Casts of rational numbers into characteristic zero fields (or division rings). #

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@[simp]

theorem Rat.cast_inj {α : Type u_3} [DivisionRing α] [CharZero α] {m : ℚ} {n : ℚ} :

↑m = ↑n ↔ m = n

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theorem Rat.cast_injective {α : Type u_3} [DivisionRing α] [CharZero α] :

Function.Injective Rat.cast

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@[simp]

theorem Rat.cast_eq_zero {α : Type u_3} [DivisionRing α] [CharZero α] {n : ℚ} :

↑n = 0 ↔ n = 0

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theorem Rat.cast_ne_zero {α : Type u_3} [DivisionRing α] [CharZero α] {n : ℚ} :

↑n ≠ 0 ↔ n ≠ 0

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@[simp]

theorem Rat.cast_add {α : Type u_3} [DivisionRing α] [CharZero α] (m : ℚ) (n : ℚ) :

↑(m + n) = ↑m + ↑n

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@[simp]

theorem Rat.cast_sub {α : Type u_3} [DivisionRing α] [CharZero α] (m : ℚ) (n : ℚ) :

↑(m - n) = ↑m - ↑n

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@[simp]

theorem Rat.cast_mul {α : Type u_3} [DivisionRing α] [CharZero α] (m : ℚ) (n : ℚ) :

↑(m * n) = ↑m * ↑n

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@[simp]

theorem Rat.cast_bit0 {α : Type u_3} [DivisionRing α] [CharZero α] (n : ℚ) :

↑(bit0 n) = bit0 ↑n

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@[simp]

theorem Rat.cast_bit1 {α : Type u_3} [DivisionRing α] [CharZero α] (n : ℚ) :

↑(bit1 n) = bit1 ↑n

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def Rat.castHom (α : Type u_3) [DivisionRing α] [CharZero α] :

ℚ →+* α

Coercion ℚ → α as a RingHom.

Equations

One or more equations did not get rendered due to their size.

Instances For

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@[simp]

theorem Rat.coe_cast_hom {α : Type u_3} [DivisionRing α] [CharZero α] :

↑(Rat.castHom α) = Rat.cast

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@[simp]

theorem Rat.cast_inv {α : Type u_3} [DivisionRing α] [CharZero α] (n : ℚ) :

↑n⁻¹ = (↑n)⁻¹

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@[simp]

theorem Rat.cast_div {α : Type u_3} [DivisionRing α] [CharZero α] (m : ℚ) (n : ℚ) :

↑(m / n) = ↑m / ↑n

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@[simp]

theorem Rat.cast_zpow {α : Type u_3} [DivisionRing α] [CharZero α] (q : ℚ) (n : ℤ) :

↑(q ^ n) = ↑q ^ n

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theorem Rat.cast_mk {α : Type u_3} [DivisionRing α] [CharZero α] (a : ℤ) (b : ℤ) :

↑(Rat.divInt a b) = ↑a / ↑b

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@[simp]

theorem Rat.cast_pow {α : Type u_3} [DivisionRing α] [CharZero α] (q : ℚ) (k : ℕ) :

↑(q ^ k) = ↑q ^ k