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.
Data
.
Nat
.
Prime
.
Factorial
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Imports
Init
Mathlib.Data.Nat.Factorial.Basic
Mathlib.Data.Nat.Prime.Basic
Imported by
Nat
.
Prime
.
dvd_factorial
Nat
.
coprime_factorial_iff
Nat
.
Prime
.
coprime_factorial_of_lt
Nat
.
Prime
.
coprime_descFactorial_of_lt_of_le
Prime natural numbers and the factorial operator
#
source
theorem
Nat
.
Prime
.
dvd_factorial
{
n
p
:
ℕ
}
:
Prime
p
→ (
p
∣
n
.
factorial
↔
p
≤
n
)
source
theorem
Nat
.
coprime_factorial_iff
{
m
n
:
ℕ
}
(
hm
:
m
≠
1
)
:
m
.
Coprime
n
.
factorial
↔
n
<
m
.
minFac
source
theorem
Nat
.
Prime
.
coprime_factorial_of_lt
{
p
n
:
ℕ
}
(
hp
:
Prime
p
)
(
hn
:
n
<
p
)
:
p
.
Coprime
n
.
factorial
source
theorem
Nat
.
Prime
.
coprime_descFactorial_of_lt_of_le
{
p
n
k
:
ℕ
}
(
hp
:
Prime
p
)
(
hn
:
n
<
p
)
(
hk
:
k
≤
n
)
:
p
.
Coprime
(
n
.
descFactorial
k
)