mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
V'' : Type v''
inst✝⁶ : Semiring K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : AddCommMonoid V'
inst✝² : Module K V'
inst✝¹ : AddCommMonoid V''
inst✝ : Module K V''
g : V →ₗ[K] V'
f : V' →ₗ[K] V''
⊢ (f ∘ₗ g).rank ≤ f.rank

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
V'' : Type v''
inst✝⁶ : Semiring K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : AddCommMonoid V'
inst✝² : Module K V'
inst✝¹ : AddCommMonoid V''
inst✝ : Module K V''
g : V →ₗ[K] V'
f : V' →ₗ[K] V''
⊢ lift.{v', v''} (f ∘ₗ g).rank ≤ lift.{v'', v'} g.rank

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' V'₁ : Type v'
inst✝⁶ : Semiring K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : AddCommMonoid V'
inst✝² : Module K V'
inst✝¹ : AddCommGroup V'₁
inst✝ : Module K V'₁
g : V →ₗ[K] V'
f : V' →ₗ[K] V'₁
⊢ (f ∘ₗ g).rank ≤ g.rank

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' V'₁ : Type v'
inst✝⁶ : Semiring K
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module K V
inst✝³ : AddCommMonoid V'
inst✝² : Module K V'
inst✝¹ : AddCommGroup V'₁
inst✝ : Module K V'₁
g : V →ₗ[K] V'
f : V' →ₗ[K] V'₁
⊢ (f ∘ₗ g).rank ≤ min f.rank g.rank

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
f g : V →ₗ[K] V'
⊢ (f + g).rank ≤ Module.rank K ↥(f.range ⊔ g.range)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
η : Type u_1
s : Finset η
f : η → V →ₗ[K] V'
x✝² : η
x✝¹ : V →ₗ[K] V'
x✝ : Cardinal.{v'}
h : x✝¹.rank ≤ x✝
⊢ (f x✝²).rank + x✝¹.rank ≤ (f x✝²).rank + x✝

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
f : V →ₗ[K] V'
g : ↥f.range →ₗ[K] V
hg : f.rangeRestrict ∘ₗ g = id
fg : LeftInverse ⇑f.rangeRestrict ⇑g
s : Set ↥f.range
si : LinearIndepOn K _root_.id s
h : #↑s ≤ f.rank
⊢ ∀ (x : ↥f.range), f (g x) = ↑x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
f : V →ₗ[K] V'
g : ↥f.range →ₗ[K] V
hg : f.rangeRestrict ∘ₗ g = id
s : Set ↥f.range
si : LinearIndepOn K _root_.id s
h : #↑s ≤ f.rank
fg : ∀ (x : ↥f.range), f (g x) = ↑x
⊢ LinearIndepOn K (fun x ↦ f (g x)) s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
c : Cardinal.{v'}
f : V →ₗ[K] V'
g : ↥f.range →ₗ[K] V
hg : f.rangeRestrict ∘ₗ g = id
fg : LeftInverse ⇑f.rangeRestrict ⇑g
s : Set V
hsc : lift.{v', v} #↑s = lift.{v, v'} c
si : LinearIndepOn K (⇑f) s
⊢ LinearIndependent (ι := ↑s) K (⇑f.range.subtype ∘ fun x ↦ f.rangeRestrict ↑x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
c : Cardinal.{v'}
f : V →ₗ[K] V'
⊢ c ≤ f.rank ↔ ∃ s, lift.{v', v} #↑s = lift.{v, v'} c ∧ LinearIndepOn K (⇑f) s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.LinearMap

Goal state before:
pp: K : Type u
V : Type v
V' : Type v'
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V'
inst✝ : Module K V'
n : ℕ
f : V →ₗ[K] V'
⊢ ↑n ≤ f.rank ↔ ∃ s, s.card = n ∧ LinearIndependent K fun (x : ↑↑s) ↦ f ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : Module.Finite R L
inst✝³ : Free R L
inst✝² : Module.Finite R M
inst✝¹ : Free R M
x y : L
inst✝ : Nontrivial R
⊢ (lieCharpoly R M x y).natDegree = finrank R M

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : BilinForm R M
f g : Module.End R M
hf : IsSkewAdjoint B ⇑f
hg : IsSkewAdjoint B ⇑g
⊢ IsAdjointPair B B ⇑(f * g) ⇑(g * f)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : BilinForm R M
f g : Module.End R M
hf : IsSkewAdjoint B ⇑f
hg : IsSkewAdjoint B ⇑g
hfg : IsAdjointPair B B ⇑(f * g) ⇑(g * f)
⊢ IsAdjointPair B B ⇑(g * f) ⇑(f * g)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
b : Basis (ChooseBasisIndex R L) R L := chooseBasis R L
⊢ (fun i ↦ (b.repr y) i * r + (b.repr x) i) = ⇑(b.repr (r • y + x))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : BilinForm R M
f g : Module.End R M
hf : f ∈ skewAdjointSubmodule B
hg : g ∈ skewAdjointSubmodule B
⊢ ⁅f, g⁆ ∈ skewAdjointSubmodule B

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : BilinForm R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
e : N ≃ₗ[R] M
⊢ ↥(skewAdjointLieSubalgebra (LinearMap.compl₁₂ B ↑e ↑e)) ≃ₗ⁅R⁆ ↥(skewAdjointLieSubalgebra B)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : BilinForm R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
e : N ≃ₗ[R] M
f : ↥(skewAdjointLieSubalgebra (LinearMap.compl₁₂ B ↑e ↑e))
⊢ ↑((skewAdjointLieSubalgebraEquiv B e) f) = e.lieConj ↑f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : BilinForm R M
N : Type w
inst✝¹ : AddCommGroup N
inst✝ : Module R N
e : N ≃ₗ[R] M
f : ↥(skewAdjointLieSubalgebra B)
⊢ ↑((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj ↑f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
⊢ Polynomial.map (evalRingHom r) (lieCharpoly R M x y) = charpoly (φ (r • y + x))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
J : Matrix n n R
inst✝ : DecidableEq n
A B : Matrix n n R
hA : A ∈ skewAdjointMatricesSubmodule J
hB : B ∈ skewAdjointMatricesSubmodule J
⊢ ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
J : Matrix n n R
inst✝ : DecidableEq n
P : Matrix n n R
h : Invertible P
A : Matrix n n R
this : (P.lieConj h) A ∈ skewAdjointMatricesSubmodule J ↔ A ∈ skewAdjointMatricesSubmodule (Pᵀ * J * P)
⊢ A ∈ LieSubalgebra.map (P.lieConj h).symm.toLieHom (skewAdjointMatricesLieSubalgebra J) ↔
    A ∈ skewAdjointMatricesLieSubalgebra (Pᵀ * J * P)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
J : Matrix n n R
inst✝ : DecidableEq n
P : Matrix n n R
h : Invertible P
⊢ LieSubalgebra.map (P.lieConj h).symm.toLieHom (skewAdjointMatricesLieSubalgebra J) =
    skewAdjointMatricesLieSubalgebra (Pᵀ * J * P)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
J : Matrix n n R
inst✝ : DecidableEq n
P : Matrix n n R
h : Invertible P
A : ↥(skewAdjointMatricesLieSubalgebra J)
⊢ ↑((skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P⁻¹ * ↑A * P

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : Fintype n
J : Matrix n n R
inst✝² : DecidableEq n
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
A : Matrix m m R
this : J.IsSkewAdjoint (e.symm A) ↔ (e J).IsSkewAdjoint A
⊢ A ∈ LieSubalgebra.map e.toLieEquiv.toLieHom (skewAdjointMatricesLieSubalgebra J) ↔
    A ∈ skewAdjointMatricesLieSubalgebra (e J)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : Fintype n
J : Matrix n n R
inst✝² : DecidableEq n
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
⊢ LieSubalgebra.map e.toLieEquiv.toLieHom (skewAdjointMatricesLieSubalgebra J) = skewAdjointMatricesLieSubalgebra (e J)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : Module.Finite R L
inst✝³ : Free R L
inst✝² : Module.Finite R M
inst✝¹ : Free R M
x y : L
inst✝ : Nontrivial R
i j : ℕ
hij : i + j = finrank R M
⊢ ((lieCharpoly R M x y).coeff i).natDegree ≤ j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.SkewAdjoint

Goal state before:
pp: R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
u : Rˣ
J A : Matrix n n R
⊢ A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
⊢ ∀ {x_1 : ↥U} {m : L}, m ∈ (engel K x).carrier → ⁅x_1, m⁆ ∈ (engel K x).carrier

Tactic:
by

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

Goal state before:
pp: m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ (toLin' (diagonal w)).ker = ⨆ i ∈ {i | w i = 0}, (LinearMap.single K (fun x ↦ K) i).range

Tactic:
by

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

Goal state before:
pp: m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ (toLin' (diagonal w)).range = ⨆ i ∈ {i | w i ≠ 0}, (LinearMap.single K (fun x ↦ K) i).range

Tactic:
dsimp only [mem_ofPred_eq]

Goal state after:
m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ (toLin' (diagonal w)).range = ⨆ i, ⨆ (_ : w i ≠ 0), (LinearMap.single K (fun x ↦ K) i).range
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r = finrank K L
this : engel K x = ⊤
⊢ engel K y ≤ engel K x

Tactic:
by

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

Goal state before:
pp: m : Type u_1
inst✝² : Fintype m
K : Type u
inst✝¹ : Semifield K
inst✝ : DecidableEq m
w : m → K
⊢ (toLin' (diagonal w)).range = ⨆ i ∈ {i | w i ≠ 0}, (LinearMap.single K (fun x ↦ K) i).range

Tactic:
by

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

Goal state before:
pp: m : Type u_1
inst✝³ : Fintype m
K : Type u
inst✝² : Field K
inst✝¹ : DecidableEq m
inst✝ : DecidableEq K
w : m → K
⊢ (toLin' (Matrix.diagonal w)).rank = ↑(Fintype.card { i // w i ≠ 0 })

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : ∀ (m : ↥E), ∃ n, ((toEnd K (↥U) L) y' ^ n) ↑m = ↑0
⊢ Ex ≤ Ey

Tactic:
ZeroMemClass.coe_zero

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : ∀ (m : ↥E), ∃ n, ((toEnd K (↥U) L) y' ^ n) ↑m = 0
⊢ Ex ≤ Ey
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : χ = X ^ r
⊢ Ex ≤ Ey

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
p : Type u_2
q : Type u_3
l : Type u_4
R : Type u₂
inst✝⁵ : DecidableEq p
inst✝⁴ : DecidableEq q
inst✝³ : DecidableEq l
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j k : n
r : R
⊢ (traceLinearMap n R R) ((singleLinearMap R i i) r) = (traceLinearMap n R R) ((singleLinearMap R j j) r)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j k : n
r : R
⊢ (singleSubSingle i j) r + (singleSubSingle j k) r = (singleSubSingle i k) r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j k : n
r : R
⊢ (singleSubSingle i k) r - (singleSubSingle i j) r = (singleSubSingle j k) r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j k : n
r : R
⊢ (singleSubSingle i k) r - (singleSubSingle j k) r = (singleSubSingle i j) r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
h : 1 < Fintype.card n
i j : n
hij : i ≠ j
A : ↥(sl n R) := (single i j hij) 1
B : ↥(sl n R) := (single j i ⋯) 1
c : IsLieAbelian ↥(sl n R)
⊢ ↑0 = 0

Tactic:
ZeroMemClass.coe_zero

Goal state after:
n : Type u_1
R : Type u₂
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
h : 1 < Fintype.card n
i j : n
hij : i ≠ j
A : ↥(sl n R) := (single i j hij) 1
B : ↥(sl n R) := (single j i ⋯) 1
c : IsLieAbelian ↥(sl n R)
⊢ 0 = 0
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
h : 1 < Fintype.card n
i j : n
hij : i ≠ j
A : ↥(sl n R) := (single i j hij) 1
B : ↥(sl n R) := (single j i ⋯) 1
c : IsLieAbelian ↥(sl n R)
⊢ ↑A * ↑B = ↑B * ↑A

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Nontrivial R
h : 1 < Fintype.card n
⊢ ¬IsLieAbelian ↥(sl n R)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
⊢ A ∈ so n R ↔ Aᵀ = -A

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: case inl.inr
p : Type u_2
q : Type u_3
R : Type u₂
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : CommRing R
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
x : p
y : q
⊢ (Pso p q R i * Pso p q R (-i)) (Sum.inl x) (Sum.inr y) = 1 (Sum.inl x) (Sum.inr y)

Tactic:
simp [Pso]

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: case inr.inl
p : Type u_2
q : Type u_3
R : Type u₂
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : CommRing R
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
x : q
y : p
⊢ (Pso p q R i * Pso p q R (-i)) (Sum.inr x) (Sum.inl y) = 1 (Sum.inr x) (Sum.inl y)

Tactic:
simp [Pso]

Goal state after:
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
this : ∀ i < r, χ.coeff i = 0
⊢ χ = X ^ r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: p : Type u_2
q : Type u_3
R : Type u₂
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : CommRing R
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
⊢ Pso p q R i * Pso p q R (-i) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: p : Type u_2
q : Type u_3
R : Type u₂
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : CommRing R
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
⊢ (Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝⁵ : DecidableEq l
inst✝⁴ : CommRing R
F : Type u_3
S : Type u_4
inst✝³ : CommRing S
inst✝² : FunLike F R S
inst✝¹ : AddMonoidHomClass F R S
inst✝ : OneHomClass F R S
f : F
⊢ (J l R).map ⇑f = J l S

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: n : Type u_1
p : Type u_2
q : Type u_3
l : Type u_4
R : Type u₂
inst✝⁶ : DecidableEq p
inst✝⁵ : DecidableEq q
inst✝⁴ : DecidableEq l
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
⊢ ↥(so' p q R) ≃ₗ⁅R⁆ ↥(so (p ⊕ q) R)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝¹ : DecidableEq l
inst✝ : CommRing R
⊢ (J l R)ᵀ = -J l R

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
⊢ J l R * J l R = -1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
⊢ (J l R)⁻¹ = -J l R

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: p : Type u_2
q : Type u_3
R : Type u₂
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : CommRing R
inst✝¹ : Fintype p
inst✝ : Fintype q
i : R
hi : i * i = -1
A : ↥(so' p q R)
⊢ ↑((soIndefiniteEquiv p q R hi) A) = (Pso p q R i)⁻¹ * ↑A * Pso p q R i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
⊢ (J l R).det * (J l R).det = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: l : Type u_4
R : Type u₂
inst✝¹ : DecidableEq l
inst✝ : CommRing R
⊢ S l R = fromBlocks 1 0 0 (-1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Classical

Goal state before:
pp: l : Type u_4
R : Type u₂
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
⊢ (PD l R)ᵀ * JD l R = fromBlocks 1 1 1 (-1)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
a b : Matrix (l ⊕ l) (l ⊕ l) R
ha : a ∈ {A | A * J l R * Aᵀ = J l R}
hb : b ∈ {A | A * J l R * Aᵀ = J l R}
⊢ a * b ∈ {A | A * J l R * Aᵀ = J l R}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype l
⊢ 1 ∈ {A | A * J l R * Aᵀ = J l R}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
⊢ A ∈ symplecticGroup l R ↔ A * J l R * Aᵀ = J l R

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
⊢ J l R ∈ symplecticGroup l R

Tactic:
by

Goal state after: