Lemmas about divisibility in rings #
Note that this file is imported by basic tactics like linarith
and so must have only minimal
imports. Further results about divisibility in rings may be found in
Mathlib/Algebra/Ring/Divisibility/Lemmas.lean
which is not subject to this import constraint.
If G
is a LeftCancelSemiGroup
, left multiplication by g
yields an equivalence between G
and the set of elements of G
divisible by g
.
Equations
Instances For
Alias of the reverse direction of neg_dvd
.
The negation of an element a
of a semigroup with a distributive negation divides another
element b
iff a
divides b
.
Alias of the forward direction of neg_dvd
.
The negation of an element a
of a semigroup with a distributive negation divides another
element b
iff a
divides b
.
Alias of the forward direction of dvd_neg
.
An element a
of a semigroup with a distributive negation divides the negation of an element
b
iff a
divides b
.
Alias of the reverse direction of dvd_neg
.
An element a
of a semigroup with a distributive negation divides the negation of an element
b
iff a
divides b
.
Alias of dvd_sub
.
If an element a
divides another element c
in a ring, a
divides the sum of another element
b
with c
iff a
divides b
.
If an element a
divides another element b
in a ring, a
divides the sum of b
and another
element c
iff a
divides c
.
If an element a
divides another element c
in a ring, a
divides the difference of another
element b
with c
iff a
divides b
.
If an element a
divides another element b
in a ring, a
divides the difference of b
and
another element c
iff a
divides c
.