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;