Article INT_2, MML version 4.99.1005
:: INT_2:funcnot 1 => INT_2:func 1
definition
let a1 be integer set;
redefine func abs a1 -> Element of NAT;
projectivity;
:: for a1 being integer set holds
:: abs abs a1 = abs a1;
end;
:: INT_2:th 1
theorem
for b1, b2, b3 being integer set
st b1 divides b2 & b1 divides b2 + b3
holds b1 divides b3;
:: INT_2:th 2
theorem
for b1, b2, b3 being integer set
st b1 divides b2
holds b1 divides b2 * b3;
:: INT_2:th 3
theorem
for b1 being integer set holds
0 divides b1
iff
b1 = 0;
:: INT_2:funcnot 2 => INT_2:func 2
definition
let a1, a2 be integer set;
func A1 lcm A2 -> natural set means
a1 divides it &
a2 divides it &
(for b1 being integer set
st a1 divides b1 & a2 divides b1
holds it divides b1);
commutativity;
:: for a1, a2 being integer set holds
:: a1 lcm a2 = a2 lcm a1;
end;
:: INT_2:def 2
theorem
for b1, b2 being integer set
for b3 being natural set holds
b3 = b1 lcm b2
iff
b1 divides b3 &
b2 divides b3 &
(for b4 being integer set
st b1 divides b4 & b2 divides b4
holds b3 divides b4);
:: INT_2:funcnot 3 => INT_2:func 2
notation
let a1, a2 be integer set;
synonym a1 lcm' a2 for a1 lcm a2;
end;
:: INT_2:th 4
theorem
for b1, b2 being integer set holds
(b1 = 0 or b2 = 0)
iff
b1 lcm b2 = 0;
:: INT_2:funcnot 4 => INT_2:func 3
definition
let a1, a2 be integer set;
func A1 gcd A2 -> natural set means
it divides a1 &
it divides a2 &
(for b1 being integer set
st b1 divides a1 & b1 divides a2
holds b1 divides it);
commutativity;
:: for a1, a2 being integer set holds
:: a1 gcd a2 = a2 gcd a1;
end;
:: INT_2:def 3
theorem
for b1, b2 being integer set
for b3 being natural set holds
b3 = b1 gcd b2
iff
b3 divides b1 &
b3 divides b2 &
(for b4 being integer set
st b4 divides b1 & b4 divides b2
holds b4 divides b3);
:: INT_2:th 5
theorem
for b1, b2 being integer set holds
b1 = 0 & b2 = 0
iff
b1 gcd b2 = 0;
:: INT_2:th 8
theorem
for b1 being natural set holds
- b1 is Element of NAT
iff
b1 = 0;
:: INT_2:th 9
theorem
- 1 is not Element of NAT;
:: INT_2:th 10
theorem
for b1 being integer set holds
0 divides b1
iff
b1 = 0;
:: INT_2:th 11
theorem
for b1 being integer set holds
b1 divides - b1 & - b1 divides b1;
:: INT_2:th 12
theorem
for b1, b2, b3 being integer set
st b1 divides b2
holds b1 divides b2 * b3;
:: INT_2:th 13
theorem
for b1, b2, b3 being integer set
st b1 divides b2 & b2 divides b3
holds b1 divides b3;
:: INT_2:th 14
theorem
for b1, b2 being integer set holds
(b1 divides b2 implies b1 divides - b2) & (b1 divides - b2 implies b1 divides b2) & (b1 divides b2 implies - b1 divides b2) & (- b1 divides b2 implies b1 divides b2) & (b1 divides b2 implies - b1 divides - b2) & (- b1 divides - b2 implies b1 divides b2) & (b1 divides - b2 implies - b1 divides b2) & (- b1 divides b2 implies b1 divides - b2);
:: INT_2:th 15
theorem
for b1, b2 being integer set
st b1 divides b2 & b2 divides b1 & b1 <> b2
holds b1 = - b2;
:: INT_2:th 16
theorem
for b1 being integer set holds
b1 divides 0 & 1 divides b1 & - 1 divides b1;
:: INT_2:th 17
theorem
for b1 being integer set
st (b1 divides 1 or b1 divides - 1) & b1 <> 1
holds b1 = - 1;
:: INT_2:th 18
theorem
for b1 being integer set
st (b1 = 1 or b1 = - 1)
holds b1 divides 1 & b1 divides - 1;
:: INT_2:th 19
theorem
for b1, b2, b3 being integer set holds
b1,b2 are_congruent_mod b3
iff
b3 divides b1 - b2;
:: INT_2:th 20
theorem
for b1 being integer set holds
abs b1 is Element of NAT;
:: INT_2:th 21
theorem
for b1, b2 being integer set holds
b1 divides b2
iff
abs b1 divides abs b2;
:: INT_2:th 23
theorem
for b1, b2 being integer set holds
b1 lcm b2 is Element of NAT;
:: INT_2:th 25
theorem
for b1, b2 being integer set holds
b1 divides b1 lcm b2;
:: INT_2:th 26
theorem
for b1, b2 being integer set holds
b1 divides b2 lcm b1;
:: INT_2:th 27
theorem
for b1, b2, b3 being integer set
st b1 divides b3 & b2 divides b3
holds b1 lcm b2 divides b3;
:: INT_2:th 29
theorem
for b1, b2 being integer set holds
b1 gcd b2 is Element of NAT;
:: INT_2:th 31
theorem
for b1, b2 being integer set holds
b1 gcd b2 divides b1;
:: INT_2:th 32
theorem
for b1, b2 being integer set holds
b1 gcd b2 divides b2;
:: INT_2:th 33
theorem
for b1, b2, b3 being integer set
st b3 divides b1 & b3 divides b2
holds b3 divides b1 gcd b2;
:: INT_2:th 34
theorem
for b1, b2 being integer set holds
(b1 = 0 or b2 = 0)
iff
b1 lcm b2 = 0;
:: INT_2:th 35
theorem
for b1, b2 being integer set holds
b1 = 0 & b2 = 0
iff
b1 gcd b2 = 0;
:: INT_2:prednot 1 => INT_2:pred 1
definition
let a1, a2 be integer set;
pred A1,A2 are_relative_prime means
a1 gcd a2 = 1;
symmetry;
:: for a1, a2 being integer set
:: st a1,a2 are_relative_prime
:: holds a2,a1 are_relative_prime;
end;
:: INT_2:dfs 3
definiens
let a1, a2 be integer set;
To prove
a1,a2 are_relative_prime
it is sufficient to prove
thus a1 gcd a2 = 1;
:: INT_2:def 4
theorem
for b1, b2 being integer set holds
b1,b2 are_relative_prime
iff
b1 gcd b2 = 1;
:: INT_2:th 38
theorem
for b1, b2 being integer set
st (b1 = 0 implies b2 <> 0)
holds ex b3, b4 being integer set st
b1 = (b1 gcd b2) * b3 & b2 = (b1 gcd b2) * b4 & b3,b4 are_relative_prime;
:: INT_2:th 39
theorem
for b1, b2, b3 being integer set
st b1,b2 are_relative_prime
holds (b3 * b1) gcd (b3 * b2) = abs b3 & (b3 * b1) gcd (b2 * b3) = abs b3 & (b1 * b3) gcd (b3 * b2) = abs b3 & (b1 * b3) gcd (b2 * b3) = abs b3;
:: INT_2:th 40
theorem
for b1, b2, b3 being integer set
st b1 divides b2 * b3 & b2,b1 are_relative_prime
holds b1 divides b3;
:: INT_2:th 41
theorem
for b1, b2, b3 being integer set
st b1,b2 are_relative_prime & b3,b2 are_relative_prime
holds b1 * b3,b2 are_relative_prime;
:: INT_2:attrnot 1 => INT_2:attr 1
definition
let a1 be natural set;
attr a1 is prime means
1 < a1 &
(for b1 being natural set
st b1 divides a1 & b1 <> 1
holds b1 = a1);
end;
:: INT_2:dfs 4
definiens
let a1 be natural set;
To prove
a1 is prime
it is sufficient to prove
thus 1 < a1 &
(for b1 being natural set
st b1 divides a1 & b1 <> 1
holds b1 = a1);
:: INT_2:def 5
theorem
for b1 being natural set holds
b1 is prime
iff
1 < b1 &
(for b2 being natural set
st b2 divides b1 & b2 <> 1
holds b2 = b1);
:: INT_2:th 43
theorem
for b1, b2 being integer set
st 0 < b1 & b2 divides b1
holds b2 <= b1;
:: INT_2:th 44
theorem
2 is prime;
:: INT_2:th 46
theorem
4 is not prime;
:: INT_2:exreg 1
registration
cluster ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer prime set;
end;
:: INT_2:exreg 2
registration
cluster ext-real non negative non empty epsilon-transitive epsilon-connected ordinal natural complex real integer non prime set;
end;
:: INT_2:th 47
theorem
for b1, b2 being natural set
st b1 is prime & b2 is prime & not b1,b2 are_relative_prime
holds b1 = b2;
:: INT_2:th 48
theorem
for b1 being natural set
st 2 <= b1
holds ex b2 being Element of NAT st
b2 is prime & b2 divides b1;
:: INT_2:th 49
theorem
for b1, b2 being integer set
st 0 <= b1 & 0 <= b2
holds (abs b1) mod abs b2 = b1 mod b2 & (abs b1) div abs b2 = b1 div b2;
:: INT_2:th 50
theorem
for b1, b2 being integer set holds
b1 lcm b2 = (abs b1) lcm abs b2;
:: INT_2:th 51
theorem
for b1, b2 being integer set holds
b1 gcd b2 = (abs b1) gcd abs b2;