mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι → M
f : ι → Dual R M
h1 : Pairwise fun i j ↦ (f i) (v j) = 0
h2 : ∀ (i : ι), (f i) (v i) = 1
s : Finset ι
g : ι → R
hrel : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
aux : ∀ j ∈ s, j ≠ i → g j * (f i) (v j) = 0
⊢ i ∉ s → g i * (f i) (v i) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι → M
f : ι → Dual R M
h1 : Pairwise fun i j ↦ (f i) (v j) = 0
h2 : ∀ (i : ι), (f i) (v i) = 1
⊢ LinearIndependent R v

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ φ ψ : AddChar A M
h : φ.toFun = ψ.toFun
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : Monoid M
f : A → M
map_zero_eq_one' : f 0 = 1
map_add_eq_mul' : ∀ (a b : A), f (a + b) = f a * f b
⊢ ⇑{ toFun := f, map_zero_eq_one' := map_zero_eq_one', map_add_eq_mul' := map_add_eq_mul' } = f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : Monoid M
ψ : AddChar A M
⊢ ⇑ψ = 1 ↔ ψ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_3
N : Type u_4
inst✝² : AddMonoid A
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ : AddChar A M
hψ : Surjective ⇑ψ
⊢ Injective fun f ↦ f.compAddChar ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: B : Type u_2
M : Type u_3
N : Type u_4
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
f : M →* N
hf : Injective ⇑f
⊢ Injective fun ψ ↦ f.compAddChar ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
B : Type u_2
M : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : Monoid M
f : A →+ B
hf : Surjective ⇑f
⊢ Injective fun ψ ↦ ψ.compAddMonoidHom f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
B : Type u_2
M : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : Monoid M
ψ : AddChar B M
hψ : Injective ⇑ψ
⊢ Injective fun f ↦ ψ.compAddMonoidHom f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
s : Finset ι
ψ : ι → AddChar A M
⊢ ⇑(∏ i ∈ s, ψ i) = ∏ i ∈ s, ⇑(ψ i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
s : Finset ι
ψ : ι → AddChar A M
⊢ ⇑(∑ i ∈ s, ψ i) = ∏ i ∈ s, ⇑(ψ i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
s : Finset ι
ψ : ι → AddChar A M
a : A
⊢ (∏ i ∈ s, ψ i) a = ∏ i ∈ s, (ψ i) a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
s : Finset ι
ψ : ι → AddChar A M
a : A
⊢ (∑ i ∈ s, ψ i) a = ∏ i ∈ s, (ψ i) a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
φ ψ : AddChar A M
⊢ toMonoidHomEquiv.toFun (φ * ψ) = toMonoidHomEquiv.toFun φ * toMonoidHomEquiv.toFun ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
φ ψ : Additive (AddChar A M)
⊢ toAddMonoidHomEquiv.toFun (φ + ψ) = toAddMonoidHomEquiv.toFun φ + toAddMonoidHomEquiv.toFun ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
a : A
⊢ 0 a = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
a : A
⊢ ∀ (a_1 b : AddChar A M), (a_1 + b) a = a_1 a * b a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
⊢ { toFun := fun ψ ↦ ψ 0, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ } = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: ι : Type u_1
A : Type u_2
M : Type u_3
inst✝¹ : AddMonoid A
inst✝ : CommMonoid M
x✝¹ x✝ : A
⊢ { toFun := fun ψ ↦ ψ (x✝¹ + x✝), map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ } =
    { toFun := fun ψ ↦ ψ x✝¹, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ } +
      { toFun := fun ψ ↦ ψ x✝, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
R : Type u_2
inst✝⁴ : AddGroup A
inst✝³ : Fintype A
inst✝² : CommSemiring R
inst✝¹ : IsDomain R
ψ : AddChar A R
inst✝ : Decidable (ψ = 0)
⊢ ∑ a, ψ a = if ψ = 0 then ↑(card A) else 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
R : Type u_2
inst✝⁴ : AddGroup A
inst✝³ : Fintype A
inst✝² : CommSemiring R
inst✝¹ : IsDomain R
ψ : AddChar A R
inst✝ : CharZero R
⊢ ∑ x, ψ x = 0 ↔ ψ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddCommGroup A
inst✝ : CommMonoid M
ψ : AddChar A M
⊢ ψ.compAddMonoidHom negAddMonoidHom * ψ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddGroup A
inst✝ : DivisionMonoid M
ψ : AddChar A M
a : A
⊢ ψ (-a) = (ψ a)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddCommGroup A
inst✝ : DivisionCommMonoid M
ψ : AddChar A M
a : A
⊢ ψ⁻¹ a = (ψ a)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddCommGroup A
inst✝ : DivisionCommMonoid M
ψ χ : AddChar A M
a : A
⊢ (ψ / χ) a = ψ a / χ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddCommGroup A
inst✝ : DivisionCommMonoid M
ψ χ : AddChar A M
a : A
⊢ (ψ - χ) a = ψ a / χ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M : Type u_2
inst✝¹ : AddCommGroup A
inst✝ : DivisionCommMonoid M
n : ℤ
ψ : AddChar A M
a : A
⊢ (n • ψ) a = ψ a ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: A : Type u_1
M₀ : Type u_2
inst✝² : AddGroup A
inst✝¹ : MonoidWithZero M₀
inst✝ : Nontrivial M₀
ψ : AddChar A M₀
h : ψ 0 = 0 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
R : Type u_2
M : Type u_4
inst✝³ : DecidableEq ι
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ι → M
hf : LinearIndependent R f
i : ι
m : M
hg : ∃ r ∈ nonZeroDivisors R, ∃ l, l i ∈ nonZeroDivisors R ∧ r • m = (linearCombination R f) l
⊢ LinearIndependent R (Function.update f i m)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
y : V
a : K
z : V
hz : z ∈ span K s
h : a • y + z ∉ span K s
⊢ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
⊢ ψ⁻¹ = ψ.mulShift (-1)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
x y : V
⊢ x ∈ span K (insert y s) → x ∉ span K s → y ∈ span K (insert x s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
n : ℕ
x : R
⊢ (ψ.mulShift ↑n) x = ψ x ^ n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v : ι → V
s : Set ι
x : ι
hs : LinearIndepOn K v s
hx : v x ∉ span K (v '' s)
⊢ LinearIndepOn K v (insert x s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
n : ℕ
⊢ ψ ^ n = ψ.mulShift ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
r s : R
⊢ ψ.mulShift r * ψ.mulShift s = ψ.mulShift (r + s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
r s : R
⊢ (ψ.mulShift r).mulShift s = ψ.mulShift (r * s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
⊢ ψ.mulShift 0 = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
⊢ ψ.mulShift 1 = ψ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.AddChar

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
u : R
hu : IsUnit u
⊢ ψ.mulShift u = 1 ↔ ψ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Lift

Goal state before:
pp: k : Type u₁
G : Type u₂
R : Type u_2
inst✝⁴ : Semiring k
inst✝³ : Mul G
inst✝² : Semiring R
g_hom : Type u_6
inst✝¹ : FunLike g_hom G R
inst✝ : MulHomClass g_hom G R
f : k →+* R
g : g_hom
a b : k[G]
h_comm : ∀ {x y : G}, y ∈ a.coeff.support → Commute (f (b.coeff x)) (g y)
⊢ (liftNC ↑f ⇑g) (a * b) = (liftNC ↑f ⇑g) a * (liftNC ↑f ⇑g) b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v : ι → V
x : V
⊢ (LinearIndependent K fun o ↦ o.casesOn' x v) ↔ LinearIndependent K v ∧ x ∉ span K (range v)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Lift

Goal state before:
pp: k : Type u₁
G : Type u₂
R : Type u_2
inst✝⁴ : NonAssocSemiring R
inst✝³ : Semiring k
inst✝² : One G
g_hom : Type u_6
inst✝¹ : FunLike g_hom G R
inst✝ : OneHomClass g_hom G R
f : k →+* R
g : g_hom
⊢ (liftNC ↑f ⇑g) 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v : Option ι → V
⊢ LinearIndependent K v ↔ LinearIndependent K (v ∘ some) ∧ v none ∉ span K (range (v ∘ some))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set ι
a : ι
f : ι → V
has : a ∉ s
⊢ LinearIndepOn K f (insert a s) ↔ LinearIndepOn K f s ∧ f a ∉ span K (f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
x : V
hxs : x ∉ s
⊢ LinearIndepOn K id s ∧ id x ∉ span K (id '' s) ↔ LinearIndepOn K id s ∧ x ∉ span K s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set ι
a : ι
f : ι → V
⊢ LinearIndepOn K f (insert a s) ↔ LinearIndepOn K f s ∧ (f a ∈ span K (f '' s) → a ∈ s)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
a : V
s : Set V
⊢ LinearIndepOn K id (insert a s) ↔ LinearIndepOn K id s ∧ (a ∈ span K s → a ∈ s)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set ι
a : ι
f : ι → V
h : LinearIndepOn K f s
⊢ f a ∈ span K (f '' s) ↔ LinearIndepOn K f (insert a s) → a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set ι
a : ι
f : ι → V
h : LinearIndepOn K f s
⊢ f a ∉ span K (f '' s) ↔ LinearIndepOn K f (insert a s) ∧ a ∉ s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
a : V
h : LinearIndepOn K id s
⊢ a ∈ span K s ↔ LinearIndepOn K id (insert a s) → a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Lift

Goal state before:
pp: k : Type u₁
G : Type u₂
R : Type u_2
inst✝⁴ : Semiring k
inst✝³ : Add G
inst✝² : Semiring R
g_hom : Type u_6
inst✝¹ : FunLike g_hom (Multiplicative G) R
inst✝ : MulHomClass g_hom (Multiplicative G) R
f : k →+* R
g : g_hom
a b : k[G]
h_comm : ∀ {x y : G}, y ∈ a.coeff.support → Commute (f (b.coeff x)) (g (Multiplicative.ofAdd y))
⊢ (liftNC ↑f ⇑g) (a * b) = (liftNC ↑f ⇑g) a * (liftNC ↑f ⇑g) b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝² : Monoid S
inst✝¹ : Semiring R
inst✝ : DistribMulAction S R
f : M → N
s : S
x : R[M]
⊢ mapDomain f (s • x) = s • mapDomain f x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Module R S
f : M → N
s : S
m : M
⊢ (mapDomainLinearMap R S f) (single m s) = single (f m) s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Module R S
f : M → N
g : N → O
⊢ mapDomainLinearMap R S (g ∘ f) = mapDomainLinearMap R S g ∘ₗ mapDomainLinearMap R S f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Module R S
e : M ≃ N
s : S
m : M
⊢ (mapDomainLinearEquiv R S e) (single m s) = single (e m) s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Module R S
e₁ : M ≃ N
e₂ : N ≃ O
⊢ mapDomainLinearEquiv R S (e₁.trans e₂) = mapDomainLinearEquiv R S e₁ ≪≫ₗ mapDomainLinearEquiv R S e₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
G : Type u_6
inst✝⁴ : Semiring R
inst✝³ : Semiring S
inst✝² : Module R S
s t : Set M
x✝ : S[M]
inst✝¹ : One M
inst✝ : Subsingleton M
r : R
x : S[M]
⊢ (coeffAddEquiv.trans (uniqueAddEquiv 1)).toFun (r • x) =
    (RingHom.id R) r • (coeffAddEquiv.trans (uniqueAddEquiv 1)).toFun x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
a : V
h : LinearIndepOn K id s
⊢ a ∉ span K s ↔ LinearIndepOn K id (insert a s) ∧ a ∉ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Module R S
s : Set M
x : S[M]
⊢ x ∈ supported R S s ↔ ∀ m ∉ s, x.coeff m = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝ : Semiring R
s : Set M
⊢ supported R R s = Submodule.span R ((fun m ↦ single m 1) '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝¹ : Semiring R
inst✝ : Nontrivial R
m : M
s : Set M
⊢ single m 1 ∈ Submodule.span R ((fun x ↦ single x 1) '' s) ↔ m ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S✝ : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
G : Type u_6
inst✝² : Semiring S✝
inst✝¹ : Monoid R
inst✝ : DistribMulAction R S✝
a : M
S : R
m : S✝
⊢ (↑(singleAddHom a)).toFun (S • m) = (MonoidHom.id R) S • (↑(singleAddHom a)).toFun m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝¹ : Semiring R
inst✝ : MulOneClass M
m : M
r : R
⊢ r • (of R M) m = single m r

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
hy : ∀ (a : K), a • x ≠ y
⊢ y ∉ K ∙ x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : MulOneClass M
s : Set M
m : M
inst✝ : Nontrivial R
h : (of R M) m ∈ Submodule.span R ↑(Submonoid.closure (⇑(of R M) '' s))
⊢ m ∈ Submonoid.closure s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
hy : ∀ (a : K), a • x ≠ y
⊢ LinearIndepOn K id {x, y}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : MulOneClass M
f : S →+* R
g : M →* R
c : S
φ : S[M]
⊢ (liftNC ↑f ⇑g) (c • φ) = f c * (liftNC ↑f ⇑g) φ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: ι : Type u'
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
i j : ι
v : ι → V
hij : i ≠ j
hi : v i ≠ 0
⊢ LinearIndepOn K v {i, j} ↔ ∀ (c : K), c • v i ≠ v j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
G : Type u_6
inst✝³ : Semiring S
inst✝² : DistribSMul R S
inst✝¹ : Mul M
inst✝ : IsScalarTower R S S
t : R
a b : S[M]
⊢ (t • a) • b = t • a • b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
G : Type u_6
inst✝³ : Semiring S
inst✝² : DistribSMul R S
inst✝¹ : Mul M
inst✝ : SMulCommClass R S S
t : R
a b : S[M]
⊢ t • a • b = a • t • b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
⊢ 0 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
⊢ ![x, y] 0 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
⊢ LinearIndependent K ![x, y] ↔ ∀ (a : K), a • x ≠ y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : V
n : ℕ
v : Fin n → V
⊢ LinearIndependent K (Fin.cons x v) ↔ LinearIndependent K v ∧ x ∉ span K (range v)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
G : Type u_6
inst✝⁵ : CommSemiring S
inst✝⁴ : Monoid M
V : Type u_7
inst✝³ : AddCommMonoid V
inst✝² : Module S V
inst✝¹ : Module S[M] V
inst✝ : IsScalarTower S S[M] V
W : Submodule S V
h : ∀ (g : M), ∀ v ∈ W, (of S M) g • v ∈ W
f : S[M]
v : V
hv : v ∈ ↑W
⊢ f • v ∈ ↑W

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : V
n : ℕ
v : Fin n → V
⊢ LinearIndependent K (Fin.snoc v x) ↔ LinearIndependent K v ∧ x ∉ span K (range v)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddMonoid M
inst✝ : Nontrivial R
m : M
s : Set M
h : of' R M m ∈ Submodule.span R ↑(Submonoid.closure (of' R M '' s))
this : Multiplicative.ofAdd m ∈ Submonoid.closure (⇑Multiplicative.toAdd ⁻¹' s)
⊢ m ∈ AddSubmonoid.closure s

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: R : Type u_6
K : Type u_7
M : Type u_8
inst✝⁷ : CommRing R
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R K
inst✝³ : Module K M
inst✝² : Module R M
inst✝¹ : IsScalarTower R K M
inst✝ : FaithfulSMul R K
n : ℕ
v : Fin n → M
hv : LinearIndependent R v
x : M
hx : x ∉ span K (range v)
c : R
y : M
hcy : y ∈ span R (range v)
heq : c • x + y = 0
hc : ¬c = 0
⊢ (algebraMap R K) c ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: R : Type u_6
K : Type u_7
M : Type u_8
inst✝⁷ : CommRing R
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R K
inst✝³ : Module K M
inst✝² : Module R M
inst✝¹ : IsScalarTower R K M
inst✝ : FaithfulSMul R K
n : ℕ
v : Fin n → M
hv : LinearIndependent R v
x : M
hx : x ∉ span K (range v)
⊢ LinearIndependent R (Fin.snoc v x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas

Goal state before:
pp: K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
n : ℕ
v : Fin (n + 1) → V
⊢ LinearIndependent K v ↔ LinearIndependent K (Fin.tail v) ∧ v 0 ∉ span K (range (Fin.tail v))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddMonoid M
inst✝ : Nontrivial R
m : M
s : Set M
h : of' R M m ∈ Submodule.span R ↑(Submonoid.closure (of' R M '' s))
⊢ m ∈ AddSubmonoid.closure s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : AddZeroClass M
f : S →+* R
g : Multiplicative M →* R
c : S
φ : S[M]
⊢ (liftNC ↑f ⇑g) (c • φ) = f c * (liftNC ↑f ⇑g) φ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Group.Pointwise.Set.ListOfFn

Goal state before:
pp: α : Type u_1
inst✝ : Monoid α
n : ℕ
a : α
s : Fin n → Set α
⊢ a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a

Tactic:
by

Goal state after: