mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
⊢ inverse Q hfg hg ∘ₗ rTensor Q g = (rTensor Q f).range.mkQ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
h : P → N
hgh : RightInverse h ⇑g
y : N ⊗[R] Q ⧸ (rTensor Q f).range
⊢ (inverse_of_rightInverse Q hfg hgh) ((toFun Q hfg).toFun y) = y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
h : P → N
hgh : RightInverse h ⇑g
z : P ⊗[R] Q
⊢ (toFun Q hfg).toFun ((inverse_of_rightInverse Q hfg hgh) z) = z

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
⊢ Exact ⇑(rTensor Q f) ⇑(rTensor Q g)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
N : Submodule R M
⊢ (rTensor Q N.mkQ).ker = (rTensor Q N.subtype).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
I : Ideal R
⊢ ((TensorProduct.mk R (R ⧸ I) Q) 1).ker = I • ⊤

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
⊢ 0 = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) 0).sum

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M
y : N
⊢ x ⊗ₜ[R] y = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) {(x, y)}).sum

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : Module R M
inst✝⁷ : Module R N
inst✝⁶ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
M' : Type u_6
N' : Type u_7
P' : Type u_8
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup N'
inst✝³ : AddCommGroup P'
inst✝² : Module R M'
inst✝¹ : Module R N'
inst✝ : Module R P'
f' : M' →ₗ[R] N'
g' : N' →ₗ[R] P'
hfg' : Exact ⇑f' ⇑g'
hg' : Surjective ⇑g'
⊢ (map g g').ker = (lTensor N f').range ⊔ (rTensor N' f).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x y : M ⊗[R] N
Sx : Multiset (M × N)
hx : x = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) Sx).sum
Sy : Multiset (M × N)
hy : y = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) Sy).sum
⊢ x + y = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) (Sx + Sy)).sum

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M ⊗[R] N
⊢ ∃ S, x = (Multiset.map (fun i ↦ i.1 ⊗ₜ[R] i.2) S).sum

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
⊢ 0 = Finsupp.sum 0 fun m n ↦ m ⊗ₜ[R] n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M
y : N
⊢ x ⊗ₜ[R] y = (Finsupp.single x y).sum fun m n ↦ m ⊗ₜ[R] n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x y : M ⊗[R] N
Sx : M →₀ N
hx : x = Sx.sum fun m n ↦ m ⊗ₜ[R] n
Sy : M →₀ N
hy : y = Sy.sum fun m n ↦ m ⊗ₜ[R] n
⊢ ∀ (a : M), a ⊗ₜ[R] 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M ⊗[R] N
⊢ ∃ S, x = S.sum fun m n ↦ m ⊗ₜ[R] n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M ⊗[R] N
⊢ ∃ S, x = S.sum fun n m ↦ m ⊗ₜ[R] n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M ⊗[R] N
⊢ ∃ S, x = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal A
a : ↥(Submodule.restrictScalars R I)
b : B
this : ↑a ⊗ₜ[R] b = 1 ⊗ₜ[R] b * ↑a ⊗ₜ[R] 1
⊢ ↑a ⊗ₜ[R] b ∈ ↑(Submodule.restrictScalars R (map includeLeft I))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal A
⊢ Submodule.restrictScalars R (map includeLeft I) = (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
s : Set (M ⊗[R] N)
hs : s.Finite
⊢ ∃ M' N', Module.Finite R ↥M' ∧ Module.Finite R ↥N' ∧ s ⊆ ↑(mapIncl M' N').range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
a : A
b : ↥(Submodule.restrictScalars R I)
this : a ⊗ₜ[R] ↑b = a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] ↑b
⊢ a ⊗ₜ[R] ↑b ∈ ↑(Submodule.restrictScalars R (map includeRight I))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
s : Set (M ⊗[R] N)
hs : s.Finite
⊢ ∃ M', Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor N M'.subtype).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
s : Set (M ⊗[R] N)
hs : s.Finite
⊢ ∃ N', Module.Finite R ↥N' ∧ s ⊆ ↑(LinearMap.lTensor M N'.subtype).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
⊢ Submodule.restrictScalars R (map includeRight I) = (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
S : Type u_4
inst✝³ : CommSemiring S
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
I : Ideal B
⊢ ((lTensor S A) (↑R (Submodule.subtype I))).range = Submodule.restrictScalars S (Ideal.map includeRight I)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_4
inst✝⁶ : CommRing R
A : Type u_6
C : Type u_8
D : Type u_9
inst✝⁵ : Ring A
inst✝⁴ : Ring C
inst✝³ : Ring D
inst✝² : Algebra R A
inst✝¹ : Algebra R C
inst✝ : Algebra R D
g : C →ₐ[R] D
hg : Function.Surjective ⇑g
⊢ RingHom.ker (map (AlgHom.id R A) g) = Ideal.map includeRight (RingHom.ker g)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
M₁ : Submodule R M
N₁ : Submodule R N
s : Set (↥M₁ ⊗[R] ↥N₁)
hs : s.Finite
⊢ ∃ M' N',
    ∃ (hM : M' ≤ M₁) (hN : N' ≤ N₁),
      Module.Finite R ↥M' ∧ Module.Finite R ↥N' ∧ s ⊆ ↑(map (inclusion hM) (inclusion hN)).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
M₁ : Submodule R M
N₁ : Submodule R N
s : Set (↥M₁ ⊗[R] ↥N₁)
hs : s.Finite
⊢ ∃ M', ∃ (hM : M' ≤ M₁), Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor (↥N₁) (inclusion hM)).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
A : Type u_6
B : Type u_7
C : Type u_8
inst✝⁹ : Ring A
inst✝⁸ : Ring B
inst✝⁷ : Ring C
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra R C
inst✝³ : Algebra S A
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
hf : Function.Surjective ⇑f
⊢ RingHom.ker (map f (AlgHom.id R C)) = Ideal.map includeLeft (RingHom.ker f)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
M₁ : Submodule R M
N₁ : Submodule R N
s : Set (↥M₁ ⊗[R] ↥N₁)
hs : s.Finite
⊢ ∃ N', ∃ (hN : N' ≤ N₁), Module.Finite R ↥N' ∧ s ⊆ ↑(LinearMap.lTensor (↥M₁) (inclusion hN)).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
⊢ 0 = ∑ j, (fun a ↦ ⋯.elim a) j ⊗ₜ[R] (fun a ↦ ⋯.elim a) j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M
y : N
⊢ x ⊗ₜ[R] y = ∑ j, (fun x_1 ↦ x) j ⊗ₜ[R] (fun x ↦ y) j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Finiteness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
x : M ⊗[R] N
⊢ ∃ k m n, x = ∑ j, m j ⊗ₜ[R] n j

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Adjoin.FGBaseChange

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A ⊗[R] B
⊢ ∃ C, C.FG ∧ x ∈ Subalgebra.baseChange A C

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : MonoidWithZero R
inst✝² : MonoidWithZero S
r : R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : MonoidWithZeroHomClass F R S
hr : IsNilpotent r
f : F
⊢ IsNilpotent (f r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : MonoidWithZero R
inst✝² : MonoidWithZero S
r : R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : MonoidWithZeroHomClass F R S
f : F
hf : Function.Injective ⇑f
x✝ : IsNilpotent (f r)
k : ℕ
hk : f r ^ k = 0
⊢ f (r ^ k) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
inst✝ : MonoidWithZero R
r u : R
hu : IsUnit u
h_comm : Commute r u
n : ℕ
⊢ (r * u) ^ n = 0 ↔ r ^ n = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝² : Zero R
inst✝¹ : Pow R ℕ
inst✝ : Subsingleton R
s : Set ℕ := {k | x ^ k = 0}
this : s = univ
⊢ nilpotencyClass x = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝² : Zero R
inst✝¹ : Pow R ℕ
inst✝ : Subsingleton R
⊢ nilpotencyClass x = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝¹ : Zero R
inst✝ : Pow R ℕ
hx : 0 < nilpotencyClass x
⊢ IsNilpotent x

Tactic:
let s : Set ℕ := {k | x ^ k = 0}
  change s.Nonempty
  change 0 < sInf s at hx
  by_contra contra
  simp [not_nonempty_iff_eq_empty.mp contra] at hx

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝¹ : Zero R
inst✝ : Pow R ℕ
hx : 0 < nilpotencyClass x
⊢ IsNilpotent x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝ : MonoidWithZero R
k : ℕ
⊢ nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
⊢ 0 ^ (0 + 1) = 0 ∧ 0 ^ 0 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
⊢ 0 < nilpotencyClass x ↔ IsNilpotent x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x : R
inst✝ : MonoidWithZero R
hx : nilpotencyClass x = 1
⊢ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : MonoidWithZero R
inst✝³ : MonoidWithZero S
F : Type u_3
inst✝² : FunLike F R S
inst✝¹ : MonoidWithZeroHomClass F R S
f : F
hf : Function.Injective ⇑f
inst✝ : IsReduced S
⊢ IsReduced R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
h : IsNilpotent x
⊢ IsNilpotent (x * y)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
h : IsNilpotent y
⊢ IsNilpotent (x * y)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
A : Type u_6
B : Type u_7
C : Type u_8
D : Type u_9
inst✝¹¹ : Ring A
inst✝¹⁰ : Ring B
inst✝⁹ : Ring C
inst✝⁸ : Ring D
inst✝⁷ : Algebra R A
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra R C
inst✝⁴ : Algebra R D
inst✝³ : Algebra S A
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
g : C →ₐ[R] D
hf : Function.Surjective ⇑f
hg : Function.Surjective ⇑g
⊢ RingHom.ker (map f g) = Ideal.map includeLeft (RingHom.ker f) ⊔ Ideal.map includeRight (RingHom.ker g)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Central.TensorProduct

Goal state before:
pp: K : Type u_1
B : Type u_2
C : Type u_3
inst✝⁴ : CommSemiring K
inst✝³ : Semiring B
inst✝² : Semiring C
inst✝¹ : Algebra K B
inst✝ : Algebra K C
⊢ Subalgebra.map includeLeft (Subalgebra.center K B) ≤ Subalgebra.center K (B ⊗[K] C)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Central.TensorProduct

Goal state before:
pp: K : Type u_1
B : Type u_2
C : Type u_3
inst✝⁴ : CommSemiring K
inst✝³ : Semiring B
inst✝² : Semiring C
inst✝¹ : Algebra K B
inst✝ : Algebra K C
x : B ⊗[K] C
hx : x ∈ Subalgebra.map includeRight (Subalgebra.center K C)
⊢ x ∈ Subalgebra.center K (B ⊗[K] C)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
f : N →ₗ[R] P
h :
  ∀ (N' : Submodule R N) (P' : Submodule R P),
    N'.FG → P'.FG → ∀ (h : N' ≤ comap f P'), Injective ⇑(rTensor M (f.restrict h))
x y : N ⊗[R] M
eq : (rTensor M f) x = (rTensor M f) y
⊢ {x, y}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
f : N →ₗ[R] P
h :
  ∀ (N' : Submodule R N) (P' : Submodule R P),
    N'.FG → P'.FG → ∀ (h : N' ≤ comap f P'), Injective ⇑(rTensor M (f.restrict h))
x y : N ⊗[R] M
eq : (rTensor M f) x = (rTensor M f) y
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
Q : Type u_3
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : AddCommMonoid Q
inst✝ : Module R Q
f : N →ₗ[R] P
hf : Injective ⇑f
e : P ≃ₗ[R] Q
⊢ Injective ⇑(rTensor M f) ↔ Injective ⇑(rTensor M (↑e ∘ₗ f).range.subtype)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
inst✝² : AddCommMonoid P
inst✝¹ : Module R P
inst✝ : Flat R M
f : N →ₗ[R] P
hf : Injective ⇑f
⊢ Injective ⇑(rTensor M f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Small.{v', u} R
⊢ Flat R M →
    ∀ ⦃N N' : Type v'⦄ [inst : AddCommMonoid N] [inst_1 : AddCommMonoid N'] [inst_2 : Module R N] [inst_3 : Module R N']
      (f : N →ₗ[R] N'), Injective ⇑f → Injective ⇑(rTensor M f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Small.{v', u} R
h :
  ∀ ⦃N N' : Type v'⦄ [inst : AddCommMonoid N] [inst_1 : AddCommMonoid N'] [inst_2 : Module R N] [inst_3 : Module R N']
    (f : N →ₗ[R] N'), Injective ⇑f → Injective ⇑(rTensor M f)
P : Type u
x✝⁴ : AddCommMonoid P
x✝³ : Module R P
x✝² : Module.Finite R P
x✝¹ : Submodule R P
x✝ : x✝¹.FG
⊢ Injective ⇑(rTensor M x✝¹.subtype)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Small.{v', u} R
⊢ Flat R M ↔
    ∀ ⦃N N' : Type v'⦄ [inst : AddCommMonoid N] [inst_1 : AddCommMonoid N'] [inst_2 : Module R N] [inst_3 : Module R N']
      (f : N →ₗ[R] N'), Injective ⇑f → Injective ⇑(lTensor M f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Flat R M ↔
    ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Injective ⇑(lTensor M N.subtype)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
Q : Type u_3
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : AddCommMonoid Q
inst✝ : Module R Q
x✝⁴ : Type u
x✝³ : AddCommMonoid x✝⁴
x✝² : Module R x✝⁴
x✝¹ : Module.Finite R x✝⁴
I : Submodule R x✝⁴
x✝ : I.FG
⊢ Injective ⇑(rTensor R I.subtype)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
f : Flat R M
i : N →ₗ[R] M
r : M →ₗ[R] N
h : r ∘ₗ i = LinearMap.id
⊢ Flat R N

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : Flat R M
e : N ≃ₗ[R] M
⊢ ↑e.symm ∘ₗ ↑e = LinearMap.id

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (M i)
inst✝ : (i : ι) → Module R (M i)
⊢ Flat R (⨁ (i : ι), M i) ↔ ∀ (i : ι), Flat R (M i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P : Type u_2
Q : Type u_3
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : AddCommMonoid Q
inst✝ : Module R Q
ι : Type v
⊢ Flat R (ι →₀ R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.Basic

Goal state before:
pp: R : Type u
M : Type v
N : Type u_1
P✝ : Type u_2
Q : Type u_3
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
inst✝¹¹ : AddCommMonoid N
inst✝¹⁰ : Module R N
inst✝⁹ : AddCommMonoid P✝
inst✝⁸ : Module R P✝
inst✝⁷ : AddCommMonoid Q
inst✝⁶ : Module R Q
S : Type u_4
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : Flat S M
inst✝ : Flat R N
P : Type u_4
x✝¹ : AddCommMonoid P
x✝ : Module S P
I : Submodule S P
⊢ Injective ⇑(rTensor (M ⊗[R] N) I.subtype)

Tactic:
by

Goal state after: