Article NAT_4, MML version 4.99.1005

:: NAT_4:th 1
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 * b2 < b1 * b1
   holds b2 < b1;

:: NAT_4:th 2
theorem
for b1, b2 being real set
      st 1 < b1 & b1 * b1 <= b2
   holds b1 < b2;

:: NAT_4:th 3
theorem
for b1, b2 being natural set
      st 1 < b1
   holds b2 < b1 |^ b2;

:: NAT_4:th 4
theorem
for b1, b2, b3 being natural set
      st b2 <= b1 & b3 = [\b1 / 2/]
   holds b1 choose b2 <= b1 choose b3;

:: NAT_4:th 5
theorem
for b1, b2 being natural set
      st b2 = [\b1 / 2/] & 2 <= b1
   holds (2 |^ b1) / b1 <= b1 choose b2;

:: NAT_4:th 6
theorem
for b1 being natural set holds
   (4 |^ b1) / (2 * b1) <= (2 * b1) choose b1;

:: NAT_4:th 7
theorem
for b1, b2 being natural set
      st 0 < b2 & b1 divides b2 & b1 <> 1 & b1 <> b2
   holds 1 < b1 & b1 < b2;

:: NAT_4:th 8
theorem
for b1 being natural set
      st ex b2 being Element of NAT st
           b2 divides b1 & 1 < b2 & b2 < b1
   holds ex b2 being Element of NAT st
      b2 divides b1 & 1 < b2 & b2 * b2 <= b1;

:: NAT_4:th 9
theorem
for b1, b2, b3, b4 being natural set
      st b1 = (b2 * b3) + b4 & b4 < b2 & 0 < b4
   holds not b2 divides b1;

:: NAT_4:th 10
theorem
for b1, b2, b3 being natural set
      st b2 hcf b3 = 1 & b2 <> 0 & b3 <> 0
   holds (b2 |^ b1) hcf b3 = 1;

:: NAT_4:th 11
theorem
for b1, b2, b3 being natural set holds
(b1 |^ (2 * b2)) mod b3 = (((b1 |^ b2) mod b3) * ((b1 |^ b2) mod b3)) mod b3;

:: NAT_4:th 12
theorem
for b1 being natural set holds
      b1 is not prime
   iff
      (1 < b1 implies ex b2 being Element of NAT st
         b2 divides b1 & 1 < b2 & b2 < b1);

:: NAT_4:th 13
theorem
for b1, b2 being natural set
      st b1 divides b2 & 1 < b1
   holds ex b3 being Element of NAT st
      b3 divides b2 & b3 <= b1 & b3 is prime;

:: NAT_4:th 14
theorem
for b1 being natural set holds
      b1 is prime
   iff
      1 < b1 &
       (for b2 being Element of NAT
             st 1 < b2 & b2 * b2 <= b1 & b2 is prime
          holds not b2 divides b1);

:: NAT_4:th 15
theorem
for b1, b2, b3 being natural set
      st (b1 |^ b3) mod b2 = 1 & 1 <= b3 & b2 is prime
   holds b1,b2 are_relative_prime;

:: NAT_4:th 16
theorem
for b1 being natural prime set
for b2 being Element of NAT
for b3 being set
      st b2 <> 0 & b3 = b1 |^ (b1 |-count b2)
   holds ex b4 being Element of NAT st
      b4 = b3 & 1 <= b4 & b4 <= b2;

:: NAT_4:th 17
theorem
for b1, b2, b3, b4 being Element of NAT
      st b2 is prime & b4 divides b1 * (b2 |^ (b3 + 1)) & not b4 divides b1 * (b2 |^ b3)
   holds b2 |^ (b3 + 1) divides b4;

:: NAT_4:th 18
theorem
for b1, b2, b3 being Element of NAT
      st b1 divides b2 |^ b3 & b2 is prime & b1 is prime & 0 < b3
   holds b2 = b1;

:: NAT_4:th 19
theorem
for b1 being natural prime set
for b2 being Element of NAT
      st b2 < b1
   holds not b1 divides b2 !;

:: NAT_4:th 20
theorem
for b1, b2 being non empty natural set
      st for b3 being Element of NAT
              st b3 is prime
           holds b3 |-count b1 <= b3 |-count b2
   holds ex b3 being Element of NAT st
      b2 = b1 * b3;

:: NAT_4:th 21
theorem
for b1, b2 being non empty natural set
      st for b3 being Element of NAT
              st b3 is prime
           holds b3 |-count b1 = b3 |-count b2
   holds b1 = b2;

:: NAT_4:th 22
theorem
for b1, b2 being natural prime set
for b3 being non empty Element of NAT
      st b1 |^ (b1 |-count b3) = b2 |^ (b2 |-count b3) &
         0 < b1 |-count b3
   holds b1 = b2;

:: NAT_4:th 23
theorem
for b1, b2, b3, b4, b5, b6, b7 being Element of NAT
      st b1 - 1 = b2 * (b3 |^ b5) &
         0 < b2 &
         0 < b5 &
         b3 is prime &
         (b7 |^ (b1 -' 1)) mod b1 = 1 &
         b6 is prime &
         b6 divides b1 &
         not b6 divides (b7 |^ ((b1 -' 1) div b3)) -' 1
   holds b6 mod (b3 |^ b5) = 1;

:: NAT_4:th 24
theorem
for b1, b2, b3 being Element of NAT
      st b1 - 1 = b2 * b3 &
         b3 < b2 &
         0 < b3 &
         b2 hcf b3 = 1 &
         (for b4 being Element of NAT
               st b4 divides b2 & b4 is prime
            holds ex b5 being Element of NAT st
               (b5 |^ (b1 -' 1)) mod b1 = 1 &
                ((b5 |^ ((b1 -' 1) div b4)) -' 1) hcf b1 = 1)
   holds b1 is prime;

:: NAT_4:th 25
theorem
for b1, b2, b3, b4, b5, b6 being Element of NAT
      st b1 - 1 = (b6 |^ b4) * b3 &
         b3 < b6 |^ b4 &
         0 < b3 &
         b6 hcf b3 = 1 &
         b6 is prime &
         (b5 |^ (b1 -' 1)) mod b1 = 1 &
         ((b5 |^ ((b1 -' 1) div b6)) -' 1) hcf b1 = 1
   holds b1 is prime;

:: NAT_4:th 26
theorem
7 is prime;

:: NAT_4:th 27
theorem
11 is prime;

:: NAT_4:th 28
theorem
13 is prime;

:: NAT_4:th 29
theorem
19 is prime;

:: NAT_4:th 30
theorem
23 is prime;

:: NAT_4:th 31
theorem
37 is prime;

:: NAT_4:th 32
theorem
43 is prime;

:: NAT_4:th 33
theorem
83 is prime;

:: NAT_4:th 34
theorem
139 is prime;

:: NAT_4:th 35
theorem
163 is prime;

:: NAT_4:th 36
theorem
317 is prime;

:: NAT_4:th 37
theorem
631 is prime;

:: NAT_4:th 38
theorem
1259 is prime;

:: NAT_4:th 39
theorem
2503 is prime;

:: NAT_4:th 40
theorem
4001 is prime;

:: NAT_4:th 41
theorem
for b1, b2, b3 being FinSequence of REAL
      st b1 = b2 + b3
   holds dom b1 = (dom b2) /\ dom b3;

:: NAT_4:th 42
theorem
for b1 being FinSequence of REAL
      st for b2 being Element of NAT
              st b2 in dom b1
           holds 0 < b1 . b2
   holds 0 < Product b1;

:: NAT_4:th 43
theorem
for b1 being set
for b2 being finite set
      st b1 c= b2 & b2 c= NAT & not {} in b2
   holds Product Sgm b1 <= Product Sgm b2;

:: NAT_4:th 44
theorem
for b1, b2 being Element of NAT
for b3 being set
for b4 being FinSequence of SetPrimes
for b5 being natural prime set
      st b3 c= SetPrimes & b3 c= Seg b2 & b4 = Sgm b3 & b1 = Product b4
   holds (b5 in rng b4 implies b5 |-count b1 = 1) & (b5 in rng b4 or b5 |-count b1 = 0);

:: NAT_4:th 45
theorem
for b1 being Element of NAT holds
   Product Sgm {b2 where b2 is prime Element of NAT: b2 <= b1 + 1} <= 4 to_power b1;

:: NAT_4:th 46
theorem
for b1 being Element of REAL
      st 2 <= b1
   holds Product Sgm {b2 where b2 is prime Element of NAT: b2 <= b1} <= 4 to_power (b1 - 1);

:: NAT_4:th 47
theorem
for b1 being Element of NAT
for b2 being natural prime set
      st b1 <> 0
   holds ex b3 being FinSequence of NAT st
      len b3 = b1 &
       (for b4 being Element of NAT
             st b4 in dom b3
          holds (b3 . b4 = 1 implies b2 |^ b4 divides b1) &
           (b2 |^ b4 divides b1 implies b3 . b4 = 1) &
           (b3 . b4 = 0 implies not b2 |^ b4 divides b1) &
           (b2 |^ b4 divides b1 or b3 . b4 = 0)) &
       b2 |-count b1 = Sum b3;

:: NAT_4:th 48
theorem
for b1 being Element of NAT
for b2 being natural prime set holds
   ex b3 being FinSequence of NAT st
      len b3 = b1 &
       (for b4 being Element of NAT
             st b4 in dom b3
          holds b3 . b4 = [\b1 / (b2 |^ b4)/]) &
       b2 |-count (b1 !) = Sum b3;

:: NAT_4:th 49
theorem
for b1 being Element of NAT
for b2 being natural prime set holds
   ex b3 being FinSequence of REAL st
      len b3 = 2 * b1 &
       (for b4 being Element of NAT
             st b4 in dom b3
          holds b3 . b4 = [\(2 * b1) / (b2 |^ b4)/] - (2 * [\b1 / (b2 |^ b4)/])) &
       b2 |-count ((2 * b1) choose b1) = Sum b3;

:: NAT_4:funcnot 1 => NAT_4:func 1
definition
  let a1 be FinSequence of NAT;
  let a2 be natural prime set;
  func A2 |-count A1 -> FinSequence of NAT means
    len it = len a1 &
     (for b1 being set
           st b1 in dom it
        holds it . b1 = a2 |-count (a1 . b1));
end;

:: NAT_4:def 1
theorem
for b1 being FinSequence of NAT
for b2 being natural prime set
for b3 being FinSequence of NAT holds
      b3 = b2 |-count b1
   iff
      len b3 = len b1 &
       (for b4 being set
             st b4 in dom b3
          holds b3 . b4 = b2 |-count (b1 . b4));

:: NAT_4:th 50
theorem
for b1 being natural prime set
for b2 being FinSequence of NAT
      st b2 = {}
   holds b1 |-count b2 = {};

:: NAT_4:th 51
theorem
for b1 being natural prime set
for b2, b3 being FinSequence of NAT holds
b1 |-count (b2 ^ b3) = (b1 |-count b2) ^ (b1 |-count b3);

:: NAT_4:th 52
theorem
for b1 being natural prime set
for b2 being non empty Element of NAT holds
   b1 |-count <*b2*> = <*b1 |-count b2*>;

:: NAT_4:th 53
theorem
for b1 being FinSequence of NAT
for b2 being natural prime set
      st Product b1 <> 0
   holds b2 |-count Product b1 = Sum (b2 |-count b1);

:: NAT_4:th 54
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2 &
         (for b3 being Element of NAT
               st b3 in dom b1
            holds b1 . b3 <= b2 . b3 & 0 < b1 . b3)
   holds Product b1 <= Product b2;

:: NAT_4:th 55
theorem
for b1 being Element of NAT
for b2 being Element of REAL
      st 0 < b2
   holds Product (b1 |-> b2) = b2 to_power b1;

:: NAT_4:sch 1
scheme NAT_4:sch 1
for b1 being natural prime set
for b2 being Element of NAT
for b3 being non empty Element of NAT
for b4 being set
      st b4 = {b5 |^ (b5 |-count b3) where b5 is prime Element of NAT: P1[b2, b3, b5]}
   holds 0 < Product Sgm b4


:: NAT_4:sch 2
scheme NAT_4:sch 2
for b1 being natural prime set
for b2 being Element of NAT
for b3 being non empty Element of NAT
for b4 being set
      st b4 = {b5 |^ (b5 |-count b3) where b5 is prime Element of NAT: P1[b2, b3, b5]} &
         not b1 |^ (b1 |-count b3) in b4
   holds b1 |-count Product Sgm b4 = 0


:: NAT_4:sch 3
scheme NAT_4:sch 3
for b1 being natural prime set
for b2 being Element of NAT
for b3 being non empty Element of NAT
for b4 being set
      st b4 = {b5 |^ (b5 |-count b3) where b5 is prime Element of NAT: P1[b2, b3, b5]} &
         b1 |^ (b1 |-count b3) in b4
   holds b1 |-count Product Sgm b4 = b1 |-count b3


:: NAT_4:sch 4
scheme NAT_4:sch 4
{F1 -> set}:
for b1, b2 being Element of NAT
for b3 being Element of REAL
for b4 being finite set
      st b4 = {F1(b5, b2) where b5 is prime Element of NAT: b5 <= b3 & P1[b5, b2]} &
         0 <= b3
   holds card b4 <= [\b3/]


:: NAT_4:th 56
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds ex b2 being natural prime set st
      b1 < b2 & b2 <= 2 * b1;