mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h₁ : Submodule.span A (Set.range solution.var) = ⊤
h₂ : solution.π.ker = Submodule.span A (Set.range relations.relation)
⊢ solution.IsPresentation

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Fintype n
⊢ M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Fintype n
⊢ M.SeparatingLeft ↔ ∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M *ᵥ w = 0) → v = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Fintype n
⊢ M.Nondegenerate ↔
    (∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M *ᵥ w = 0) → v = 0) ∧ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Finite n
⊢ M.SeparatingLeft ↔ ∀ (v : m → R), v ᵥ* M = 0 → v = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.CharP.Reduced

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsReduced R
p n : ℕ
inst✝ : ExpChar R p
x y : R
H : (iterateFrobenius R p n) x = (iterateFrobenius R p n) y
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.CharP.Reduced

Goal state before:
pp: R : Type u_1
inst✝³ : Finite R
inst✝² : CommRing R
inst✝¹ : IsReduced R
inst✝ : CharP R 2
a : R
⊢ IsSquare a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.CharP.Reduced

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsReduced R
p k m : ℕ
inst✝ : ExpChar R p
x : R
⊢ x ^ (p ^ k * m) = 1 ↔ x ^ m = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Fintype n
⊢ M.SeparatingRight ↔ ∀ (v : n → R), M *ᵥ v = 0 → v = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Fintype n
⊢ M.Nondegenerate ↔ (∀ (v : m → R), v ᵥ* M = 0 → v = 0) ∧ ∀ (v : n → R), M *ᵥ v = 0 → v = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
M : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
e : Basis ι R M
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
v : ι → M
⊢ e.toMatrix v = (LinearMap.toMatrix e e) ((e.constr ℕ) v)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Finite n
⊢ Mᵀ.SeparatingLeft ↔ M.SeparatingRight

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
inst✝¹ : CommSemiring R
inst✝ : Finite ι
⊢ (Pi.basisFun R ι).toMatrix = transpose

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
M : Type u_6
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
e : Basis ι R M
inst✝ : DecidableEq ι
⊢ e.toMatrix ⇑e = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Finite n
⊢ Mᵀ.SeparatingRight ↔ M.SeparatingLeft

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Finite n
⊢ Mᵀ.Nondegenerate ↔ M.Nondegenerate

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m R
hM : M.det ∈ R⁰
⊢ M.SeparatingLeft

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m R
hM : M.det ∈ R⁰
⊢ Mᵀ.det ∈ R⁰

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
⊢ { X₁ := AddCommGrpCat.of (Ext S.X₂ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₁ Y n₀),
      X₃ := AddCommGrpCat.of (Ext S.X₃ Y n₁), f := AddCommGrpCat.ofHom ((mk₀ S.f).precomp Y ⋯),
      g := AddCommGrpCat.ofHom (hS.extClass.precomp Y h), zero := ⋯ }.Exact

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m R
hM : M.det ∈ R⁰
x✝¹ x✝ : m → R
hxy : M *ᵥ x✝¹ = M *ᵥ x✝
⊢ M *ᵥ (x✝¹ - x✝) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
e : Basis ι R M
v : ι' → M
j : ι'
inst✝ : DecidableEq ι'
x : M
⊢ e.toMatrix (update v j x) = (e.toMatrix v).updateCol j ⇑(e.repr x)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
⊢ AddCommGrpCat.ofHom (hS.extClass.precomp Y h) ≫ AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R₂ : Type u_7
M₂ : Type u_8
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
inst✝ : DecidableEq ι
e : Basis ι R₂ M₂
w : ι → R₂ˣ
⊢ e.toMatrix ⇑(e.unitsSMul w) = Matrix.diagonal (Units.val ∘ w)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
e : Basis ι R M
v : ι' → M
j : ι'
inst✝ : Fintype ι
⊢ ∑ i, e.toMatrix v i j • e i = v j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: ι : Type u_1
κ : Type u_2
R : Type u_3
M : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : Finite κ
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι → M
hv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0
A : Matrix κ ι R
hA : A.Nondegenerate
this✝ : Fintype κ
w : κ → R
hw : ∑ i, w i • ∑ j, A i j • v j = 0
this : w = 0
⊢ ∀ (i : κ), w i = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R₁ : Type u_9
S : Type u_10
inst✝⁴ : CommSemiring R₁
inst✝³ : Semiring S
inst✝² : Algebra R₁ S
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : S
b : Basis ι R₁ S
w : ι → S
⊢ b.toMatrix (x • w) = (Algebra.leftMulMatrix b) x * b.toMatrix w

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: ι : Type u_1
κ : Type u_2
R : Type u_3
M : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : Finite κ
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι → M
A : Matrix κ ι R
hA : A.Nondegenerate
this : Fintype κ
w : κ → R
hw : ∑ x, (∑ i, w i • A i x) • v x = 0
hv : w ᵥ* A = 0
w' : ι → R
⊢ w ⬝ᵥ A *ᵥ w' = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: ι : Type u_1
κ : Type u_2
R : Type u_3
M : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : Finite κ
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι → M
hv : LinearIndependent R v
A : Matrix κ ι R
hA : A.Nondegenerate
⊢ LinearIndependent R fun i ↦ ∑ j, A i j • v j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
inst✝³ : CommSemiring R
S : Type u_9
inst✝² : Semiring S
inst✝¹ : Algebra R S
inst✝ : Fintype ι
b : Basis ι R S
v : ι' → S
⊢ ⇑b ᵥ* (b.toMatrix v).map ⇑(algebraMap R S) = v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
e : Basis ι R M
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
v : Basis ι' R M
i : ι'
⊢ ((toLin v e) (e.toMatrix ⇑v)) (v i) = LinearMap.id (v i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
R₂ : Type u_7
M₂ : Type u_8
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
e✝ : Basis ι R M
v✝ : ι' → M
i : ι
j : ι'
inst✝ : Fintype ι
e : Basis ι R M
v w : ι → M
⊢ e.toMatrix (v + w) = e.toMatrix v + e.toMatrix w

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
R₂ : Type u_7
M₂ : Type u_8
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
e✝ : Basis ι R M
v : ι' → M
i : ι
j : ι'
inst✝ : Fintype ι
e : Basis ι R M
⊢ ∀ (m : R) (x : ι → M), e.toMatrix (m • x) = (RingHom.id R) m • e.toMatrix x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
R₂ : Type u_7
M₂ : Type u_8
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
e✝ : Basis ι R M
v : ι' → M
i : ι
j : ι'
inst✝ : Fintype ι
e : Basis ι R M
⊢ LeftInverse (fun m j ↦ ∑ i, m i j • e i) e.toMatrix

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
R₂ : Type u_7
M₂ : Type u_8
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
e✝ : Basis ι R M
v : ι' → M
i : ι
j : ι'
inst✝ : Fintype ι
e : Basis ι R M
⊢ RightInverse (fun m j ↦ ∑ i, m i j • e i) e.toMatrix

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R₂ : Type u_7
M₂ : Type u_8
inst✝¹¹ : CommRing R₂
inst✝¹⁰ : AddCommGroup M₂
inst✝⁹ : Module R₂ M₂
inst✝⁸ : Fintype ι
inst✝⁷ : DecidableEq ι
S : Type u_9
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : Algebra R₂ S
inst✝³ : Module S M₂
inst✝² : IsScalarTower R₂ S M₂
inst✝¹ : IsDomain R₂
inst✝ : IsTorsionFree R₂ S
b : Basis ι S M₂
v : ι → ↥(span R₂ (Set.range ⇑b))
⊢ (algebraMap R₂ S).mapMatrix ((restrictScalars R₂ b).toMatrix v) = b.toMatrix fun i ↦ ↑(v i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
b : Basis ι R M
b' : Basis ι' R M
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Finite ι'
⊢ (toMatrix b b') id = b'.toMatrix ⇑b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: M✝ : Type u_1
inst✝⁶ : MonoidWithZero M✝
inst✝⁵ : Finite M✝
M : Type u_2
inst✝⁴ : MonoidWithZero M
inst✝³ : IsLeftCancelMulZero M
inst✝² : DecidableEq M
inst✝¹ : Fintype M
inst✝ : Nontrivial M
⊢ (if h : 0 = 0 then 0 else bijInv ⋯ 1) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: M✝ : Type u_1
inst✝⁶ : MonoidWithZero M✝
inst✝⁵ : Finite M✝
M : Type u_2
inst✝⁴ : MonoidWithZero M
inst✝³ : IsLeftCancelMulZero M
inst✝² : DecidableEq M
inst✝¹ : Fintype M
inst✝ : Nontrivial M
a : M
ha : a ≠ 0
⊢ (a * if h : a = 0 then 0 else bijInv ⋯ 1) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : GCDMonoid R
inst✝ : Subsingleton Rˣ
a b c : R
n : ℕ
cp : IsCoprime a b
h : a * b = c ^ n
⊢ ∃ d, a = d ^ n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
N : Type u_9
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
b' : Basis ι' R M
c : Basis κ R N
c' : Basis κ' R N
f : M →ₗ[R] N
inst✝³ : Fintype ι'
inst✝² : Finite κ
inst✝¹ : Fintype κ'
inst✝ : DecidableEq ι'
⊢ (toLin b' c) (c.toMatrix ⇑c' * (toMatrix b' c') f) = (toLin b' c) ((toMatrix b' c) f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: ι : Type u_2
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : GCDMonoid R
inst✝ : Subsingleton Rˣ
n : ℕ
c : R
s : Finset ι
f : ι → R
h : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → IsCoprime (f i) (f j)
hprod : ∏ i ∈ s, f i = c ^ n
⊢ ∀ i ∈ s, ∃ d, f i = d ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_3
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : Finite R
⊢ IsField R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
f : G →* R
hf : Injective ⇑f
n : ℕ
hn : 0 < n
g₀ : G
this : DecidableEq R
⊢ Set.MapsTo ⇑f ↑{g | g ^ n = g₀} ↑(nthRoots n (f g₀)).toFinset

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
f : G →* R
hf : Injective ⇑f
n : ℕ
hn : 0 < n
g₀ : G
⊢ #{g | g ^ n = g₀} ≤ (nthRoots n (f g₀)).card

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
R : Type u_5
M : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
N : Type u_9
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Fintype ι'
inst✝² : Fintype κ
inst✝¹ : Finite ι
inst✝ : DecidableEq κ
b₁ : Basis ι R M
b₂ : Basis ι' R M
b₃ : Basis κ R N
A : Matrix ι' κ R
⊢ b₁.toMatrix ⇑b₂ * A = (toMatrix b₃ b₁) ((toLin b₃ b₂) A)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Finite G
f : G →* R
hf : Injective ⇑f
⊢ IsCyclic G

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
S : Subgroup Rˣ
inst✝ : Finite ↥S
⊢ ∀ (x y : ↥S), ↑↑(x * y) = ↑↑x * ↑↑y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
N : Type u_9
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R N
b : Basis ι R M
b' : Basis ι' R M
c' : Basis κ' R N
f : M →ₗ[R] N
inst✝⁴ : Fintype ι'
inst✝³ : Fintype ι
inst✝² : Finite κ'
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
⊢ f ∘ₗ (toLin b b') (b'.toMatrix ⇑b) = f

Tactic:
b'.toLin_toMatrix,

Goal state after:
ι : Type u_1
ι' : Type u_2
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
N : Type u_9
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R N
b : Basis ι R M
b' : Basis ι' R M
c' : Basis κ' R N
f : M →ₗ[R] N
inst✝⁴ : Fintype ι'
inst✝³ : Fintype ι
inst✝² : Finite κ'
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
⊢ f ∘ₗ LinearMap.id = f
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
N : Type u_9
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R N
b : Basis ι R M
b' : Basis ι' R M
c' : Basis κ' R N
f : M →ₗ[R] N
inst✝⁴ : Fintype ι'
inst✝³ : Fintype ι
inst✝² : Finite κ'
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
⊢ (toLin b c') ((toMatrix b' c') f * b'.toMatrix ⇑b) = (toLin b c') ((toMatrix b c') f)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
κ' : Type u_4
R : Type u_5
M : Type u_6
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
N : Type u_9
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R N
b : Basis ι R M
b' : Basis ι' R M
c : Basis κ R N
c' : Basis κ' R N
f : M →ₗ[R] N
inst✝⁵ : Fintype ι'
inst✝⁴ : Finite κ
inst✝³ : Fintype ι
inst✝² : Fintype κ'
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_3
inst✝² : Field K
inst✝¹ : Algebra R[X] K
inst✝ : IsFractionRing R[X] K
f g : R[X]
hg : g.Monic
⊢ ∃ q r,
    r.degree < g.degree ∧
      (algebraMap R[X] K) f / (algebraMap R[X] K) g =
        (algebraMap R[X] K) q + (algebraMap R[X] K) r / (algebraMap R[X] K) g

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
κ : Type u_3
R : Type u_5
M : Type u_6
inst✝⁹ : CommSemiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
N : Type u_9
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R N
inst✝⁴ : Fintype ι'
inst✝³ : Finite κ
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
b₁ : Basis ι R M
b₂ : Basis ι' R M
b₃ : Basis κ R N
A : Matrix κ ι R
⊢ A * b₁.toMatrix ⇑b₂ = (toMatrix b₂ b₃) ((toLin b₁ b₃) A)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
inst✝¹ : CommSemiring R
inst✝ : Fintype ι
b : Basis ι R (ι → R)
A : Matrix ι ι R
⊢ b.toMatrix ⇑(Pi.basisFun R ι) * A = of fun i j ↦ (b.repr (A.col j)) i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Fintype G
f : G →* R
hf : f ≠ 1
x : ↥f.toHomUnits.range
hx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x
⊢ ↑↑x - 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Fintype ι'
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
b : Basis ι R M
v : ι' → M
e : ι ≃ ι'
⊢ (b.reindex e).toMatrix v = (reindexAlgEquiv R R e) (b.toMatrix (v ∘ ⇑e))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
b' : Basis ι' R M
inst✝¹ : Fintype ι
inst✝ : Finite ι'
m : M
⊢ b'.toMatrix ⇑b *ᵥ ⇑(b.repr m) = ⇑(b'.repr m)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Fintype G
f : G →* R
hf : f ≠ 1
x : ↥f.toHomUnits.range
hx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x
hx1 : ↑↑x - 1 ≠ 0
c : ℕ := #{g | f.toHomUnits g = 1}
⊢ ∀ (x : Rˣ), x ∈ image (⇑f.toHomUnits) univ ↔ x ∈ f.toHomUnits.range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
b : Basis ι R M
b' : Basis ι' R M
ι'' : Type u_10
inst✝ : Fintype ι'
b'' : ι'' → M
⊢ b.toMatrix ⇑b' * b'.toMatrix b'' = b.toMatrix b''

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
b' : Basis ι' R M
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
⊢ b.toMatrix ⇑b' * b'.toMatrix ⇑b = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
v : ι' → M
e : ι ≃ ι'
⊢ (b.reindex e).toMatrix v = (b.toMatrix v).submatrix (⇑e.symm) _root_.id

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
M : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
b : Basis ι R M
f : M ≃ₗ[R] N
v : ι → N
⊢ (b.map f).toMatrix v = b.toMatrix (⇑f.symm ∘ v)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_5
M : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
b : Basis ι R M
f : M ≃ₗ[R] N
v : ι → N
⊢ (b.map f).toMatrix v = b.toMatrix (⇑f.symm ∘ v)

Tactic:
by
  ext
  simp only [toMatrix_apply, Basis.map, LinearEquiv.trans_apply, (· ∘ ·)]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Basis

Goal state before:
pp: ι : Type u_1
κ : Type u_3
R : Type u_5
M : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
N : Type u_9
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
b : Basis ι R M
c : Basis κ R N
f : M →ₗ[R] N
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Finite κ
⊢ (LinearMap.toMatrix b c) f = c.toMatrix (⇑f ∘ ⇑b)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Fintype G
f : G →* R
hf : f ≠ 1
x : ↥f.toHomUnits.range
hx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x
hx1 : ↑↑x - 1 ≠ 0
c : ℕ := #{g | f.toHomUnits g = 1}
⊢ ∀ a ∈ range (orderOf x), x ^ a ∈ univ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Fintype G
f : G →* R
hf : f ≠ 1
x : ↥f.toHomUnits.range
hx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x
hx1 : ↑↑x - 1 ≠ 0
c : ℕ := #{g | f.toHomUnits g = 1}
⊢ Set.InjOn (fun x_1 ↦ x ^ x_1) ↑(range (orderOf x))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : Group G
inst✝ : Fintype G
f : G →* R
hf : f ≠ 1
x : ↥f.toHomUnits.range
hx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x
hx1 : ↑↑x - 1 ≠ 0
c : ℕ := #{g | f.toHomUnits g = 1}
b : ↥f.toHomUnits.range
x✝ : b ∈ ↑univ
n : ℕ
hn : (fun x_1 ↦ x ^ x_1) n = b
⊢ (fun x_1 ↦ x ^ x_1) (n % orderOf x) = b

Tactic:
by

Goal state after: