Documentation

Mathlib.Data.Nat.Cast.Field

Cast of naturals into fields #

This file concerns the canonical homomorphism ℕ → F, where F is a field.

Main results #

@[simp]
theorem Nat.cast_div {α : Type u_1} [DivisionSemiring α] {m : } {n : } (n_dvd : n m) (n_nonzero : n 0) :
↑(m / n) = m / n
theorem Nat.cast_div_div_div_cancel_right {α : Type u_1} [DivisionSemiring α] [CharZero α] {m : } {n : } {d : } (hn : d n) (hm : d m) :
↑(m / d) / ↑(n / d) = m / n
theorem Nat.cast_div_le {α : Type u_1} [LinearOrderedSemifield α] {m : } {n : } :
↑(m / n) m / n

Natural division is always less than division in the field.

theorem Nat.inv_pos_of_nat {α : Type u_1} [LinearOrderedSemifield α] {n : } :
0 < (n + 1)⁻¹
theorem Nat.one_div_pos_of_nat {α : Type u_1} [LinearOrderedSemifield α] {n : } :
0 < 1 / (n + 1)
theorem Nat.one_div_le_one_div {α : Type u_1} [LinearOrderedSemifield α] {n : } {m : } (h : n m) :
1 / (m + 1) 1 / (n + 1)
theorem Nat.one_div_lt_one_div {α : Type u_1} [LinearOrderedSemifield α] {n : } {m : } (h : n < m) :
1 / (m + 1) < 1 / (n + 1)