mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : NonUnitalSemiring A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module R A
inst✝¹³ : NonUnitalSemiring B
inst✝¹² : StarRing B
inst✝¹¹ : Module R B
inst✝¹⁰ : FunLike F A B
inst✝⁹ : NonUnitalAlgHomClass F R A B
inst✝⁸ : StarHomClass F A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝⁷ : StarRing R
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : StarModule R A
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
⊢ ∀ (a : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ (star a) =
      star (Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : NonUnitalSemiring A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module R A
inst✝¹³ : NonUnitalSemiring B
inst✝¹² : StarRing B
inst✝¹¹ : Module R B
inst✝¹⁰ : FunLike F A B
inst✝⁹ : NonUnitalAlgHomClass F R A B
inst✝⁸ : StarHomClass F A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝⁷ : StarRing R
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : StarModule R A
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
⊢ ↥T →⋆ₙₐ[R] B

Tactic:
by

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

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : StrongRankCondition R
v : M
n : v ≠ 0
h : ∀ (w : M), ∃ c, c • v = w
⊢ Module.rank R M = 1

Tactic:
by

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

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
v : M
h : ∀ (w : M), ∃ c, c • v = w
⊢ Module.rank R M ≤ 1

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
⊢ LinearIndepOn K _root_.id n

Tactic:
by

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

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
v : M
h : ∀ (w : M), ∃ c, c • v = w
⊢ finrank R M ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : ↥(K i)
h : K i ≤ T
⊢ (iSupLift K dir f hf T hT) ((inclusion h) x) = (f i) x

Tactic:
by

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

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_1
⊢ Module.Finite R (ι →₀ M) → IsEmpty ι ∨ Subsingleton M ∨ Module.Finite R M ∧ Finite ι

Tactic:
by

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

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_1
h : Module.Finite R M ∧ Finite ι
⊢ Module.Finite R (ι →₀ M)

Tactic:
cases h; infer_instance

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

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Type u_1
h : Module.Finite R M ∧ Finite ι
⊢ Module.Finite R (ι →₀ M)

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : Semiring R
ι : Type u_1
⊢ Module.Finite R (ι →₀ R) ↔ Subsingleton R ∨ Finite ι

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
⊢ ⊤ ≤ span K n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
h : K i ≤ T
⊢ (iSupLift K dir f hf T hT).comp (inclusion h) = f i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Defs

Goal state before:
pp: K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
⊢ FiniteDimensional K ↥S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Finiteness.Lattice

Goal state before:
pp: R : Type u_2
V : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup V
inst✝² : Module R V
ι : Sort u_1
inst✝¹ : Finite ι
S : ι → Submodule R V
inst✝ : ∀ (i : ι), Module.Finite R ↥(S i)
⊢ Module.Finite R ↥(⨆ i, S i)

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
⊢ ⊤ ≤ span K (Set.range fun x ↦ _root_.id ↑x)

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
b : Basis (↑n) K V := Basis.mk H₁ ⋯
⊢ ↑v ∈ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : ↥(K i)
hx : ↑x ∈ T
⊢ (iSupLift K dir f hf T hT) ⟨↑x, hx⟩ = (f i) x

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
b : Basis (↑n) K V := Basis.mk H₁ ⋯
⊢ w ∈ n

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
b : Basis (↑n) K V := Basis.mk H₁ ⋯
⊢ ↑v = b ⟨↑v, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : ↥T
hx : ↑x ∈ K i
⊢ (iSupLift K dir f hf T hT) x = (f i) ⟨↑x, hx⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : NonUnitalSemiring A
inst✝³ : StarRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
s : Set A
z : A
⊢ z ∈ centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : NonUnitalSemiring A
inst✝³ : StarRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
⊢ ↑(centralizer R Set.univ) = ↑(center R A)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : NonUnitalSemiring A
inst✝³ : StarRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
s : Set A
⊢ ↑(centralizer R ↑(centralizer R s)) = (s ∪ star s).centralizer.centralizer

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
s : Set A
⊢ adjoin R s ≤ centralizer R ↑(centralizer R s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
a b✝ : A
s : Set A
hb✝ : b✝ ∈ adjoin R s
h : ∀ b ∈ s, Commute a b
h_star : ∀ b ∈ s, Commute a (star b)
b : A
hb : b ∈ s ∪ star s
⊢ b ∈ star s → Commute a b

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this✝ : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this✝.extend ⋯)) K ↥f.ker := Basis.extend this✝
w : V
hw : f w = 1
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := insert w s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
b : Basis (↑n) K V := Basis.mk H₁ ⋯
x : ↥f.ker
hx : x ∈ Set.range ⇑b₁
this : ↑x ≠ w
⊢ f ((Basis.mk H₁ ⋯) ⟨f.ker.subtype x, ⋯⟩) = (Finsupp.single ⟨f.ker.subtype x, ⋯⟩ 1) ⟨w, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
a b c : A
hc : c ∈ adjoin R {b}
h : Commute a b
h_star : Commute a (star b)
⊢ ∀ b_1 ∈ {b}, Commute a b_1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
a b c : A
hc : c ∈ adjoin R {b}
h : Commute a b
h_star : Commute a (star b)
⊢ ∀ b_1 ∈ {b}, Commute a (star b_1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
s : Set A
hcomm : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x
hcomm_star : ∀ a ∈ s, ∀ b ∈ s, a * star b = star b * a
⊢ IsMulCommutative ↥(adjoin R s)

Tactic:
by

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

Goal state before:
pp: K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
v : V
hfv : f v = 0
hf : f ≠ 0
hv : v ≠ 0
⊢ ∃ n b i j, i ≠ j ∧ v = b i ∧ f = b.coord j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : StarRing A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
a : A
inst✝ : IsStarNormal a
⊢ ∀ x ∈ {a}, ∀ y ∈ {a}, x * y = y * x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : NonUnitalSemiring A
inst✝⁵ : StarRing A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
a : A
inst✝ : IsStarNormal a
⊢ ∀ a_1 ∈ {a}, ∀ b ∈ {a}, a_1 * star b = star b * a_1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
f g : E →ₛₗ.[σ] F
h : f.domain = g.domain
h' : ∀ ⦃x : E⦄ ⦃hf : x ∈ f.domain⦄ ⦃hg : x ∈ g.domain⦄, ↑f ⟨x, hf⟩ = ↑g ⟨x, hg⟩
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
f g : E →ₛₗ.[σ] F
⊢ f = g → f.domain = g.domain ∧ ∀ ⦃x : E⦄ ⦃hf : x ∈ f.domain⦄ ⦃hg : x ∈ g.domain⦄, ↑f ⟨x, hf⟩ = ↑g ⟨x, hg⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
f g : E →ₛₗ.[σ] F
EQ : f = g
x y : ↥f.domain
h : ↑x = ↑y
⊢ ↑f x = ↑f y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
x : E
y : F
H : ∀ (c : R), c • x = 0 → σ c • y = 0
⊢ ∀ (c₁ c₂ : R), c₁ • x = c₂ • x → σ c₁ • y = σ c₂ • y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
x : E
y : F
H✝ : ∀ (c : R), c • x = 0 → σ c • y = 0
H : ∀ (c₁ c₂ : R), c₁ • x = c₂ • x → σ c₁ • y = σ c₂ • y
y' z' : ↥(R ∙ x)
⊢ σ (Classical.choose ⋯) • y = σ (Classical.choose ⋯) • y + σ (Classical.choose ⋯) • y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
x : E
y : F
H✝ : ∀ (c : R), c • x = 0 → σ c • y = 0
H : ∀ (c₁ c₂ : R), c₁ • x = c₂ • x → σ c₁ • y = σ c₂ • y
c : R
z : ↥(R ∙ x)
⊢ σ (Classical.choose ⋯) • y = σ c • σ (Classical.choose ⋯) • y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
x : E
y : F
H : ∀ (c : R), c • x = 0 → σ c • y = 0
c : R
h : c • x ∈ (mkSpanSingleton' x y H).domain
⊢ ↑(mkSpanSingleton' x y H) ⟨c • x, h⟩ = σ c • y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
x : E
y : F
H : ∀ (c : R), c • x = 0 → σ c • y = 0
h : x ∈ (mkSpanSingleton' x y H).domain
⊢ ↑(mkSpanSingleton' x y H) ⟨x, h⟩ = y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁴ : Ring R
inst✝¹³ : Ring S
inst✝¹² : Ring T
σ✝ : R →+* S
τ : S →+* T
E✝ : Type u_4
inst✝¹¹ : AddCommGroup E✝
inst✝¹⁰ : Module R E✝
F✝ : Type u_5
inst✝⁹ : AddCommGroup F✝
inst✝⁸ : Module S F✝
G : Type u_6
inst✝⁷ : AddCommGroup G
inst✝⁶ : Module T G
K : Type u_7
L : Type u_8
E : Type u_9
F : Type u_10
inst✝⁵ : DivisionRing K
inst✝⁴ : DivisionRing L
σ : K →+* L
inst✝³ : AddCommGroup E
inst✝² : Module K E
inst✝¹ : AddCommGroup F
inst✝ : Module L F
x : E
y : F
hx : x ≠ 0
c : K
hc✝ : c • x = 0
hc : c = 0
⊢ σ c • y = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f g : E →ₛₗ.[σ] F
x y : E
x✝¹ : x ∈ {x | ∃ (hf : x ∈ f.domain) (hg : x ∈ g.domain), ↑f ⟨x, hf⟩ = ↑g ⟨x, hg⟩}
x✝ : y ∈ {x | ∃ (hf : x ∈ f.domain) (hg : x ∈ g.domain), ↑f ⟨x, hf⟩ = ↑g ⟨x, hg⟩}
hfx : x ∈ f.domain
hgx : x ∈ g.domain
hx : ↑f ⟨x, hfx⟩ = ↑g ⟨x, hgx⟩
hfy : y ∈ f.domain
hgy : y ∈ g.domain
hy : ↑f ⟨y, hfy⟩ = ↑g ⟨y, hgy⟩
⊢ ↑f ⟨x + y, ⋯⟩ = ↑g ⟨x + y, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f✝ g : E →ₛₗ.[σ] F
c : R
x : E
x✝ : x ∈ {x | ∃ (hf : x ∈ f✝.domain) (hg : x ∈ g.domain), ↑f✝ ⟨x, hf⟩ = ↑g ⟨x, hg⟩}
hfx✝ : x ∈ f✝.domain
hgx : x ∈ g.domain
hx : ↑f✝ ⟨x, hfx✝⟩ = ↑g ⟨x, hgx⟩
f : E →ₛₗ.[σ] F
hfx : x ∈ f.domain
⊢ ⟨c • x, ⋯⟩ = c • ⟨x, hfx⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f g : E →ₛₗ.[σ] F
c : R
x : E
x✝ : x ∈ {x | ∃ (hf : x ∈ f.domain) (hg : x ∈ g.domain), ↑f ⟨x, hf⟩ = ↑g ⟨x, hg⟩}
hfx : x ∈ f.domain
hgx : x ∈ g.domain
hx : ↑f ⟨x, hfx⟩ = ↑g ⟨x, hgx⟩
⊢ ↑f ⟨c • x, ⋯⟩ = ↑g ⟨c • x, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
⊢ ∀ (a b c : E →ₛₗ.[σ] F), a ≤ b → a ≤ c → a ≤ { domain := b.eqLocus c, toFun := b.toFun ∘ₛₗ inclusion ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f : E →ₛₗ.[σ] F
x : ↥⊥.domain
y : ↥f.domain
h : ↑x = ↑y
⊢ ↑⊥ x = ↑f y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n : ℕ
⊢ finrank K V ≤ n ↔ Module.rank K V ≤ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n : ℕ
⊢ finrank K V < n ↔ Module.rank K V < ↑n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
⊢ Disjoint (span K (Set.range ⇑bS)) S.subtype.ker

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
⊢ LinearIndepOn K id (Subtype.val '' Basis.ofVectorSpaceIndex K ↥S)

Tactic:
by

Goal state after: