state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h : ¬Finite a b
n : ℕ
⊢ a ^ n ∣ b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact not_finite_iff_forall.mp h n | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
... | Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Semiring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≤ multiplicity p b
⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [min_eq_left h, multiplicity_le_multiplicity_iff] | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
| Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8 | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Semiring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≤ multiplicity p b
⊢ ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn) | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
| Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8 | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Semiring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b ≤ multiplicity p a
⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [min_eq_right h, multiplicity_le_multiplicity_iff] | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_... | Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8 | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Semiring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b ≤ multiplicity p a
⊢ ∀ (n : ℕ), p ^ n ∣ b → p ^ n ∣ a + b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_... | Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8 | theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
⊢ (multiplicity a (-b)).Dom ↔ (multiplicity a b).Dom | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp only [multiplicity, PartENat.find, dvd_neg] | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by | Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8 | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h₁ : (multiplicity a (-b)).Dom
h₂ : (multiplicity a b).Dom
⊢ ↑(Part.get (multiplicity a (-b)) h₁) = ↑(Part.get (multiplicity a b) h₂) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [PartENat.natCast_get] | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
| Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8 | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
h₁ : (multiplicity a (-b)).Dom
h₂ : (multiplicity a b).Dom
⊢ multiplicity a (-b) = ↑(Part.get (multiplicity a b) h₂) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))) | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
| Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8 | @[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | cases' Int.natAbs_eq b with h h | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
| Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = ↑(_root_.Int.natAbs b)
⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | conv_rhs => rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = ↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = ↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = ↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
case inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = -↑(_root_.Int.natAbs b)
⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | conv_rhs => rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = -↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = -↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = -↑(_root_.Int.natAbs b)
| multiplicity (↑a) b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [h] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = ↑(_root_.Int.natAbs b)
⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity ↑a ↑(_root_.Int.natAbs b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [Int.coe_nat_multiplicity] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
case inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : ℕ
b : ℤ
h : b = -↑(_root_.Int.natAbs b)
⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) (-↑(_root_.Int.natAbs b)) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [multiplicity.neg, Int.coe_nat_multiplicity] | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· | Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8 | theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (a + b) = multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply le_antisymm | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
| Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (a + b) ≤ multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply PartENat.le_of_lt_add_one | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (a + b) < multiplicity p b + 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
| Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ multiplicity p (a + b) < multiplicity p b + 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [hk] | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
| Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ multiplicity p (a + b) < ↑k + 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
| Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ ¬p ^ (k + 1) ∣ b
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multip... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | intro h_dvd | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply multiplicity.is_greatest _ h_dvd | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ multiplicity p b < ↑(k + 1) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [hk, ← Nat.succ_eq_add_one] | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ ↑k < ↑(succ k) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | norm_cast | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ k < succ k | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply Nat.lt_succ_self k | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ p ^ (k + 1) ∣ a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ multiplicity p b + 1 ≤ multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact PartENat.add_one_le_of_lt h | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p b ≤ multiplicity p (a + b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have := @min_le_multiplicity_add α _ _ p a b | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
this : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
⊢ multiplicity p b ≤ multiplicity p (a + b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rwa [← min_eq_right (le_of_lt h)] | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dv... | Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (a - b) = multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [sub_eq_add_neg, multiplicity_add_of_gt] | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
| Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (-b) = multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [multiplicity.neg] | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> | Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (-b) < multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [multiplicity.neg] | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> | Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p b < multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | assumption | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; | Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
| Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a < multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· | Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a < multiplicity p b
⊢ multiplicity p a ≤ multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact le_of_lt hab | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
| Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
case inr.inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a = multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | contradiction | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le... | Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p b < multiplicity p a
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [multiplicity_add_of_gt hab, min_eq_right] | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le... | Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p b < multiplicity p a
⊢ multiplicity p b ≤ multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact le_of_lt hab | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le... | Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
⊢ a * b = p * (p ^ (n + m) * s) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ a
x : α
hx : a = p * x
hn0 : n = 0
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hx, hn0] at ha | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
x✝ : p ^ (n - 1 + 1) ∣ x
y : α
hy : x = p ^ (n - ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
x✝ : p ^ (n - 1 + 1) ∣ x
y : α
hy : x = p ^ n * y... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this✝ : p ∣ a * b
x✝ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x *... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this✝ : p ∣ a * b
x✝ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x *... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ b
x : α
hx : b = p * x
hm0 : m = 0
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hx, hm0] at hb | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
x✝ : p ^ (m - 1 + 1) ∣ x
y : α
hy : x = p ^ (m - ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
x✝ : p ^ (m - 1 + 1) ∣ x
y : α
hy : x = p ^ m * y... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
hpx : ¬p ^ (m - 1 + 1) ∣ x
⊢ a * x * p = p ^ (n + ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
hpx : ¬p ^ (m - 1 + 1) ∣ x
⊢ a * x * p = p ^ (n + ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add] | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>... | Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8 | theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
x✝ : Finite p a
⊢ ¬p ^ (0 + 1) ∣ a ^ 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [mt isUnit_iff_dvd_one.2 hp.2.1] | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by | Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8 | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
k : ℕ
ha : Finite p a
⊢ Finite p (a ^ (k + 1)) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [_root_.pow_succ] | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by | Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8 | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
k : ℕ
ha : Finite p a
⊢ Finite p (a * a ^ k) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact finite_mul hp ha (finite_pow hp ha) | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; | Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8 | theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ multiplicity a a = 1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← Nat.cast_one] | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
| Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ multiplicity a a = ↑1 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
| Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ a ^ 1 ∣ a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by | Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : a = a ^ (1 + 1) * b
⊢ a * 1 = a * (a * b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simpa [_root_.pow_succ, mul_assoc] using hb | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by | Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
⊢ a ^ 1 ∣ a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by | Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : a = a ^ (1 + 1) * b
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
| Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ False | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩ | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
| Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ 1 = a * b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp_all | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact... | Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8 | @[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
⊢ Part.get (multiplicity p (a * b)) h =
Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
| Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
⊢ Part.get (multiplicity p (a * b)) h =
Part.get (multiplicity p a) (_ : Finite p a) + Part.get... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
⊢ Part.get (multiplicity p (a * b)) h ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add] | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
⊢ p ^ (Part.get (multiplicity p a) (_ ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [pow_add] | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [hpoweq] | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply mul_dvd_mul | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multi... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | assumption | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
case a
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multi... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | assumption | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_s... | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff] | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
p ^ (Part.get (multiplicity... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | exact ⟨hdiv, hsucc⟩ | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ ... | Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8 | protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : Finite p a ∧ Finite p b
⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp] | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
| Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8 | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : ¬(Finite p a ∧ Finite p b)
⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (... | Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8 | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : ¬(Finite p a ∧ Finite p b)
⊢ ⊤ = multiplicity p a + multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | cases' not_and_or.1 h with h h | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (... | Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8 | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | Mathlib_RingTheory_Multiplicity |
case inl
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h✝ : ¬(Finite p a ∧ Finite p b)
h : ¬Finite p a
⊢ ⊤ = multiplicity p a + multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [eq_top_iff_not_finite.2 h] | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (... | Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8 | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | Mathlib_RingTheory_Multiplicity |
case inr
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h✝ : ¬(Finite p a ∧ Finite p b)
h : ¬Finite p b
⊢ ⊤ = multiplicity p a + multiplicity p b | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [eq_top_iff_not_finite.2 h] | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (... | Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8 | protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
s : Finset β
f : β → α
⊢ multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
| Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
s : Finset β
f : β → α
⊢ multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | induction' s using Finset.induction with a s has ih h | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
| Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
case empty
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
f : β → α
⊢ multiplicity p (∏ x in ∅, f x) = ∑ x in ∅, multiplicity p (f x) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp only [Finset.sum_empty, Finset.prod_empty] | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· | Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
case empty
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
f : β → α
⊢ multiplicity p 1 = 0 | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | convert one_right hp.not_unit | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
| Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
case insert
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
f : β → α
a : β
s : Finset β
has : a ∉ s
ih : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
⊢ multiplicity p (∏ x in insert a s, f x) = ∑ x in insert a s... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [has, ← ih] | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
... | Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
case insert
α : Type u_1
β✝ : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
β : Type u_3
p : α
hp : Prime p
f : β → α
a : β
s : Finset β
has : a ∉ s
ih : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
⊢ multiplicity p (f a * ∏ x in s, f x) = multiplicity p (f a)... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | convert multiplicity.mul hp | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
... | Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
⊢ ∀ {k : ℕ}, Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | intro k | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
| Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
k : ℕ
⊢ Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | induction' k with k hk | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
| Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
case zero
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
⊢ Part.get (multiplicity p (a ^ zero)) (_ : Finite p (a ^ zero)) = zero * Part.get (multiplicity p a) ha | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [one_right hp.not_unit] | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· | Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
case succ
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
k : ℕ
hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha
⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· | Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
k : ℕ
hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha
⊢ multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [_root_.pow_succ] | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by | Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
case succ
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
ha : Finite p a
k : ℕ
hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha
this : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k)... | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_roo... | Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8 | protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
⊢ multiplicity p (a ^ 0) = 0 • multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [one_right hp.not_unit] | theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by | Mathlib.RingTheory.Multiplicity.614_0.uTHZeAJqYiw3Jx8 | theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a : α
hp : Prime p
k : ℕ
⊢ multiplicity p (a ^ succ k) = succ k • multiplicity p a | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] | theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by | Mathlib.RingTheory.Multiplicity.614_0.uTHZeAJqYiw3Jx8 | theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p : α
h0 : p ≠ 0
hu : ¬IsUnit p
n : ℕ
⊢ multiplicity p (p ^ n) = ↑n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [eq_coe_iff] | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
| Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n | Mathlib_RingTheory_Multiplicity |
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p : α
h0 : p ≠ 0
hu : ¬IsUnit p
n : ℕ
⊢ p ^ n ∣ p ^ n ∧ ¬p ^ (n + 1) ∣ p ^ n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | use dvd_rfl | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
| Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n | Mathlib_RingTheory_Multiplicity |
case right
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p : α
h0 : p ≠ 0
hu : ¬IsUnit p
n : ℕ
⊢ ¬p ^ (n + 1) ∣ p ^ n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | rw [pow_dvd_pow_iff h0 hu] | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
| Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n | Mathlib_RingTheory_Multiplicity |
case right
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p : α
h0 : p ≠ 0
hu : ¬IsUnit p
n : ℕ
⊢ ¬n + 1 ≤ n | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import r... | apply Nat.not_succ_le_self | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
| Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8 | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n | Mathlib_RingTheory_Multiplicity |
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