Documentation

Mathlib.Data.Int.CharZero

Injectivity of Int.Cast into characteristic zero rings and fields. #

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@[simp]
theorem Int.cast_eq_zero {α : Type u_1} [AddGroupWithOne α] [CharZero α] {n : ℤ} :
↑n = 0 ↔ n = 0
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@[simp]
theorem Int.cast_inj {α : Type u_1} [AddGroupWithOne α] [CharZero α] {m : ℤ} {n : ℤ} :
↑m = ↑n ↔ m = n
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theorem Int.cast_injective {α : Type u_1} [AddGroupWithOne α] [CharZero α] :
Function.Injective Int.cast
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theorem Int.cast_ne_zero {α : Type u_1} [AddGroupWithOne α] [CharZero α] {n : ℤ} :
↑n ≠ 0 ↔ n ≠ 0
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@[simp]
theorem Int.cast_eq_one {α : Type u_1} [AddGroupWithOne α] [CharZero α] {n : ℤ} :
↑n = 1 ↔ n = 1
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theorem Int.cast_ne_one {α : Type u_1} [AddGroupWithOne α] [CharZero α] {n : ℤ} :
↑n ≠ 1 ↔ n ≠ 1
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@[simp]
theorem Int.cast_div_charZero {k : Type u_3} [DivisionRing k] [CharZero k] {m : ℤ} {n : ℤ} (n_dvd : n ∣ m) :
↑(m / n) = ↑m / ↑n
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@[simp]
theorem Int.cast_div_ofNat_charZero {k : Type u_3} [DivisionRing k] [CharZero k] {m : ℕ} {n : ℕ} (n_dvd : n ∣ m) :
↑(↑m / ↑n) = ↑m / ↑n
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theorem RingHom.injective_int {α : Type u_3} [NonAssocRing α] (f : ℤ →+* α) [CharZero α] :
Function.Injective ⇑f
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theorem Function.support_int_cast {α : Type u_1} {β : Type u_2} [AddGroupWithOne β] [CharZero β] {n : ℤ} (hn : n ≠ 0) :
Function.support ↑n = Set.univ
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theorem Function.mulSupport_int_cast {α : Type u_1} {β : Type u_2} [AddGroupWithOne β] [CharZero β] {n : ℤ} (hn : n ≠ 1) :
Function.mulSupport ↑n = Set.univ