Lemmas about division (semi)rings and (semi)fields

theorem add_div {α : Type u_1} [DivisionSemiring α] (a : α) (b : α) (c : α) : (a + b) / c = a / c + b / c

theorem div_add_div_same {α : Type u_1} [DivisionSemiring α] (a : α) (b : α) (c : α) : a / c + b / c = (a + b) / c

theorem same_add_div {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (h : b ≠ 0) : (b + a) / b = 1 + a / b

theorem div_add_same {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (h : b ≠ 0) : (a + b) / b = a / b + 1

theorem one_add_div {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (h : b ≠ 0) : 1 + a / b = (b + a) / b

theorem div_add_one {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (h : b ≠ 0) : a / b + 1 = (a + b) / b

theorem inv_add_inv' {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ + b⁻¹ = a⁻¹ * (a + b) * b⁻¹

See inv_add_inv for the more convenient version when K is commutative.

theorem one_div_mul_add_mul_one_div_eq_one_div_add_one_div {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a * (a + b) * (1 / b) = 1 / a + 1 / b

theorem add_div_eq_mul_add_div {α : Type u_1} [DivisionSemiring α] {c : α} (a : α) (b : α) (hc : c ≠ 0) : a + b / c = (a * c + b) / c

theorem add_div' {α : Type u_1} [DivisionSemiring α] (a : α) (b : α) (c : α) (hc : c ≠ 0) : b + a / c = (b * c + a) / c

theorem div_add' {α : Type u_1} [DivisionSemiring α] (a : α) (b : α) (c : α) (hc : c ≠ 0) : a / c + b = (a + b * c) / c

theorem Commute.div_add_div {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} {c : α} {d : α} (hbc : Commute b c) (hbd : Commute b d) (hb : b ≠ 0) (hd : d ≠ 0) : a / b + c / d = (a * d + b * c) / (b * d)

theorem Commute.one_div_add_one_div {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (hab : Commute a b) (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b)

theorem Commute.inv_add_inv {α : Type u_1} [DivisionSemiring α] {a : α} {b : α} (hab : Commute a b) (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ + b⁻¹ = (a + b) / (a * b)

theorem one_div_neg_one_eq_neg_one {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] : 1 / -1 = -1

theorem one_div_neg_eq_neg_one_div {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] (a : K) : 1 / -a = -(1 / a)

theorem div_neg_eq_neg_div {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] (a : K) (b : K) : b / -a = -(b / a)

theorem neg_div {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] (a : K) (b : K) : -b / a = -(b / a)

theorem neg_div' {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] (a : K) (b : K) : -(b / a) = -b / a

@[simp]

theorem neg_div_neg_eq {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] (a : K) (b : K) : -a / -b = a / b

theorem neg_inv {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] {a : K} : -a⁻¹ = (-a)⁻¹

theorem div_neg {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] {b : K} (a : K) : a / -b = -(a / b)

theorem inv_neg {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] {a : K} : (-a)⁻¹ = -a⁻¹

theorem inv_neg_one {K : Type u_3} [DivisionMonoid K] [HasDistribNeg K] : (-1)⁻¹ = -1

@[simp]

theorem div_neg_self {K : Type u_3} [DivisionRing K] {a : K} (h : a ≠ 0) : a / -a = -1

@[simp]

theorem neg_div_self {K : Type u_3} [DivisionRing K] {a : K} (h : a ≠ 0) : -a / a = -1

theorem div_sub_div_same {K : Type u_3} [DivisionRing K] (a : K) (b : K) (c : K) : a / c - b / c = (a - b) / c

theorem same_sub_div {K : Type u_3} [DivisionRing K] {a : K} {b : K} (h : b ≠ 0) : (b - a) / b = 1 - a / b

theorem one_sub_div {K : Type u_3} [DivisionRing K] {a : K} {b : K} (h : b ≠ 0) : 1 - a / b = (b - a) / b

theorem div_sub_same {K : Type u_3} [DivisionRing K] {a : K} {b : K} (h : b ≠ 0) : (a - b) / b = a / b - 1

theorem div_sub_one {K : Type u_3} [DivisionRing K] {a : K} {b : K} (h : b ≠ 0) : a / b - 1 = (a - b) / b

theorem sub_div {K : Type u_3} [DivisionRing K] (a : K) (b : K) (c : K) : (a - b) / c = a / c - b / c

theorem inv_sub_inv' {K : Type u_3} [DivisionRing K] {a : K} {b : K} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ - b⁻¹ = a⁻¹ * (b - a) * b⁻¹

See inv_sub_inv for the more convenient version when K is commutative.

theorem one_div_mul_sub_mul_one_div_eq_one_div_add_one_div {K : Type u_3} [DivisionRing K] {a : K} {b : K} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a * (b - a) * (1 / b) = 1 / a - 1 / b

@[instance 100]

instance DivisionRing.isDomain {K : Type u_3} [DivisionRing K] : IsDomain K

Equations

• ⋯ = ⋯

theorem Commute.div_sub_div {K : Type u_3} [DivisionRing K] {a : K} {b : K} {c : K} {d : K} (hbc : Commute b c) (hbd : Commute b d) (hb : b ≠ 0) (hd : d ≠ 0) : a / b - c / d = (a * d - b * c) / (b * d)

theorem Commute.inv_sub_inv {K : Type u_3} [DivisionRing K] {a : K} {b : K} (hab : Commute a b) (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ - b⁻¹ = (b - a) / (a * b)

theorem div_add_div {α : Type u_1} [Semifield α] {b : α} {d : α} (a : α) (c : α) (hb : b ≠ 0) (hd : d ≠ 0) : a / b + c / d = (a * d + b * c) / (b * d)

theorem one_div_add_one_div {α : Type u_1} [Semifield α] {a : α} {b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b)

theorem inv_add_inv {α : Type u_1} [Semifield α] {a : α} {b : α} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ + b⁻¹ = (a + b) / (a * b)

theorem div_sub_div {K : Type u_3} [Field K] (a : K) {b : K} (c : K) {d : K} (hb : b ≠ 0) (hd : d ≠ 0) : a / b - c / d = (a * d - b * c) / (b * d)

theorem inv_sub_inv {K : Type u_3} [Field K] {a : K} {b : K} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ - b⁻¹ = (b - a) / (a * b)

theorem sub_div' {K : Type u_3} [Field K] (a : K) (b : K) (c : K) (hc : c ≠ 0) : b - a / c = (b * c - a) / c

theorem div_sub' {K : Type u_3} [Field K] (a : K) (b : K) (c : K) (hc : c ≠ 0) : a / c - b = (a - c * b) / c

@[instance 100]

instance Field.isDomain {K : Type u_3} [Field K] : IsDomain K

Equations

• ⋯ = ⋯

theorem RingHom.injective {α : Type u_1} {β : Type u_2} [DivisionRing α] [Semiring β] [Nontrivial β] (f : α →+* β) : Function.Injective ⇑f

@[reducible, inline]

noncomputable abbrev DivisionRing.ofIsUnitOrEqZero {R : Type u_4} [Nontrivial R] [Ring R] (h : ∀ (a : R), IsUnit a ∨ a = 0) : DivisionRing R

Constructs a DivisionRing structure on a Ring consisting only of units and 0.

Equations

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

@[reducible, inline]

noncomputable abbrev Field.ofIsUnitOrEqZero {R : Type u_4} [Nontrivial R] [CommRing R] (h : ∀ (a : R), IsUnit a ∨ a = 0) : Field R

Constructs a Field structure on a CommRing consisting only of units and 0.

Equations

• Field.ofIsUnitOrEqZero h = let __spread.0 := DivisionRing.ofIsUnitOrEqZero h; Field.mk ⋯ DivisionRing.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionRing.nnqsmul ⋯ ⋯ DivisionRing.qsmul ⋯

@[reducible, inline]

abbrev Function.Injective.divisionSemiring {α : Type u_1} {β : Type u_2} [Zero α] [Add α] [One α] [Mul α] [Inv α] [Div α] [SMul ℕ α] [SMul ℚ≥0 α] [Pow α ℕ] [Pow α ℤ] [NatCast α] [NNRatCast α] (f : α → β) (hf : Function.Injective f) [DivisionSemiring β] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (inv : ∀ (x : α), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : α), f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x : α), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x : α), f (q • x) = q • f x) (npow : ∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : α) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (nnratCast : ∀ (q : ℚ≥0), f ↑q = ↑q) : DivisionSemiring α

Pullback a DivisionSemiring along an injective function.

Equations

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

@[reducible, inline]

abbrev Function.Injective.divisionRing {α : Type u_1} {β : Type u_2} [Zero α] [Add α] [Neg α] [Sub α] [One α] [Mul α] [Inv α] [Div α] [SMul ℕ α] [SMul ℤ α] [SMul ℚ≥0 α] [SMul ℚ α] [Pow α ℕ] [Pow α ℤ] [NatCast α] [IntCast α] [NNRatCast α] [RatCast α] (f : α → β) (hf : Function.Injective f) [DivisionRing β] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (inv : ∀ (x : α), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : α), f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x : α), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : α), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x : α), f (q • x) = q • f x) (qsmul : ∀ (q : ℚ) (x : α), f (q • x) = q • f x) (npow : ∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : α) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) (nnratCast : ∀ (q : ℚ≥0), f ↑q = ↑q) (ratCast : ∀ (q : ℚ), f ↑q = ↑q) : DivisionRing α

Pullback a DivisionSemiring along an injective function.

Equations

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

@[reducible, inline]

abbrev Function.Injective.semifield {α : Type u_1} {β : Type u_2} [Zero α] [Add α] [One α] [Mul α] [Inv α] [Div α] [SMul ℕ α] [SMul ℚ≥0 α] [Pow α ℕ] [Pow α ℤ] [NatCast α] [NNRatCast α] (f : α → β) (hf : Function.Injective f) [Semifield β] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (inv : ∀ (x : α), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : α), f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x : α), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x : α), f (q • x) = q • f x) (npow : ∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : α) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (nnratCast : ∀ (q : ℚ≥0), f ↑q = ↑q) : Semifield α

Pullback a Field along an injective function.

Equations

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

@[reducible, inline]

abbrev Function.Injective.field {α : Type u_1} {β : Type u_2} [Zero α] [Add α] [Neg α] [Sub α] [One α] [Mul α] [Inv α] [Div α] [SMul ℕ α] [SMul ℤ α] [SMul ℚ≥0 α] [SMul ℚ α] [Pow α ℕ] [Pow α ℤ] [NatCast α] [IntCast α] [NNRatCast α] [RatCast α] (f : α → β) (hf : Function.Injective f) [Field β] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : α), f (x + y) = f x + f y) (mul : ∀ (x y : α), f (x * y) = f x * f y) (neg : ∀ (x : α), f (-x) = -f x) (sub : ∀ (x y : α), f (x - y) = f x - f y) (inv : ∀ (x : α), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : α), f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x : α), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x : α), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x : α), f (q • x) = q • f x) (qsmul : ∀ (q : ℚ) (x : α), f (q • x) = q • f x) (npow : ∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : α) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (intCast : ∀ (n : ℤ), f ↑n = ↑n) (nnratCast : ∀ (q : ℚ≥0), f ↑q = ↑q) (ratCast : ∀ (q : ℚ), f ↑q = ↑q) : Field α

Pullback a Field along an injective function.

Equations

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

Order dual

instance OrderDual.instRatCast {α : Type u_1} [RatCast α] : RatCast αᵒᵈ

Equations

• OrderDual.instRatCast = inst

instance OrderDual.instDivisionSemiring {α : Type u_1} [DivisionSemiring α] : DivisionSemiring αᵒᵈ

Equations

• OrderDual.instDivisionSemiring = inst

instance OrderDual.instDivisionRing {α : Type u_1} [DivisionRing α] : DivisionRing αᵒᵈ

Equations

• OrderDual.instDivisionRing = inst

instance OrderDual.instSemifield {α : Type u_1} [Semifield α] : Semifield αᵒᵈ

Equations

• OrderDual.instSemifield = inst

instance OrderDual.instField {α : Type u_1} [Field α] : Field αᵒᵈ

Equations

• OrderDual.instField = inst

@[simp]

theorem toDual_ratCast {α : Type u_1} [RatCast α] (n : ℚ) : OrderDual.toDual ↑n = ↑n

@[simp]

theorem ofDual_ratCast {α : Type u_1} [RatCast α] (n : ℚ) : OrderDual.ofDual ↑n = ↑n

Lexicographic order

instance Lex.instRatCast {α : Type u_1} [RatCast α] : RatCast (Lex α)

Equations

• Lex.instRatCast = inst

instance Lex.instDivisionSemiring {α : Type u_1} [DivisionSemiring α] : DivisionSemiring (Lex α)

Equations

• Lex.instDivisionSemiring = inst

instance Lex.instDivisionRing {α : Type u_1} [DivisionRing α] : DivisionRing (Lex α)

Equations

• Lex.instDivisionRing = inst

instance Lex.instSemifield {α : Type u_1} [Semifield α] : Semifield (Lex α)

Equations

• Lex.instSemifield = inst

instance Lex.instField {α : Type u_1} [Field α] : Field (Lex α)

Equations

• Lex.instField = inst

@[simp]

theorem toLex_ratCast {α : Type u_1} [RatCast α] (n : ℚ) : toLex ↑n = ↑n

@[simp]

theorem ofLex_ratCast {α : Type u_1} [RatCast α] (n : ℚ) : ofLex ↑n = ↑n