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Mathlib.Algebra.Order.Sub.Unbundled.Hom

Lemmas about subtraction in unbundled canonically ordered monoids #

theorem AddHom.le_map_tsub {α : Type u_1} {β : Type u_2} [Preorder α] [Add α] [Sub α] [OrderedSub α] [Preorder β] [Add β] [Sub β] [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_3} [Distrib R] [Preorder R] [Sub R] [OrderedSub R] [CovariantClass R R (fun (x1 x2 : R) => x1 * x2) fun (x1 x2 : R) => x1 x2] {a : R} {b : R} {c : R} :
a * b - a * c a * (b - c)
theorem le_tsub_mul {R : Type u_3} [CommSemiring R] [Preorder R] [Sub R] [OrderedSub R] [CovariantClass R R (fun (x1 x2 : R) => x1 * x2) fun (x1 x2 : R) => x1 x2] {a : R} {b : R} {c : R} :
a * c - b * c (a - b) * c
theorem OrderIso.map_tsub {M : Type u_3} {N : Type u_4} [Preorder M] [Add M] [Sub M] [OrderedSub M] [PartialOrder N] [Add N] [Sub N] [OrderedSub N] (e : M ≃o N) (h_add : ∀ (a b : M), e (a + b) = e a + e b) (a : M) (b : M) :
e (a - b) = e a - e b

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

Preorder #

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