# mathlib3documentation

algebra.order.smul

# Ordered scalar product #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file we define

• ordered_smul R M : an ordered additive commutative monoid M is an ordered_smul over an ordered_semiring R if the scalar product respects the order relation on the monoid and on the ring. There is a correspondence between this structure and convex cones, which is proven in analysis/convex/cone.lean.

## Implementation notes #

• We choose to define ordered_smul as a Prop-valued mixin, so that it can be used for actions, modules, and algebras (the axioms for an "ordered algebra" are exactly that the algebra is ordered as a module).
• To get ordered modules and ordered vector spaces, it suffices to replace the order_add_comm_monoid and the ordered_semiring as desired.

## Tags #

ordered module, ordered scalar, ordered smul, ordered action, ordered vector space

@[class]
structure ordered_smul (R : Type u_1) (M : Type u_2) [ M] :
Prop

The ordered scalar product property is when an ordered additive commutative monoid with a partial order has a scalar multiplication which is compatible with the order.

Instances of this typeclass
@[protected, instance]
def order_dual.smul_with_zero {R : Type u_3} {M : Type u_4} [has_zero R] [h : M] :
Equations
@[protected, instance]
def order_dual.mul_action {R : Type u_3} {M : Type u_4} [monoid R] [ M] :
Equations
@[protected, instance]
def order_dual.mul_action_with_zero {R : Type u_3} {M : Type u_4} [add_monoid M] [ M] :
Equations
@[protected, instance]
def order_dual.distrib_mul_action {R : Type u_3} {M : Type u_4} [add_monoid M] [ M] :
Equations
@[protected, instance]
def order_dual.ordered_smul {R : Type u_3} {M : Type u_4} [ M] [ M] :
theorem smul_lt_smul_of_pos {R : Type u_3} {M : Type u_4} [ M] [ M] {a b : M} {c : R} :
a < b 0 < c c a < c b
theorem smul_le_smul_of_nonneg {R : Type u_3} {M : Type u_4} [ M] [ M] {a b : M} {c : R} (h₁ : a b) (h₂ : 0 c) :
c a c b
theorem smul_nonneg {R : Type u_3} {M : Type u_4} [ M] [ M] {a : M} {c : R} (hc : 0 c) (ha : 0 a) :
0 c a
theorem smul_nonpos_of_nonneg_of_nonpos {R : Type u_3} {M : Type u_4} [ M] [ M] {a : M} {c : R} (hc : 0 c) (ha : a 0) :
c a 0
theorem eq_of_smul_eq_smul_of_pos_of_le {R : Type u_3} {M : Type u_4} [ M] [ M] {a b : M} {c : R} (h₁ : c a = c b) (hc : 0 < c) (hle : a b) :
a = b
theorem lt_of_smul_lt_smul_of_nonneg {R : Type u_3} {M : Type u_4} [ M] [ M] {a b : M} {c : R} (h : c a < c b) (hc : 0 c) :
a < b
theorem smul_lt_smul_iff_of_pos {R : Type u_3} {M : Type u_4} [ M] [ M] {a b : M} {c : R} (hc : 0 < c) :
c a < c b a < b
theorem smul_pos_iff_of_pos {R : Type u_3} {M : Type u_4} [ M] [ M] {a : M} {c : R} (hc : 0 < c) :
0 < c a 0 < a
theorem smul_pos {R : Type u_3} {M : Type u_4} [ M] [ M] {a : M} {c : R} (hc : 0 < c) :
0 < a 0 < c a

Alias of the reverse direction of smul_pos_iff_of_pos.

theorem monotone_smul_left {R : Type u_3} {M : Type u_4} [ M] [ M] {c : R} (hc : 0 c) :
theorem strict_mono_smul_left {R : Type u_3} {M : Type u_4} [ M] [ M] {c : R} (hc : 0 < c) :
theorem smul_lower_bounds_subset_lower_bounds_smul {R : Type u_3} {M : Type u_4} [ M] [ M] {s : set M} {c : R} (hc : 0 c) :
theorem smul_upper_bounds_subset_upper_bounds_smul {R : Type u_3} {M : Type u_4} [ M] [ M] {s : set M} {c : R} (hc : 0 c) :
theorem bdd_below.smul_of_nonneg {R : Type u_3} {M : Type u_4} [ M] [ M] {s : set M} {c : R} (hs : bdd_below s) (hc : 0 c) :
theorem bdd_above.smul_of_nonneg {R : Type u_3} {M : Type u_4} [ M] [ M] {s : set M} {c : R} (hs : bdd_above s) (hc : 0 c) :
theorem ordered_smul.mk'' {𝕜 : Type u_2} {M : Type u_4} [ M] (h : ⦃c : 𝕜⦄, 0 < c strict_mono (λ (a : M), c a)) :
M

To prove that a linear ordered monoid is an ordered module, it suffices to verify only the first axiom of ordered_smul.

@[protected, instance]
def nat.ordered_smul {M : Type u_4}  :
@[protected, instance]
def int.ordered_smul {M : Type u_4}  :
@[protected, instance]
theorem smul_max {R : Type u_3} {M : Type u_4} [ M] [ M] {a : R} (ha : 0 a) (b₁ b₂ : M) :
a b₂ = linear_order.max (a b₁) (a b₂)
theorem smul_min {R : Type u_3} {M : Type u_4} [ M] [ M] {a : R} (ha : 0 a) (b₁ b₂ : M) :
a b₂ = linear_order.min (a b₁) (a b₂)
theorem ordered_smul.mk' {𝕜 : Type u_2} {M : Type u_4} [ M] (h : ⦃a b : M⦄ ⦃c : 𝕜⦄, a < b 0 < c c a c b) :
M

To prove that a vector space over a linear ordered field is ordered, it suffices to verify only the first axiom of ordered_smul.

@[protected, instance]
def prod.ordered_smul {𝕜 : Type u_2} {M : Type u_4} {N : Type u_5} [ M] [ N] [ M] [ N] :
(M × N)
@[protected, instance]
def pi.ordered_smul {ι : Type u_1} {𝕜 : Type u_2} {M : ι Type u_3} [Π (i : ι), ] [Π (i : ι), (M i)] [ (i : ι), (M i)] :
(Π (i : ι), M i)
@[protected, instance]
def pi.ordered_smul' {ι : Type u_1} {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] :
M)
@[protected, instance]
def pi.ordered_smul'' {ι : Type u_1} {𝕜 : Type u_2}  :
𝕜)
theorem smul_le_smul_iff_of_pos {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {a b : M} {c : 𝕜} (hc : 0 < c) :
c a c b a b
theorem inv_smul_le_iff {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {a b : M} {c : 𝕜} (h : 0 < c) :
c⁻¹ a b a c b
theorem inv_smul_lt_iff {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {a b : M} {c : 𝕜} (h : 0 < c) :
c⁻¹ a < b a < c b
theorem le_inv_smul_iff {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {a b : M} {c : 𝕜} (h : 0 < c) :
a c⁻¹ b c a b
theorem lt_inv_smul_iff {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {a b : M} {c : 𝕜} (h : 0 < c) :
a < c⁻¹ b c a < b
@[simp]
theorem order_iso.smul_left_apply {𝕜 : Type u_2} (M : Type u_4) [ M] [ M] {c : 𝕜} (hc : 0 < c) (b : M) :
hc) b = c b
def order_iso.smul_left {𝕜 : Type u_2} (M : Type u_4) [ M] [ M] {c : 𝕜} (hc : 0 < c) :
M ≃o M

Left scalar multiplication as an order isomorphism.

Equations
@[simp]
theorem order_iso.smul_left_symm_apply {𝕜 : Type u_2} (M : Type u_4) [ M] [ M] {c : 𝕜} (hc : 0 < c) (b : M) :
@[simp]
theorem lower_bounds_smul_of_pos {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {s : set M} {c : 𝕜} (hc : 0 < c) :
@[simp]
theorem upper_bounds_smul_of_pos {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {s : set M} {c : 𝕜} (hc : 0 < c) :
@[simp]
theorem bdd_below_smul_iff_of_pos {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {s : set M} {c : 𝕜} (hc : 0 < c) :
@[simp]
theorem bdd_above_smul_iff_of_pos {𝕜 : Type u_2} {M : Type u_4} [ M] [ M] {s : set M} {c : 𝕜} (hc : 0 < c) :

Extension for the positivity tactic: scalar multiplication is nonnegative/positive/nonzero if both sides are.