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.divInt.

Tags #

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

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@[simp]

theorem NNRat.cast_natCast {α : Type u_3} [DivisionSemiring α] (n : ℕ) :

↑↑n = ↑n

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@[simp]

theorem NNRat.cast_ofNat {α : Type u_3} [DivisionSemiring α] (n : ℕ) [Nat.AtLeastTwo n] :

↑(OfNat.ofNat n) = OfNat.ofNat n

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@[simp]

theorem NNRat.cast_zero {α : Type u_3} [DivisionSemiring α] :

↑0 = 0

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@[simp]

theorem NNRat.cast_one {α : Type u_3} [DivisionSemiring α] :

↑1 = 1

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theorem NNRat.cast_commute {α : Type u_3} [DivisionSemiring α] (q : ℚ≥0) (a : α) :

Commute (↑q) a

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theorem NNRat.commute_cast {α : Type u_3} [DivisionSemiring α] (a : α) (q : ℚ≥0) :

Commute a ↑q

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theorem NNRat.cast_comm {α : Type u_3} [DivisionSemiring α] (q : ℚ≥0) (a : α) :

↑q * a = a * ↑q

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theorem NNRat.cast_divNat_of_ne_zero {α : Type u_3} [DivisionSemiring α] (a : ℕ) {b : ℕ} (hb : ↑b ≠ 0) :

↑(NNRat.divNat a b) = ↑a / ↑b

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theorem NNRat.cast_add_of_ne_zero {α : Type u_3} [DivisionSemiring α] {q : ℚ≥0} {r : ℚ≥0} (hq : ↑q.den ≠ 0) (hr : ↑r.den ≠ 0) :

↑(q + r) = ↑q + ↑r

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theorem NNRat.cast_mul_of_ne_zero {α : Type u_3} [DivisionSemiring α] {q : ℚ≥0} {r : ℚ≥0} (hq : ↑q.den ≠ 0) (hr : ↑r.den ≠ 0) :

↑(q * r) = ↑q * ↑r

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theorem NNRat.cast_inv_of_ne_zero {α : Type u_3} [DivisionSemiring α] {q : ℚ≥0} (hq : ↑q.num ≠ 0) :

↑q⁻¹ = (↑q)⁻¹

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theorem NNRat.cast_div_of_ne_zero {α : Type u_3} [DivisionSemiring α] {q : ℚ≥0} {r : ℚ≥0} (hq : ↑q.den ≠ 0) (hr : ↑r.num ≠ 0) :

↑(q / r) = ↑q / ↑r

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@[simp]

theorem Rat.cast_intCast {α : Type u_3} [DivisionRing α] (n : ℤ) :

↑↑n = ↑n

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@[simp]

theorem Rat.cast_natCast {α : Type u_3} [DivisionRing α] (n : ℕ) :

↑↑n = ↑n

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@[deprecated Rat.cast_intCast]

theorem Rat.cast_coe_int {α : Type u_3} [DivisionRing α] (n : ℤ) :

↑↑n = ↑n

Alias of Rat.cast_intCast.

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@[deprecated Rat.cast_natCast]

theorem Rat.cast_coe_nat {α : Type u_3} [DivisionRing α] (n : ℕ) :

↑↑n = ↑n

Alias of Rat.cast_natCast.

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@[simp]

theorem Rat.cast_ofNat {α : Type u_3} [DivisionRing α] (n : ℕ) [Nat.AtLeastTwo n] :

↑(OfNat.ofNat n) = OfNat.ofNat n

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@[simp]

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

↑0 = 0

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@[simp]

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

↑1 = 1

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theorem Rat.cast_commute {α : Type u_3} [DivisionRing α] (r : ℚ) (a : α) :

Commute (↑r) a

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theorem Rat.cast_comm {α : Type u_3} [DivisionRing α] (r : ℚ) (a : α) :

↑r * a = a * ↑r

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theorem Rat.commute_cast {α : Type u_3} [DivisionRing α] (a : α) (r : ℚ) :

Commute a ↑r

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theorem Rat.cast_divInt_of_ne_zero {α : Type u_3} [DivisionRing α] (a : ℤ) {b : ℤ} (b0 : ↑b ≠ 0) :

↑(Rat.divInt a b) = ↑a / ↑b

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theorem Rat.cast_mkRat_of_ne_zero {α : Type u_3} [DivisionRing α] (a : ℤ) {b : ℕ} (hb : ↑b ≠ 0) :

↑(mkRat a b) = ↑a / ↑b

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theorem Rat.cast_add_of_ne_zero {α : Type u_3} [DivisionRing α] {q : ℚ} {r : ℚ} (hq : ↑q.den ≠ 0) (hr : ↑r.den ≠ 0) :

↑(q + r) = ↑q + ↑r

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@[simp]

theorem Rat.cast_neg {α : Type u_3} [DivisionRing α] (q : ℚ) :

↑(-q) = -↑q

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theorem Rat.cast_sub_of_ne_zero {α : Type u_3} [DivisionRing α] {p : ℚ} {q : ℚ} (hp : ↑p.den ≠ 0) (hq : ↑q.den ≠ 0) :

↑(p - q) = ↑p - ↑q

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theorem Rat.cast_mul_of_ne_zero {α : Type u_3} [DivisionRing α] {p : ℚ} {q : ℚ} (hp : ↑p.den ≠ 0) (hq : ↑q.den ≠ 0) :

↑(p * q) = ↑p * ↑q

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theorem Rat.cast_inv_of_ne_zero {α : Type u_3} [DivisionRing α] {q : ℚ} (hq : ↑q.num ≠ 0) :

↑q⁻¹ = (↑q)⁻¹

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theorem Rat.cast_div_of_ne_zero {α : Type u_3} [DivisionRing α] {p : ℚ} {q : ℚ} (hp : ↑p.den ≠ 0) (hq : ↑q.num ≠ 0) :

↑(p / q) = ↑p / ↑q

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@[simp]

theorem map_nnratCast {F : Type u_1} {α : Type u_3} {β : Type u_4} [FunLike F α β] [DivisionSemiring α] [DivisionSemiring β] [RingHomClass F α β] (f : F) (q : ℚ≥0) :

f ↑q = ↑q

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@[simp]

theorem eq_nnratCast {F : Type u_1} {α : Type u_3} [DivisionSemiring α] [FunLike F ℚ≥0 α] [RingHomClass F ℚ≥0 α] (f : F) (q : ℚ≥0) :

f q = ↑q

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@[simp]

theorem map_ratCast {F : Type u_1} {α : Type u_3} {β : Type u_4} [FunLike F α β] [DivisionRing α] [DivisionRing β] [RingHomClass F α β] (f : F) (q : ℚ) :

f ↑q = ↑q

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@[simp]

theorem eq_ratCast {F : Type u_1} {α : Type u_3} [DivisionRing α] [FunLike F ℚ α] [RingHomClass F ℚ α] (f : F) (q : ℚ) :

f q = ↑q

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theorem MonoidWithZeroHom.ext_rat' {F : Type u_1} {M₀ : Type u_5} [MonoidWithZero M₀] [FunLike F ℚ M₀] [MonoidWithZeroHomClass F ℚ M₀] {f : F} {g : F} (h : ∀ (m : ℤ), f ↑m = g ↑m) :

f = g

If f and g agree on the integers then they are equal φ.

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theorem MonoidWithZeroHom.ext_rat {M₀ : Type u_5} [MonoidWithZero M₀] {f : ℚ →*₀ M₀} {g : ℚ →*₀ M₀} (h : MonoidWithZeroHom.comp f ↑(Int.castRingHom ℚ) = MonoidWithZeroHom.comp g ↑(Int.castRingHom ℚ)) :

f = g

If f and g agree on the integers then they are equal φ.

See note [partially-applied ext lemmas] for why comp is used here.

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theorem MonoidWithZeroHom.ext_rat_on_pnat {F : Type u_1} {M₀ : Type u_5} [MonoidWithZero M₀] [FunLike F ℚ M₀] [MonoidWithZeroHomClass F ℚ M₀] {f : F} {g : F} (same_on_neg_one : f (-1) = g (-1)) (same_on_pnat : ∀ (n : ℕ), 0 < n → f ↑n = g ↑n) :

f = g

Positive integer values of a morphism φ and its value on -1 completely determine φ.

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