# mathlib3documentation

algebra.order.sub.basic

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

theorem add_hom.le_map_tsub {α : Type u_1} {β : Type u_2} [preorder α] [has_add α] [has_sub α] [preorder β] [has_add β] [has_sub β] (f : β) (hf : monotone f) (a b : α) :
f a - f b f (a - b)
theorem le_mul_tsub {R : Type u_1} [distrib R] [preorder R] [has_sub R] {a b c : R} :
a * b - a * c a * (b - c)
theorem le_tsub_mul {R : Type u_1} [preorder R] [has_sub R] {a b c : R} :
a * c - b * c (a - b) * c
theorem order_iso.map_tsub {M : Type u_1} {N : Type u_2} [preorder M] [has_add M] [has_sub M] [has_add N] [has_sub N] (e : M ≃o N) (h_add : (a b : M), e (a + b) = e a + e b) (a 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 add_monoid_hom.le_map_tsub {α : Type u_1} {β : Type u_2} [preorder α] [has_sub α] [preorder β] [has_sub β] (f : α →+ β) (hf : monotone f) (a b : α) :
f a - f b f (a - b)