Additional results about ordered Subtraction #

theorem AddHom.le_map_tsub {α : Type u_2} {β : Type u_1} [inst : Preorder α] [inst : Add α] [inst : Sub α] [inst : OrderedSub α] [inst : Preorder β] [inst : Add β] [inst : Sub β] [inst : OrderedSub β] (f : AddHom α β) (hf : Monotone ↑f) (a : α) (b : α) :

↑f a - ↑f b ≤ ↑f (a - b)

theorem le_mul_tsub {R : Type u_1} [inst : Distrib R] [inst : Preorder R] [inst : Sub R] [inst : OrderedSub R] [inst : CovariantClass R R (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : R} {b : R} {c : R} :

a * b - a * c ≤ a * (b - c)

theorem le_tsub_mul {R : Type u_1} [inst : CommSemiring R] [inst : Preorder R] [inst : Sub R] [inst : OrderedSub R] [inst : CovariantClass R R (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : R} {b : R} {c : R} :

a * c - b * c ≤ (a - b) * c

theorem OrderIso.map_tsub {M : Type u_1} {N : Type u_2} [inst : Preorder M] [inst : Add M] [inst : Sub M] [inst : OrderedSub M] [inst : PartialOrder N] [inst : Add N] [inst : Sub N] [inst : OrderedSub N] (e : M ≃o N) (h_add : ∀ (a b : M), ↑(RelIso.toRelEmbedding e).toEmbedding (a + b) = ↑(RelIso.toRelEmbedding e).toEmbedding a + ↑(RelIso.toRelEmbedding e).toEmbedding b) (a : M) (b : M) :

↑(RelIso.toRelEmbedding e).toEmbedding (a - b) = ↑(RelIso.toRelEmbedding e).toEmbedding a - ↑(RelIso.toRelEmbedding e).toEmbedding b

An order isomorphism between types with ordered subtraction preserves subtraction provided that it preserves addition.

theorem AddMonoidHom.le_map_tsub {α : Type u_2} {β : Type u_1} [inst : Preorder α] [inst : AddCommMonoid α] [inst : Sub α] [inst : OrderedSub α] [inst : Preorder β] [inst : AddCommMonoid β] [inst : Sub β] [inst : OrderedSub β] (f : α →+ β) (hf : Monotone ↑f) (a : α) (b : α) :

↑f a - ↑f b ≤ ↑f (a - b)