Absolute values in ordered groups #
The absolute value of an element in a group which is also a lattice is its supremum with its
negation. This generalizes the usual absolute value on real numbers (|x| = max x (-x)
).
Notations #
|a|
: The absolute value of an elementa
of an additive lattice ordered group|a|ₘ
: The absolute value of an elementa
of a multiplicative lattice ordered group
theorem
div_le_of_mabs_div_le_left
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b c : G}
(h : mabs (a / b) ≤ c)
:
theorem
sub_le_of_abs_sub_le_left
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b c : G}
(h : |a - b| ≤ c)
:
theorem
div_le_of_mabs_div_le_right
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b c : G}
(h : mabs (a / b) ≤ c)
:
theorem
sub_le_of_abs_sub_le_right
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b c : G}
(h : |a - b| ≤ c)
:
theorem
div_lt_of_mabs_div_lt_left
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b c : G}
(h : mabs (a / b) < c)
:
theorem
sub_lt_of_abs_sub_lt_left
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b c : G}
(h : |a - b| < c)
:
theorem
div_lt_of_mabs_div_lt_right
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b c : G}
(h : mabs (a / b) < c)
:
theorem
sub_lt_of_abs_sub_lt_right
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b c : G}
(h : |a - b| < c)
:
theorem
eq_of_mabs_div_eq_one
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b : G}
(h : mabs (a / b) = 1)
:
theorem
eq_of_abs_sub_eq_zero
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b : G}
(h : |a - b| = 0)
:
@[deprecated abs_sub_le_of_le_of_le (since := "2025-03-02")]
theorem
dist_bdd_within_interval
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b lb ub : G}
(hal : lb ≤ a)
(hau : a ≤ ub)
(hbl : lb ≤ b)
(hbu : b ≤ ub)
:
Alias of abs_sub_le_of_le_of_le
.
theorem
eq_of_mabs_div_le_one
{G : Type u_1}
[LinearOrderedCommGroup G]
{a b : G}
(h : mabs (a / b) ≤ 1)
:
theorem
eq_of_abs_sub_nonpos
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{a b : G}
(h : |a - b| ≤ 0)
:
theorem
eq_of_mabs_div_lt_all
{G : Type u_1}
[LinearOrderedCommGroup G]
{x y : G}
(h : ∀ (ε : G), ε > 1 → mabs (x / y) < ε)
:
theorem
eq_of_abs_sub_lt_all
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{x y : G}
(h : ∀ (ε : G), ε > 0 → |x - y| < ε)
:
theorem
eq_of_mabs_div_le_all
{G : Type u_1}
[LinearOrderedCommGroup G]
[DenselyOrdered G]
{x y : G}
(h : ∀ (ε : G), ε > 1 → mabs (x / y) ≤ ε)
:
theorem
eq_of_abs_sub_le_all
{G : Type u_1}
[LinearOrderedAddCommGroup G]
[DenselyOrdered G]
{x y : G}
(h : ∀ (ε : G), ε > 0 → |x - y| ≤ ε)
:
@[simp]
@[simp]
@[simp]
@[simp]
@[simp]
theorem
apply_abs_le_mul_of_one_le'
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{H : Type u_2}
[MulOneClass H]
[LE H]
[MulLeftMono H]
[MulRightMono H]
{f : G → H}
{a : G}
(h₁ : 1 ≤ f a)
(h₂ : 1 ≤ f (-a))
:
theorem
apply_abs_le_add_of_nonneg'
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{H : Type u_2}
[AddZeroClass H]
[LE H]
[AddLeftMono H]
[AddRightMono H]
{f : G → H}
{a : G}
(h₁ : 0 ≤ f a)
(h₂ : 0 ≤ f (-a))
:
theorem
apply_abs_le_mul_of_one_le
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{H : Type u_2}
[MulOneClass H]
[LE H]
[MulLeftMono H]
[MulRightMono H]
{f : G → H}
(h : ∀ (x : G), 1 ≤ f x)
(a : G)
:
theorem
apply_abs_le_add_of_nonneg
{G : Type u_1}
[LinearOrderedAddCommGroup G]
{H : Type u_2}
[AddZeroClass H]
[LE H]
[AddLeftMono H]
[AddRightMono H]
{f : G → H}
(h : ∀ (x : G), 0 ≤ f x)
(a : G)
: