot 2 => INT_7:func 2
definition
  let a1 be natural prime set;
  func multint0 A1 -> Function-like quasi_total Relation of [:Segm0 a1,Segm0 a1:],Segm0 a1 equals
    (multint a1) || Segm0 a1;
end;

:: INT_7:def 3
theorem
for b1 being natural prime set holds
   multint0 b1 = (multint b1) || Segm0 b1;

:: INT_7:th 19
theorem
for b1 being natural prime set holds
   multMagma(#Segm0 b1,multint0 b1#) is associative & multMagma(#Segm0 b1,multint0 b1#) is commutative & multMagma(#Segm0 b1,multint0 b1#) is Group-like;

:: INT_7:funcnot 3 => INT_7:func 3
definition
  let a1 be natural prime set;
  func Z/Z* A1 -> non empty Group-like associative commutative multMagma equals
    multMagma(#Segm0 a1,multint0 a1#);
end;

:: INT_7:def 4
theorem
for b1 being natural prime set holds
   Z/Z* b1 = multMagma(#Segm0 b1,multint0 b1#);

:: INT_7:th 20
theorem
for b1 being natural prime set
for b2, b3 being Element of the carrier of Z/Z* b1
for b4, b5 being Element of the carrier of INT.Ring b1
      st b2 = b4 & b3 = b5
   holds b2 * b3 = b4 * b5;

:: INT_7:th 21
theorem
for b1 being natural prime set holds
   1_ Z/Z* b1 = 1 & 1_ Z/Z* b1 = 1. INT.Ring b1;

:: INT_7:th 22
theorem
for b1 being natural prime set
for b2 being Element of the carrier of Z/Z* b1
for b3 being Element of the carrier of INT.Ring b1
      st b2 = b3
   holds b2 " = b3 ";

:: INT_7:funcreg 2
registration
  let a1 be natural prime set;
  cluster Z/Z* a1 -> non empty finite Group-like associative commutative;
end;

:: INT_7:th 23
theorem
for b1 being natural prime set holds
   ord Z/Z* b1 = b1 - 1;

:: INT_7:th 24
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being integer set
      st b2 is not being_of_order_0(b1)
   holds ex b4, b5 being Element of NAT st
      b2 |^ b3 = b2 |^ b4 & b4 = (b5 * ord b2) + b3;

:: INT_7:th 25
theorem
for b1 being non empty Group-like associative commutative multMagma
for b2, b3 being Element of the carrier of b1
for b4, b5 being natural set
      st b1 is finite & ord b2 = b4 & ord b3 = b5 & b4 gcd b5 = 1
   holds ord (b2 * b3) = b4 * b5;

:: INT_7:th 26
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st 0 <= degree b2
   holds b2 is non-zero(b1);

:: INT_7:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st 0 <= degree b2
   holds Roots b2 is finite set &
    (ex b3, b4 being Element of NAT st
       b4 = degree b2 & b3 = Card Roots b2 & b3 <= b4);

:: INT_7:th 28
theorem
for b1 being natural prime set
for b2 being Element of the carrier of Z/Z* b1
for b3 being Element of the carrier of INT.Ring b1
   st b2 = b3
for b4 being Element of NAT holds
   (power Z/Z* b1) .(b2,b4) = (power INT.Ring b1) .(b3,b4);

:: INT_7:th 29
theorem
for b1 being natural prime set
for b2, b3 being Element of the carrier of Z/Z* b1
for b4 being natural set
      st 0 < b4 & ord b2 = b4 & b3 |^ b4 = 1
   holds b3 is Element of the carrier of gr {b2};

:: INT_7:th 30
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of NAT
      st b1 is finite & ord b2 = b3 * b4
   holds ord (b2 |^ b3) = b4;

:: INT_7:th 31
theorem
for b1 being natural prime set holds
   Z/Z* b1 is non empty Group-like associative cyclic multMagma;