Loogle!

Result

Found 482 declarations mentioning Semifield. Of these, only the first 200 are shown.

• Semifield 📋 Mathlib.Algebra.Field.Defs
  (K : Type u_2) : Type u_2
• Field.toSemifield 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_1} [Field K] : Semifield K
• Semifield.toCommGroupWithZero 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : CommGroupWithZero K
• Semifield.toCommSemiring 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : CommSemiring K
• Semifield.toDiv 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : Div K
• Semifield.toDivisionSemiring 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : DivisionSemiring K
• Semifield.toInv 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : Inv K
• Semifield.toNNRatCast 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : NNRatCast K
• Semifield.toNontrivial 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : Nontrivial K
• Semifield.toZPow 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : ZPow K
• Semifield.nnqsmul 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : ℚ≥0 → K → K
• Semifield.nnqsmul_def 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (q : ℚ≥0) (a : K) : Semifield.nnqsmul q a = ↑q * a
• Semifield.zpow_zero' 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (a : K) : a ^ 0 = 1
• Semifield.div_eq_mul_inv 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (a b : K) : a / b = a * b⁻¹
• Semifield.inv_zero 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] : 0⁻¹ = 0
• Semifield.nnratCast_def 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (q : ℚ≥0) : ↑q = ↑q.num / ↑q.den
• Semifield.zpow_neg' 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (n : ℕ) (a : K) : a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹
• Semifield.zpow_succ' 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (n : ℕ) (a : K) : a ^ ↑n.succ = a ^ ↑n * a
• Semifield.mul_inv_cancel 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [self : Semifield K] (a : K) : a ≠ 0 → a * a⁻¹ = 1
• Semifield.mk 📋 Mathlib.Algebra.Field.Defs
  {K : Type u_2} [toCommSemiring : CommSemiring K] [toInv : Inv K] [toDiv : Div K] [toZPow : ZPow K] (div_eq_mul_inv : ∀ (a b : K), a / b = a * b⁻¹ := by intros; rfl) (zpow_zero' : ∀ (a : K), a ^ 0 = 1 := by intros; rfl) (zpow_succ' : ∀ (n : ℕ) (a : K), a ^ ↑n.succ = a ^ ↑n * a := by intros; rfl) (zpow_neg' : ∀ (n : ℕ) (a : K), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹ := by intros; rfl) [toNontrivial : Nontrivial K] (inv_zero : 0⁻¹ = 0) (mul_inv_cancel : ∀ (a : K), a ≠ 0 → a * a⁻¹ = 1) [toNNRatCast : NNRatCast K] (nnratCast_def : ∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den := by intros; rfl) (nnqsmul : ℚ≥0 → K → K) (nnqsmul_def : ∀ (q : ℚ≥0) (a : K), nnqsmul q a = ↑q * a := by intros; rfl) : Semifield K
• Lex.instSemifield 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} [Semifield K] : Semifield (Lex K)
• OrderDual.instSemifield 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} [Semifield K] : Semifield Kᵒᵈ
• inv_add_inv 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} [Semifield K] {a b : K} (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ + b⁻¹ = (a + b) / (a * b)
• div_add_div 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} [Semifield K] {b d : K} (a c : K) (hb : b ≠ 0) (hd : d ≠ 0) : a / b + c / d = (a * d + b * c) / (b * d)
• one_div_add_one_div 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} [Semifield K] {a b : K} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b)
• Function.Injective.semifield 📋 Mathlib.Algebra.Field.Basic
  {K : Type u_1} {L : Type u_2} [Zero K] [Add K] [One K] [Mul K] [Inv K] [Div K] [SMul ℕ K] [SMul ℚ≥0 K] [Pow K ℕ] [Pow K ℤ] [NatCast K] [NNRatCast K] (f : K → L) (hf : Function.Injective f) [Semifield L] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : K), f (x + y) = f x + f y) (mul : ∀ (x y : K), f (x * y) = f x * f y) (inv : ∀ (x : K), f x⁻¹ = (f x)⁻¹) (div : ∀ (x y : K), f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x : K), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x : K), f (q • x) = q • f x) (npow : ∀ (x : K) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x : K) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ (n : ℕ), f ↑n = ↑n) (nnratCast : ∀ (q : ℚ≥0), f ↑q = ↑q) : Semifield K
• LinearOrderedSemiField.toDenselyOrdered 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] : DenselyOrdered α
• Mathlib.Meta.Positivity.zpow_zero_pos 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_4} [Semifield α] [PartialOrder α] [IsStrictOrderedRing α] (a : α) : 0 < a ^ 0
• inv_strictAntiOn 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] : StrictAntiOn (fun x => x⁻¹) (Set.Ioi 0)
• monotone_div_right_of_nonneg 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} (ha : 0 ≤ a) : Monotone fun x => x / a
• strictMono_div_right_of_pos 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} (ha : 0 < a) : StrictMono fun x => x / a
• Monotone.div_const 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {β : Type u_3} [Preorder β] {f : β → α} (hf : Monotone f) {c : α} (hc : 0 ≤ c) : Monotone fun x => f x / c
• StrictMono.div_const 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] {β : Type u_3} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α} (hc : 0 < c) : StrictMono fun x => f x / c
• one_div_strictAntiOn 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] : StrictAntiOn (fun x => 1 / x) (Set.Ioi 0)
• div_nat_le_self_of_nonnneg 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (ha : 0 ≤ a) (n : ℕ) : a / ↑n ≤ a
• Bound.div_lt_one_of_pos_of_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (b0 : 0 < b) : a < b → a / b < 1
• Bound.one_lt_div_of_pos_of_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (b0 : 0 < b) : b < a → 1 < a / b
• div_le_one 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b
• div_lt_one 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (hb : 0 < b) : a / b < 1 ↔ a < b
• one_le_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a
• one_lt_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (hb : 0 < b) : 1 < a / b ↔ b < a
• inv_pow_anti 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 ≤ a) : Antitone fun n => (a ^ n)⁻¹
• inv_pow_strictAnti 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 < a) : StrictAnti fun n => (a ^ n)⁻¹
• two_inv_lt_one 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] : 2⁻¹ < 1
• div_nat_lt_self_of_pos_of_two_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (ha : 0 < a) {n : ℕ} (hn : 2 ≤ n) : a / ↑n < a
• div_le_self 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a
• div_lt_self 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (ha : 0 < a) (hb : 1 < b) : a / b < a
• div_two_lt_of_pos 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] : 0 < a → a / 2 < a
• half_le_self 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] : 0 ≤ a → a / 2 ≤ a
• half_lt_self 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] : 0 < a → a / 2 < a
• half_le_self_iff 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] : a / 2 ≤ a ↔ 0 ≤ a
• half_lt_self_iff 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] : a / 2 < a ↔ 0 < a
• IsGLB.mul_left 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) : IsGLB ((fun b => a * b) '' s) (a * b)
• IsGLB.mul_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) : IsGLB ((fun b => b * a) '' s) (b * a)
• IsLUB.mul_left 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) : IsLUB ((fun b => a * b) '' s) (a * b)
• IsLUB.mul_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) : IsLUB ((fun b => b * a) '' s) (b * a)
• add_div_two_lt_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] : (a + b) / 2 < b ↔ a < b
• left_lt_add_div_two 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] : a < (a + b) / 2 ↔ a < b
• exists_pos_lt_mul 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_3} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} (h : 0 < a) (b : α) : ∃ c, 0 < c ∧ b < c * a
• exists_pos_mul_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_3} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} (h : 0 < a) (b : α) : ∃ c, 0 < c ∧ b * c < a
• div_pos_iff_of_pos_left 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (ha : 0 < a) : 0 < a / b ↔ 0 < b
• div_pos_iff_of_pos_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (hb : 0 < b) : 0 < a / b ↔ 0 < a
• one_half_pos 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] : 0 < 1 / 2
• one_div_pow_anti 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 ≤ a) : Antitone fun n => 1 / a ^ n
• one_div_pow_strictAnti 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 < a) : StrictAnti fun n => 1 / a ^ n
• le_div_self 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) : a ≤ a / b
• one_half_lt_one 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] [IsStrictOrderedRing α] : 1 / 2 < 1
• half_pos 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (h : 0 < a) : 0 < a / 2
• max_div_div_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_3} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] {c : α} (hc : 0 ≤ c) (a b : α) : max (a / c) (b / c) = max a b / c
• min_div_div_right 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_3} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] {c : α} (hc : 0 ≤ c) (a b : α) : min (a / c) (b / c) = min a b / c
• le_of_one_div_le_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a
• lt_of_one_div_lt_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (h : 1 / a < 1 / b) : b < a
• mul_le_mul_of_mul_div_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b c d : α} (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c
• one_div_le_one_div_of_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a
• one_div_lt_one_div_of_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (h : a < b) : 1 / b < 1 / a
• le_iff_forall_one_lt_le_mul₀ 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_3} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} (hb : 0 ≤ b) : a ≤ b ↔ ∀ (ε : α), 1 < ε → a ≤ b * ε
• one_le_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a
• one_lt_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a
• inv_pow_le_inv_pow_of_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹
• inv_pow_lt_inv_pow_of_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 < a) {m n : ℕ} (mn : m < n) : (a ^ n)⁻¹ < (a ^ m)⁻¹
• le_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a
• lt_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a
• one_div_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a
• one_div_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a
• one_div_le_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ 1 / b ↔ b ≤ a
• one_div_lt_one_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b : α} [IsStrictOrderedRing α] (ha : 0 < a) (hb : 0 < b) : 1 / a < 1 / b ↔ b < a
• div_mul_le_div_mul_of_div_le_div 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a b c d e : α} [IsStrictOrderedRing α] (h : a / b ≤ c / d) (he : 0 ≤ e) : a / (b * e) ≤ c / (d * e)
• one_div_pow_le_one_div_pow_of_le 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : 1 / a ^ n ≤ 1 / a ^ m
• one_div_pow_lt_one_div_pow_of_lt 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [PosMulReflectLT α] {a : α} [IsStrictOrderedRing α] (a1 : 1 < a) {m n : ℕ} (mn : m < n) : 1 / a ^ n < 1 / a ^ m
• add_thirds 📋 Mathlib.Algebra.Order.Field.Basic
  {α : Type u_1} [Semifield α] [PartialOrder α] [IsStrictOrderedRing α] (a : α) : a / 3 + a / 3 + a / 3 = a
• NNRat.instSemifield 📋 Mathlib.Algebra.Field.Rat
  : Semifield ℚ≥0
• Mathlib.Tactic.Ring.Common.inv_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] {a a' b : R} : a = a' → a'⁻¹ = b → a⁻¹ = b
• Mathlib.Tactic.Ring.Common.inv_zero 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] : 0⁻¹ = 0
• Mathlib.Tactic.Ring.Common.div_congr 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] {a a' b b' c : R} : a = a' → b = b' → a' / b' = c → a / b = c
• Mathlib.Tactic.Ring.Common.div_pf 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] {a b c d : R} : b⁻¹ = c → a * c = d → a / b = d
• Mathlib.Tactic.Ring.Common.inv_single 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] {a b : R} : a⁻¹ = b → (a + 0)⁻¹ = b + 0
• Mathlib.Tactic.Ring.Common.inv_mul 📋 Mathlib.Tactic.Ring.Common
  {R : Type u_2} [Semifield R] {a₁ : R} {a₂ : ℕ} {a₃ b₁ b₃ c : R} : a₁⁻¹ = b₁ → a₃⁻¹ = b₃ → b₃ * (b₁ ^ a₂ * Nat.rawCast 1) = c → (a₁ ^ a₂ * a₃)⁻¹ = c
• CanonicallyOrderedAdd.toLinearOrderedCommGroupWithZero 📋 Mathlib.Algebra.Order.Field.Canonical
  {α : Type u_1} [Semifield α] [LinearOrder α] [CanonicallyOrderedAdd α] : LinearOrderedCommGroupWithZero α
• tsub_div 📋 Mathlib.Algebra.Order.Field.Canonical
  {α : Type u_1} [Semifield α] [LinearOrder α] [CanonicallyOrderedAdd α] [IsStrictOrderedRing α] [Sub α] [OrderedSub α] (a b c : α) : (a - b) / c = a / c - b / c
• Nonneg.semifield 📋 Mathlib.Algebra.Order.Nonneg.Field
  {α : Type u_1} [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] : Semifield { x // 0 ≤ x }
• NNRat.castOrderEmbedding 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] : ℚ≥0 ↪o K
• NNRat.cast_mono 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] : Monotone NNRat.cast
• NNRat.cast_strictMono 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] : StrictMono NNRat.cast
• NNRat.not_cast_lt_zero 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {q : ℚ≥0} : ¬↑q < 0
• NNRat.preimage_cast_uIoc 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.uIoc ↑p ↑q = Set.uIoc p q
• NNRat.preimage_cast_uIcc 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.uIcc ↑p ↑q = Set.uIcc p q
• NNRat.preimage_cast_Ici 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p : ℚ≥0) : NNRat.cast ⁻¹' Set.Ici ↑p = Set.Ici p
• NNRat.preimage_cast_Iic 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p : ℚ≥0) : NNRat.cast ⁻¹' Set.Iic ↑p = Set.Iic p
• NNRat.preimage_cast_Iio 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p : ℚ≥0) : NNRat.cast ⁻¹' Set.Iio ↑p = Set.Iio p
• NNRat.preimage_cast_Ioi 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p : ℚ≥0) : NNRat.cast ⁻¹' Set.Ioi ↑p = Set.Ioi p
• NNRat.cast_le 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p q : ℚ≥0} : ↑p ≤ ↑q ↔ p ≤ q
• NNRat.cast_lt 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p q : ℚ≥0} : ↑p < ↑q ↔ p < q
• NNRat.cast_max 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : ↑(max p q) = max ↑p ↑q
• NNRat.cast_min 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : ↑(min p q) = min ↑p ↑q
• NNRat.preimage_cast_Icc 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.Icc ↑p ↑q = Set.Icc p q
• NNRat.preimage_cast_Ico 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.Ico ↑p ↑q = Set.Ico p q
• NNRat.preimage_cast_Ioc 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.Ioc ↑p ↑q = Set.Ioc p q
• NNRat.preimage_cast_Ioo 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ≥0) : NNRat.cast ⁻¹' Set.Ioo ↑p ↑q = Set.Ioo p q
• NNRat.cast_lt_zero 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {q : ℚ≥0} : ↑q < 0 ↔ q < 0
• NNRat.cast_nonpos 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {q : ℚ≥0} : ↑q ≤ 0 ↔ q ≤ 0
• NNRat.cast_pos 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {q : ℚ≥0} : 0 < ↑q ↔ 0 < q
• NNRat.cast_le_natCast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {m : ℚ≥0} {n : ℕ} : ↑m ≤ ↑n ↔ m ≤ ↑n
• NNRat.cast_lt_natCast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {m : ℚ≥0} {n : ℕ} : ↑m < ↑n ↔ m < ↑n
• NNRat.natCast_le_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {m : ℕ} {n : ℚ≥0} : ↑m ≤ ↑n ↔ ↑m ≤ n
• NNRat.natCast_lt_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {m : ℕ} {n : ℚ≥0} : ↑m < ↑n ↔ ↑m < n
• NNRat.cast_le_one 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} : ↑p ≤ 1 ↔ p ≤ 1
• NNRat.cast_lt_one 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} : ↑p < 1 ↔ p < 1
• NNRat.one_le_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} : 1 ≤ ↑p ↔ 1 ≤ p
• NNRat.one_lt_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} : 1 < ↑p ↔ 1 < p
• NNRat.cast_le_ofNat 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} {n : ℕ} [n.AtLeastTwo] : ↑p ≤ OfNat.ofNat n ↔ p ≤ OfNat.ofNat n
• NNRat.cast_lt_ofNat 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} {n : ℕ} [n.AtLeastTwo] : ↑p < OfNat.ofNat n ↔ p < OfNat.ofNat n
• NNRat.ofNat_le_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} {n : ℕ} [n.AtLeastTwo] : OfNat.ofNat n ≤ ↑p ↔ OfNat.ofNat n ≤ p
• NNRat.ofNat_lt_cast 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] {p : ℚ≥0} {n : ℕ} [n.AtLeastTwo] : OfNat.ofNat n < ↑p ↔ OfNat.ofNat n < p
• NNRat.castOrderEmbedding_apply 📋 Mathlib.Data.Rat.Cast.Order
  {K : Type u_2} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] (a✝ : ℚ≥0) : NNRat.castOrderEmbedding a✝ = ↑a✝
• exists_pow_lt_of_lt_one 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (hx : 0 < x) (hy : y < 1) : ∃ n, y ^ n < x
• exists_nat_one_div_lt 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {ε : K} (hε : 0 < ε) : ∃ n, 1 / (↑n + 1) < ε
• exists_mem_Ico_zpow 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (hx : 0 < x) (hy : 1 < y) : ∃ n, x ∈ Set.Ico (y ^ n) (y ^ (n + 1))
• exists_mem_Ioc_zpow 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (hx : 0 < x) (hy : 1 < y) : ∃ n, x ∈ Set.Ioc (y ^ n) (y ^ (n + 1))
• exists_zpow_btwn_of_lt_mul 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] [ExistsAddOfLE K] {a b c : K} (h : a < b * c) (hb₀ : 0 < b) (hc₀ : 0 < c) (hc₁ : c < 1) : ∃ n, a < c ^ n ∧ c ^ n < b
• exists_nat_pow_near_of_lt_one 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) : ∃ n, y ^ (n + 1) < x ∧ x ≤ y ^ n
• exists_pow_btwn_of_lt_mul 📋 Mathlib.Algebra.Order.Archimedean.Basic
  {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] [ExistsAddOfLE K] {a b c : K} (h : a < b * c) (hb₀ : 0 < b) (hb₁ : b ≤ 1) (hc₀ : 0 < c) (hc₁ : c < 1) : ∃ n, a < c ^ n ∧ c ^ n < b
• Submodule.comap_smul 📋 Mathlib.Algebra.Module.Submodule.Map
  {K : Type u_7} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (p : Submodule K V₂) (a : K) (h : a ≠ 0) : Submodule.comap (a • f) p = Submodule.comap f p
• Submodule.map_smul 📋 Mathlib.Algebra.Module.Submodule.Map
  {K : Type u_7} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (p : Submodule K V) (a : K) (h : a ≠ 0) : Submodule.map (a • f) p = Submodule.map f p
• Submodule.comap_smul' 📋 Mathlib.Algebra.Module.Submodule.Map
  {K : Type u_7} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (p : Submodule K V₂) (a : K) : Submodule.comap (a • f) p = ⨅ (_ : a ≠ 0), Submodule.comap f p
• Submodule.map_smul' 📋 Mathlib.Algebra.Module.Submodule.Map
  {K : Type u_7} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (p : Submodule K V) (a : K) : Submodule.map (a • f) p = ⨆ (_ : a ≠ 0), Submodule.map f p
• LinearMap.ker_smul 📋 Mathlib.Algebra.Module.Submodule.Ker
  {K : Type u_4} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0) : (a • f).ker = f.ker
• LinearMap.ker_smul' 📋 Mathlib.Algebra.Module.Submodule.Ker
  {K : Type u_4} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) : (a • f).ker = ⨅ (_ : a ≠ 0), f.ker
• Module.ker_algebraMap_end 📋 Mathlib.Algebra.Algebra.Basic
  (K : Type u) (V : Type v) [Semifield K] [AddCommMonoid V] [Module K V] (a : K) (ha : a ≠ 0) : LinearMap.ker ((algebraMap K (Module.End K V)) a) = ⊥
• LinearMap.range_smul 📋 Mathlib.Algebra.Module.Submodule.Range
  {K : Type u_4} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0) : (a • f).range = f.range
• LinearMap.range_smul' 📋 Mathlib.Algebra.Module.Submodule.Range
  {K : Type u_4} {V : Type u_8} {V₂ : Type u_9} [Semifield K] [AddCommMonoid V] [Module K V] [AddCommMonoid V₂] [Module K V₂] (f : V →ₗ[K] V₂) (a : K) : (a • f).range = ⨆ (_ : a ≠ 0), f.range
• IsAbsoluteValue.abv_inv 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
  {S : Type u_4} [Semifield S] [LinearOrder S] {R : Type u_5} [DivisionSemiring R] (abv : R → S) [IsAbsoluteValue abv] (a : R) : abv a⁻¹ = (abv a)⁻¹
• IsAbsoluteValue.abv_div 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
  {S : Type u_4} [Semifield S] [LinearOrder S] {R : Type u_5} [DivisionSemiring R] (abv : R → S) [IsAbsoluteValue abv] (a b : R) : abv (a / b) = abv a / abv b
• AbsoluteValue.IsNontrivial.exists_abv_gt_one 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
  {R : Type u_2} {S : Type u_3} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) : ∃ x, 1 < v x
• AbsoluteValue.IsNontrivial.exists_abv_lt_one 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
  {R : Type u_2} {S : Type u_3} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) : ∃ x, x ≠ 0 ∧ v x < 1
• Finset.sq_sum_div_le_sum_sq_div 📋 Mathlib.Algebra.Order.BigOperators.Ring.Finset
  {ι : Type u_1} {R : Type u_2} [Semifield R] [LinearOrder R] [IsStrictOrderedRing R] [ExistsAddOfLE R] (s : Finset ι) (f : ι → R) {g : ι → R} (hg : ∀ i ∈ s, 0 < g i) : (∑ i ∈ s, f i) ^ 2 / ∑ i ∈ s, g i ≤ ∑ i ∈ s, f i ^ 2 / g i
• IsField.toSemifield 📋 Mathlib.Algebra.Field.IsField
  {R : Type u} [Semiring R] (h : IsField R) : Semifield R
• instIsDomain 📋 Mathlib.Algebra.Field.IsField
  {R : Type u} [Semifield R] : IsDomain R
• Semifield.toIsField 📋 Mathlib.Algebra.Field.IsField
  (R : Type u) [Semifield R] : IsField R
• Semifield.isCoprime_iff 📋 Mathlib.RingTheory.Coprime.Basic
  {R : Type u_1} [Semifield R] {m n : R} : IsCoprime m n ↔ m ≠ 0 ∨ n ≠ 0
• AddOpposite.instSemifield 📋 Mathlib.Algebra.Field.Opposite
  {α : Type u_1} [Semifield α] : Semifield αᵃᵒᵖ
• MulOpposite.instSemifield 📋 Mathlib.Algebra.Field.Opposite
  {α : Type u_1} [Semifield α] : Semifield αᵐᵒᵖ
• IsSelfAdjoint.div 📋 Mathlib.Algebra.Star.SelfAdjoint
  {R : Type u_1} [Semifield R] [StarRing R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x / y)
• algebraMap.coe_inv 📋 Mathlib.Algebra.Algebra.Field
  {R : Type u_1} (A : Type u_2) [Semifield R] [DivisionSemiring A] [Algebra R A] (r : R) : ↑r⁻¹ = (↑r)⁻¹
• algebraMap.coe_zpow 📋 Mathlib.Algebra.Algebra.Field
  {R : Type u_1} (A : Type u_2) [Semifield R] [DivisionSemiring A] [Algebra R A] (r : R) (z : ℤ) : ↑(r ^ z) = ↑r ^ z
• algebraMap.coe_div 📋 Mathlib.Algebra.Algebra.Field
  {R : Type u_1} (A : Type u_2) [Semifield R] [DivisionSemiring A] [Algebra R A] (r s : R) : ↑(r / s) = ↑r / ↑s
• spectrum.inv₀_mem 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r ∈ spectrum R ↑a⁻¹ → r⁻¹ ∈ spectrum R ↑a
• spectrum.inv₀_mem_inv 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r ∈ spectrum R ↑a → r⁻¹ ∈ spectrum R ↑a⁻¹
• spectrum.of_inv₀_mem 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a → r ∈ spectrum R ↑a⁻¹
• spectrum.of_inv₀_mem_inv 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a⁻¹ → r ∈ spectrum R ↑a
• spectrum.inv₀_mem_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a ↔ r ∈ spectrum R ↑a⁻¹
• spectrum.inv₀_mem_inv_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Basic
  {R : Type u} {A : Type v} [Semifield R] [Ring A] [Algebra R A] {r : R} {a : Aˣ} : r⁻¹ ∈ spectrum R ↑a⁻¹ ↔ r ∈ spectrum R ↑a
• SpectrumRestricts 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R A] [Algebra S A] [Algebra R S] (a : A) (f : S → R) : Prop
• quasispectrum.of_subsingleton 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {A : Type u_4} [Semifield R] [NonUnitalRing A] [Module R A] [Subsingleton A] (a : A) : quasispectrum R a = {0}
• quasispectrum.zero_eq 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {A : Type u_4} [Semifield R] [NonUnitalRing A] [Module R A] : quasispectrum R 0 = {0}
• quasispectrum_eq_spectrum_union_zero 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  (R : Type u_3) {A : Type u_4} [Semifield R] [Ring A] [Algebra R A] (a : A) : quasispectrum R a = spectrum R a ∪ {0}
• mem_quasispectrum_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {A : Type u_4} [Semifield R] [Ring A] [Algebra R A] {a : A} {x : R} : x ∈ quasispectrum R a ↔ x = 0 ∨ x ∈ spectrum R a
• SpectrumRestricts.of_spectrum_eq 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a b : A} {f : S → R} (ha : SpectrumRestricts a f) (h : spectrum S a = spectrum S b) : SpectrumRestricts b f
• quasispectrumRestricts_iff_spectrumRestricts 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} : QuasispectrumRestricts a f ↔ SpectrumRestricts a f
• SpectrumRestricts.mul_comm 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_6} {S : Type u_7} {A : Type u_8} [Semifield R] [Field S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a b : A} {f : S → R} : SpectrumRestricts (a * b) f → SpectrumRestricts (b * a) f
• SpectrumRestricts.mul_comm_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_6} {S : Type u_7} {A : Type u_8} [Semifield R] [Field S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a b : A} {f : S → R} : SpectrumRestricts (a * b) f ↔ SpectrumRestricts (b * a) f
• QuasispectrumRestricts.map_zero 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (h : QuasispectrumRestricts a f) : f 0 = 0
• SpectrumRestricts.image 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f) : f '' spectrum S a = spectrum R a
• NonnegSpectrumClass.of_spectrum_nonneg 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {𝕜 : Type u_3} {A : Type u_4} [Semifield 𝕜] [LinearOrder 𝕜] [Ring A] [PartialOrder A] [Algebra 𝕜 A] : (∀ (a : A), 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x) → NonnegSpectrumClass 𝕜 A
• SpectrumRestricts.rightInvOn 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (h : SpectrumRestricts a f) : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)
• SpectrumRestricts.subset_preimage 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f) : spectrum S a ⊆ f ⁻¹' spectrum R a
• NonnegSpectrumClass.iff_spectrum_nonneg 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {𝕜 : Type u_3} {A : Type u_4} [Semifield 𝕜] [LinearOrder 𝕜] [Ring A] [PartialOrder A] [Algebra 𝕜 A] : NonnegSpectrumClass 𝕜 A ↔ ∀ (a : A), 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x
• QuasispectrumRestricts.of_quasispectrum_eq 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a b : A} {f : S → R} (ha : QuasispectrumRestricts a f) (h : quasispectrum S a = quasispectrum S b) : QuasispectrumRestricts b f
• SpectrumRestricts.apply_mem 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f) {s : S} (hs : s ∈ spectrum S a) : f s ∈ spectrum R a
• SpectrumRestricts.algebraMap_image 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f) : ⇑(algebraMap R S) '' spectrum R a = spectrum S a
• SpectrumRestricts.of_rightInvOn 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (h₁ : Function.LeftInverse f ⇑(algebraMap R S)) (h₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)) : SpectrumRestricts a f
• spectrumRestricts_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} : SpectrumRestricts a f ↔ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a) ∧ Function.LeftInverse f ⇑(algebraMap R S)
• SpectrumRestricts.of_subset_range_algebraMap 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (hf : Function.LeftInverse f ⇑(algebraMap R S)) (h : spectrum S a ⊆ Set.range ⇑(algebraMap R S)) : SpectrumRestricts a f
• QuasispectrumRestricts.of_subset_range_algebraMap 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (hf : Function.LeftInverse f ⇑(algebraMap R S)) (h : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)) : QuasispectrumRestricts a f
• QuasispectrumRestricts.mul_comm 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] [IsScalarTower S A A] [SMulCommClass S A A] {f : S → R} {a b : A} : QuasispectrumRestricts (a * b) f → QuasispectrumRestricts (b * a) f
• QuasispectrumRestricts.mul_comm_iff 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] [IsScalarTower S A A] [SMulCommClass S A A] {f : S → R} {a b : A} : QuasispectrumRestricts (a * b) f ↔ QuasispectrumRestricts (b * a) f
• QuasispectrumRestricts.image 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] (h : QuasispectrumRestricts a f) : f '' quasispectrum S a = quasispectrum R a
• QuasispectrumRestricts.subset_preimage 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] (h : QuasispectrumRestricts a f) : quasispectrum S a ⊆ f ⁻¹' quasispectrum R a
• QuasispectrumRestricts.apply_mem 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] (h : QuasispectrumRestricts a f) {s : S} (hs : s ∈ quasispectrum S a) : f s ∈ quasispectrum R a
• Unitization.quasispectrum_eq_spectrum_inr' 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  (R : Type u_3) (S : Type u_4) {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A) : quasispectrum R a = spectrum R ↑a
• quasispectrum.preimage_algebraMap 📋 Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
  (S : Type u_3) {R : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] {a : A} : ⇑(algebraMap R S) ⁻¹' quasispectrum S a = quasispectrum R a

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c