mathlib documentation

data.rat.cast

Casts for Rational Numbers

Summary

We define the canonical injection from ℚ into an arbitrary division ring and prove various casting lemmas showing the well-behavedness of this injection.

Notations

/. is infix notation for rat.mk.

Tags

rat, rationals, field, ℚ, numerator, denominator, num, denom, cast, coercion, casting

@[instance]

def rat.cast_coe {α : Type u_1} [division_ring α] :

has_coe_t ℚ α

Construct the canonical injection from ℚ into an arbitrary division ring. If the field has positive characteristic p, we define 1 / p = 1 / 0 = 0 for consistency with our division by zero convention.

Equations

rat.cast_coe = {coe := λ (r : ℚ), ↑(r.num) / ↑(r.denom)}

@[simp]

theorem rat.cast_of_int {α : Type u_1} [division_ring α] (n : ℤ) :

↑(rat.of_int n) = ↑n

@[simp]

theorem rat.cast_coe_int {α : Type u_1} [division_ring α] (n : ℤ) :

@[simp]

theorem rat.cast_coe_nat {α : Type u_1} [division_ring α] (n : ℕ) :

@[simp]

theorem rat.cast_zero {α : Type u_1} [division_ring α] :

↑0 = 0

@[simp]

theorem rat.cast_one {α : Type u_1} [division_ring α] :

↑1 = 1

theorem rat.cast_commute {α : Type u_1} [division_ring α] (r : ℚ) (a : α) :

theorem rat.commute_cast {α : Type u_1} [division_ring α] (a : α) (r : ℚ) :

theorem rat.cast_mk_of_ne_zero {α : Type u_1} [division_ring α] (a b : ℤ) (b0 : ↑b ≠ 0) :

↑(a /. b) = ↑a / ↑b

theorem rat.cast_add_of_ne_zero {α : Type u_1} [division_ring α] {m n : ℚ} (a : ↑(m.denom) ≠ 0) (a_1 : ↑(n.denom) ≠ 0) :

↑(m + n) = ↑m + ↑n

@[simp]

theorem rat.cast_neg {α : Type u_1} [division_ring α] (n : ℚ) :

theorem rat.cast_sub_of_ne_zero {α : Type u_1} [division_ring α] {m n : ℚ} (m0 : ↑(m.denom) ≠ 0) (n0 : ↑(n.denom) ≠ 0) :

↑(m - n) = ↑m - ↑n

theorem rat.cast_mul_of_ne_zero {α : Type u_1} [division_ring α] {m n : ℚ} (a : ↑(m.denom) ≠ 0) (a_1 : ↑(n.denom) ≠ 0) :

↑m * n = (↑m) * ↑n

theorem rat.cast_inv_of_ne_zero {α : Type u_1} [division_ring α] {n : ℚ} (a : ↑(n.num) ≠ 0) (a_1 : ↑(n.denom) ≠ 0) :

↑n⁻¹ = (↑n)⁻¹

theorem rat.cast_div_of_ne_zero {α : Type u_1} [division_ring α] {m n : ℚ} (md : ↑(m.denom) ≠ 0) (nn : ↑(n.num) ≠ 0) (nd : ↑(n.denom) ≠ 0) :

↑(m / n) = ↑m / ↑n

@[simp]

theorem rat.cast_inj {α : Type u_1} [division_ring α] [char_zero α] {m n : ℚ} :

↑m = ↑n ↔ m = n

theorem rat.cast_injective {α : Type u_1} [division_ring α] [char_zero α] :

function.injective coe

@[simp]

theorem rat.cast_eq_zero {α : Type u_1} [division_ring α] [char_zero α] {n : ℚ} :

↑n = 0 ↔ n = 0

theorem rat.cast_ne_zero {α : Type u_1} [division_ring α] [char_zero α] {n : ℚ} :

↑n ≠ 0 ↔ n ≠ 0

@[simp]

theorem rat.cast_add {α : Type u_1} [division_ring α] [char_zero α] (m n : ℚ) :

↑(m + n) = ↑m + ↑n

@[simp]

theorem rat.cast_sub {α : Type u_1} [division_ring α] [char_zero α] (m n : ℚ) :

↑(m - n) = ↑m - ↑n

@[simp]

theorem rat.cast_mul {α : Type u_1} [division_ring α] [char_zero α] (m n : ℚ) :

↑m * n = (↑m) * ↑n

@[simp]

theorem rat.cast_bit0 {α : Type u_1} [division_ring α] [char_zero α] (n : ℚ) :

↑(bit0 n) = bit0 ↑n

@[simp]

theorem rat.cast_bit1 {α : Type u_1} [division_ring α] [char_zero α] (n : ℚ) :

↑(bit1 n) = bit1 ↑n

def rat.cast_hom (α : Type u_1) [division_ring α] [char_zero α] :

Coercion ℚ → α as a ring_hom.

Equations

rat.cast_hom α = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

@[simp]

theorem rat.coe_cast_hom {α : Type u_1} [division_ring α] [char_zero α] :

⇑(rat.cast_hom α) = coe

@[simp]

theorem rat.cast_inv {α : Type u_1} [division_ring α] [char_zero α] (n : ℚ) :

↑n⁻¹ = (↑n)⁻¹

@[simp]

theorem rat.cast_div {α : Type u_1} [division_ring α] [char_zero α] (m n : ℚ) :

↑(m / n) = ↑m / ↑n

theorem rat.cast_mk {α : Type u_1} [division_ring α] [char_zero α] (a b : ℤ) :

↑(a /. b) = ↑a / ↑b

@[simp]

theorem rat.cast_pow {α : Type u_1} [division_ring α] [char_zero α] (q : ℚ) (k : ℕ) :

↑(q ^ k) = ↑q ^ k

@[simp]

theorem rat.cast_nonneg {α : Type u_1} [linear_ordered_field α] {n : ℚ} :

0 ≤ ↑n ↔ 0 ≤ n

@[simp]

theorem rat.cast_le {α : Type u_1} [linear_ordered_field α] {m n : ℚ} :

↑m ≤ ↑n ↔ m ≤ n

@[simp]

theorem rat.cast_lt {α : Type u_1} [linear_ordered_field α] {m n : ℚ} :

↑m < ↑n ↔ m < n

@[simp]

theorem rat.cast_nonpos {α : Type u_1} [linear_ordered_field α] {n : ℚ} :

↑n ≤ 0 ↔ n ≤ 0

@[simp]

theorem rat.cast_pos {α : Type u_1} [linear_ordered_field α] {n : ℚ} :

0 < ↑n ↔ 0 < n

@[simp]

theorem rat.cast_lt_zero {α : Type u_1} [linear_ordered_field α] {n : ℚ} :

↑n < 0 ↔ n < 0

@[simp]

theorem rat.cast_id (n : ℚ) :

↑n = n

@[simp]

theorem rat.cast_min {α : Type u_1} [discrete_linear_ordered_field α] {a b : ℚ} :

↑(min a b) = min ↑a ↑b

@[simp]

theorem rat.cast_max {α : Type u_1} [discrete_linear_ordered_field α] {a b : ℚ} :

↑(max a b) = max ↑a ↑b

@[simp]

theorem rat.cast_abs {α : Type u_1} [discrete_linear_ordered_field α] {q : ℚ} :

↑(abs q) = abs ↑q

theorem ring_hom.eq_rat_cast {k : Type u_1} [division_ring k] (f : ℚ →+* k) (r : ℚ) :

theorem ring_hom.map_rat_cast {k : Type u_1} {k' : Type u_2} [division_ring k] [char_zero k] [division_ring k'] (f : k →+* k') (r : ℚ) :

⇑f ↑r = ↑r

theorem ring_hom.ext_rat {R : Type u_1} [semiring R] (f g : ℚ →+* R) :

f = g

@[instance]

def rat.subsingleton_ring_hom {R : Type u_1} [semiring R] :

subsingleton (ℚ →+* R)