mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' M'' : Submodule R M
N' : Submodule S N
m : M
s : ↥p
hn : Function.uncurry (IsLocalizedModule.mk' f) (m, s) ∈ N'
⊢ f m ∈ N'

Tactic:
← IsLocalizedModule.mk'_cancel' _ _ s,

Goal state after:
R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' M'' : Submodule R M
N' : Submodule S N
m : M
s : ↥p
hn : Function.uncurry (IsLocalizedModule.mk' f) (m, s) ∈ N'
⊢ s • IsLocalizedModule.mk' f m s ∈ N'
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
⊢ ⊥.colon ↑N = N.annihilator

Tactic:
ext x

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
x : R
⊢ x ∈ ⊥.colon ↑N ↔ x ∈ N.annihilator
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
x : N
a✝ : x ∈ ⊤
⊢ x ∈ localized₀ p f ⊤

Tactic:
obtain ⟨⟨x, s⟩, rfl⟩ := IsLocalizedModule.mk'_surjective p f x

Goal state after:
R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
x : M
s : ↥p
a✝ : Function.uncurry (IsLocalizedModule.mk' f) (x, s) ∈ ⊤
⊢ Function.uncurry (IsLocalizedModule.mk' f) (x, s) ∈ localized₀ p f ⊤
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : Module R M
inst✝⁶ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝⁵ : IsLocalizedModule p f
P : Type u_5
inst✝⁴ : AddCommMonoid P
inst✝³ : Module R P
Q : Type u_6
inst✝² : AddCommMonoid Q
inst✝¹ : Module R Q
f' : P →ₗ[R] Q
inst✝ : IsLocalizedModule p f'
g : M →ₗ[R] P
⊢ ((IsLocalizedModule.map p f f') g).ker = localized₀ p f g.ker

Tactic:
ext x

Goal state after:
R : Type u_1
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid N
inst✝⁷ : Module R M
inst✝⁶ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝⁵ : IsLocalizedModule p f
P : Type u_5
inst✝⁴ : AddCommMonoid P
inst✝³ : Module R P
Q : Type u_6
inst✝² : AddCommMonoid Q
inst✝¹ : Module R Q
f' : P →ₗ[R] Q
inst✝ : IsLocalizedModule p f'
g : M →ₗ[R] P
x : N
⊢ x ∈ ((IsLocalizedModule.map p f f') g).ker ↔ x ∈ localized₀ p f g.ker
Module: Mathlib.RingTheory.Localization.Ideal

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
J : Ideal S
x : S
hJ : x ∈ J
r : R
s : ↥M
hx : mk' S r s = x
⊢ x ∈ Ideal.map (algebraMap R S) (Ideal.under R J)

Tactic:
rw [← hx] at hJ ⊢

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
J : Ideal S
x : S
r : R
s : ↥M
hJ : mk' S r s ∈ J
hx : mk' S r s = x
⊢ mk' S r s ∈ Ideal.map (algebraMap R S) (Ideal.under R J)
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
inst✝⁵ : CommSemiring P
inst✝⁴ : CommSemiring Q
M : Submonoid R
T : Submonoid P
inst✝³ : Algebra R S
inst✝² : Algebra P Q
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization T Q
g : R →+* P
hT : Submonoid.map g M = T
⊢ RingHom.ker (map Q g ⋯) = Ideal.map (algebraMap R S) (RingHom.ker g)

Tactic:
ext x

Goal state after:
R : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
inst✝⁵ : CommSemiring P
inst✝⁴ : CommSemiring Q
M : Submonoid R
T : Submonoid P
inst✝³ : Algebra R S
inst✝² : Algebra P Q
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization T Q
g : R →+* P
hT : Submonoid.map g M = T
x : S
⊢ x ∈ RingHom.ker (map Q g ⋯) ↔ x ∈ Ideal.map (algebraMap R S) (RingHom.ker g)
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ IsLocalization (Submonoid.map f.toRingHom (algebraMapSubmonoid A M)) Bₚ

Tactic:
rw [AlgHom.toRingHom_eq_coe, ← Submonoid.map_coe_toMonoidHom, AlgHom.toRingHom_toMonoidHom,
    Submonoid.map_coe_toMonoidHom, algebraMapSubmonoid_map_eq M f]

Goal state after:
R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ IsLocalization (algebraMapSubmonoid B M) Bₚ
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ ↑R
        ((IsLocalizedModule.mapExtendScalars M (IsScalarTower.toAlgHom R A Aₚ).toLinearMap
            (IsScalarTower.toAlgHom R B Bₚ).toLinearMap Rₚ)
          f.toLinearMap) ∘ₗ
      (IsScalarTower.toAlgHom R A Aₚ).toLinearMap =
    ↑R (mapₐ M Rₚ Aₚ Bₚ f).toLinearMap ∘ₗ (IsScalarTower.toAlgHom R A Aₚ).toLinearMap

Tactic:
ext x

Goal state after:
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
x : A
⊢ (↑R
          ((IsLocalizedModule.mapExtendScalars M (IsScalarTower.toAlgHom R A Aₚ).toLinearMap
              (IsScalarTower.toAlgHom R B Bₚ).toLinearMap Rₚ)
            f.toLinearMap) ∘ₗ
        (IsScalarTower.toAlgHom R A Aₚ).toLinearMap)
      x =
    (↑R (mapₐ M Rₚ Aₚ Bₚ f).toLinearMap ∘ₗ (IsScalarTower.toAlgHom R A Aₚ).toLinearMap) x
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ (IsLocalizedModule.map M (IsScalarTower.toAlgHom R A Aₚ).toLinearMap (IsScalarTower.toAlgHom R B Bₚ).toLinearMap)
      f.toLinearMap =
    ↑R (mapₐ M Rₚ Aₚ Bₚ f).toLinearMap

Tactic:
ext x

Goal state after:
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
x : Aₚ
⊢ ((IsLocalizedModule.map M (IsScalarTower.toAlgHom R A Aₚ).toLinearMap (IsScalarTower.toAlgHom R B Bₚ).toLinearMap)
        f.toLinearMap)
      x =
    (↑R (mapₐ M Rₚ Aₚ Bₚ f).toLinearMap) x
Module: Mathlib.RingTheory.Localization.LocalizationLocalization

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
M N : Submonoid R
inst✝ : IsLocalization M S
h : M ≤ N
h' : ∀ (x : ↥N), ∃ m, m * ↑x ∈ M
x✝¹ x✝ : R
⊢ (∃ c, ↑c * x✝¹ = ↑c * x✝) → ∃ c, ↑c * x✝¹ = ↑c * x✝

Tactic:
exact fun ⟨x, hx⟩ => ⟨⟨_, h x.prop⟩, hx⟩

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Exactness

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)
ι : Type u_9
v : ι → M
H : ∀ (P : Ideal R) [inst : P.IsMaximal], LinearIndependent (Rₚ P) (⇑(f P) ∘ v)
P : Ideal R
x✝ : P.IsMaximal
⊢ Function.Injective ⇑(↑R (Finsupp.linearCombination (Rₚ P) (⇑(f P) ∘ v)))

Tactic:
exact H P

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Exactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
L : Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : Module R N
inst✝¹⁰ : AddCommMonoid L
inst✝⁹ : Module R L
s : Set R
spn : Ideal.span s = ⊤
Mₚ : ↑s → Type u_5
inst✝⁸ : (r : ↑s) → AddCommMonoid (Mₚ r)
inst✝⁷ : (r : ↑s) → Module R (Mₚ r)
f : (r : ↑s) → M →ₗ[R] Mₚ r
inst✝⁶ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)
Nₚ : ↑s → Type u_6
inst✝⁵ : (r : ↑s) → AddCommMonoid (Nₚ r)
inst✝⁴ : (r : ↑s) → Module R (Nₚ r)
g : (r : ↑s) → N →ₗ[R] Nₚ r
inst✝³ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (g r)
Lₚ : ↑s → Type u_7
inst✝² : (r : ↑s) → AddCommMonoid (Lₚ r)
inst✝¹ : (r : ↑s) → Module R (Lₚ r)
h : (r : ↑s) → L →ₗ[R] Lₚ r
inst✝ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (h r)
F : M →ₗ[R] N
G : N →ₗ[R] L
H :
  ∀ (r : ↑s),
    ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker =
      ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range
r : ↑s
this :
  ∀ (x : Nₚ r),
    x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker ↔
      x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range
⊢ ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker =
    ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range

Tactic:
ext x

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
L : Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : Module R N
inst✝¹⁰ : AddCommMonoid L
inst✝⁹ : Module R L
s : Set R
spn : Ideal.span s = ⊤
Mₚ : ↑s → Type u_5
inst✝⁸ : (r : ↑s) → AddCommMonoid (Mₚ r)
inst✝⁷ : (r : ↑s) → Module R (Mₚ r)
f : (r : ↑s) → M →ₗ[R] Mₚ r
inst✝⁶ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)
Nₚ : ↑s → Type u_6
inst✝⁵ : (r : ↑s) → AddCommMonoid (Nₚ r)
inst✝⁴ : (r : ↑s) → Module R (Nₚ r)
g : (r : ↑s) → N →ₗ[R] Nₚ r
inst✝³ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (g r)
Lₚ : ↑s → Type u_7
inst✝² : (r : ↑s) → AddCommMonoid (Lₚ r)
inst✝¹ : (r : ↑s) → Module R (Lₚ r)
h : (r : ↑s) → L →ₗ[R] Lₚ r
inst✝ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (h r)
F : M →ₗ[R] N
G : N →ₗ[R] L
H :
  ∀ (r : ↑s),
    ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker =
      ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range
r : ↑s
this :
  ∀ (x : Nₚ r),
    x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker ↔
      x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range
x : Nₚ r
⊢ x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (g r) (h r)) G).ker ↔
    x ∈ ((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F).range
Module: Mathlib.LinearAlgebra.TensorProduct.Quotient

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
I : Ideal R
⊢ ∀ (m : S) (x : S ⧸ Ideal.map (algebraMap R S) I),
    (↑(tensorQuotEquivQuotSMul S I ≪≫ₗ
                  (I • ⊤).quotEquivOfEq (Submodule.restrictScalars R (Ideal.map (algebraMap R S) I)) ⋯ ≪≫ₗ
                Submodule.Quotient.restrictScalarsEquiv R (Ideal.map (algebraMap R S) I)).symm).toFun
        (m • x) =
      (RingHom.id S) m •
        (↑(tensorQuotEquivQuotSMul S I ≪≫ₗ
                    (I • ⊤).quotEquivOfEq (Submodule.restrictScalars R (Ideal.map (algebraMap R S) I)) ⋯ ≪≫ₗ
                  Submodule.Quotient.restrictScalarsEquiv R (Ideal.map (algebraMap R S) I)).symm).toFun
          x

Tactic:
rintro _ ⟨_⟩

Goal state after:
case mk
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
I : Ideal R
m✝ : S
x✝ : S ⧸ Ideal.map (algebraMap R S) I
a✝ : S
⊢ (↑(tensorQuotEquivQuotSMul S I ≪≫ₗ
                (I • ⊤).quotEquivOfEq (Submodule.restrictScalars R (Ideal.map (algebraMap R S) I)) ⋯ ≪≫ₗ
              Submodule.Quotient.restrictScalarsEquiv R (Ideal.map (algebraMap R S) I)).symm).toFun
      (m✝ • Quot.mk (⇑(Submodule.quotientRel (Ideal.map (algebraMap R S) I))) a✝) =
    (RingHom.id S) m✝ •
      (↑(tensorQuotEquivQuotSMul S I ≪≫ₗ
                  (I • ⊤).quotEquivOfEq (Submodule.restrictScalars R (Ideal.map (algebraMap R S) I)) ⋯ ≪≫ₗ
                Submodule.Quotient.restrictScalarsEquiv R (Ideal.map (algebraMap R S) I)).symm).toFun
        (Quot.mk (⇑(Submodule.quotientRel (Ideal.map (algebraMap R S) I))) a✝)
Module: Mathlib.RingTheory.TensorProduct.Quotient

Goal state before:
pp: R : Type u_1
R' : Type u_2
R'' : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing R'
inst✝⁶ : CommRing R''
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R R'
inst✝³ : Algebra R R''
inst✝² : Algebra R' R''
inst✝¹ : IsScalarTower R R' R''
inst✝ : Algebra R S
e : R' ⊗[R] S
φ : R' ⊗[R] S →ₐ[R'] R'' ⊗[R] S := TensorProduct.rTensor S (ofId R' R'')
ψ : R'' ⊗[R] S →ₐ[R''] R'' ⊗[R'] (R' ⊗[R] S ⧸ Ideal.span {e}) :=
  TensorProduct.lift (ofId R'' (R'' ⊗[R'] (R' ⊗[R] S ⧸ Ideal.span {e})))
    ((AlgHom.restrictScalars R TensorProduct.includeRight).comp
      ((Ideal.Quotient.mkₐ R (Ideal.span {e})).comp TensorProduct.includeRight))
    ⋯
hψφ :
  (AlgHom.restrictScalars R' ψ).comp φ =
    (AlgHom.restrictScalars R' TensorProduct.includeRight).comp (Ideal.Quotient.mkₐ R' (Ideal.span {e}))
heψ : Ideal.span {φ e} ≤ RingHom.ker ψ
⊢ Ideal.map φ (Ideal.span {e}) ≤ Ideal.span {(TensorProduct.rTensor S (ofId R' R'')) e}

Tactic:
simp [Ideal.map_span, φ]

Goal state after:
Module: Mathlib.RingTheory.EssentialFiniteness

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
hf : f.EssFiniteType
g₁ g₂ : S →+* T
h₁ : g₁.comp f = g₂.comp f
h₂ : ∀ x ∈ hf.finset, g₁ x = g₂ x
algInst✝¹ : Algebra R S := f.toAlgebra
algInst✝ : Algebra R T := (g₁.comp f).toAlgebra
algebraizeInst✝ : Algebra.EssFiniteType R S
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
hf : f.EssFiniteType
g₁ g₂ : S →+* T
h₁ : g₁.comp f = g₂.comp f
h₂ : ∀ x ∈ hf.finset, g₁ x = g₂ x
algInst✝¹ : Algebra R S := f.toAlgebra
algInst✝ : Algebra R T := (g₁.comp f).toAlgebra
algebraizeInst✝ : Algebra.EssFiniteType R S
x : S
⊢ g₁ x = g₂ x
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ContainsIdentities fun {R S} [CommRing R] [CommRing S] ↦ P

Tactic:
by
  introv R
  exact hPl.of_bijective _ _ Function.bijective_id

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

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
        ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ
r : ↑s
Rᵣ Sᵣ : Type u
w✝⁵ : CommRing Rᵣ
w✝⁴ : CommRing Sᵣ
w✝³ : Algebra R Rᵣ
w✝² : Algebra S Sᵣ
w✝¹ : IsLocalization.Away (↑r) Rᵣ
w✝ : IsLocalization.Away (f ↑r) Sᵣ
fᵣ : Rᵣ →+* Sᵣ
hfᵣ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f
hf : P fᵣ
e₁ : Localization (Submonoid.powers ↑r) ≃+* Rᵣ := ⋯
e₂ : Sᵣ ≃+* Localization (Submonoid.powers (f ↑r)) := ⋯
⊢ (Localization.awayMap f ↑r).comp (algebraMap R (Localization.Away ↑r)) =
    ((e₂.toRingHom.comp fᵣ).comp e₁.toRingHom).comp (algebraMap R (Localization.Away ↑r))

Tactic:
ext x

Goal state after:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
        ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ
r : ↑s
Rᵣ Sᵣ : Type u
w✝⁵ : CommRing Rᵣ
w✝⁴ : CommRing Sᵣ
w✝³ : Algebra R Rᵣ
w✝² : Algebra S Sᵣ
w✝¹ : IsLocalization.Away (↑r) Rᵣ
w✝ : IsLocalization.Away (f ↑r) Sᵣ
fᵣ : Rᵣ →+* Sᵣ
hfᵣ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f
hf : P fᵣ
e₁ : Localization (Submonoid.powers ↑r) ≃+* Rᵣ := (Localization.algEquiv (Submonoid.powers ↑r) Rᵣ).toRingEquiv
e₂ : Sᵣ ≃+* Localization (Submonoid.powers (f ↑r)) :=
  (IsLocalization.algEquiv (Submonoid.powers (f ↑r)) (Localization (Submonoid.powers (f ↑r))) Sᵣ).symm.toRingEquiv
x : R
⊢ ((Localization.awayMap f ↑r).comp (algebraMap R (Localization.Away ↑r))) x =
    (((e₂.toRingHom.comp fᵣ).comp e₁.toRingHom).comp (algebraMap R (Localization.Away ↑r))) x
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
f : R[T;T⁻¹]
m n : ℤ
⊢ f * T m * T n = f * T (m + n)

Tactic:
simp [← T_add, mul_assoc]

Goal state after:
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
I p : Ideal R
inst✝¹ : p.IsPrime
inst✝ : IsLocalization.AtPrime S p
hle : ¬I ≤ p
⊢ Ideal.map (algebraMap R S) I = ⊤

Tactic:
apply IsLocalization.map_eq_top_of_not_subset p.primeCompl
  simpa [SetLike.le_def, Set.not_subset_iff_exists_mem_notMem] using hle

Goal state after:
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
motive : R[T;T⁻¹] → Prop
p : R[T;T⁻¹]
add : ∀ (p q : R[T;T⁻¹]), motive p → motive q → motive (p + q)
C_mul_T : ∀ (n : ℤ) (a : R), motive (C a * T n)
⊢ motive p

Tactic:
by
  refine p.induction_on (fun a => ?_) (fun {p q} => add p q) ?_ ?_ <;>
      try exact fun n f _ => C_mul_T _ f
  convert! C_mul_T 0 a
  exact (mul_one _).symm

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

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
⊢ A.det • 1ᵀ = A.det • 1

Tactic:
transpose_one

Goal state after:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
⊢ A.det • 1 = A.det • 1
Module: Mathlib.LinearAlgebra.Matrix.Invertible

Goal state before:
pp: m : Type u_1
n : Type u_2
α : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommSemiring α
A : Matrix n n α
inst✝ : Invertible A
⊢ 1ᵀ = 1

Tactic:
transpose_one

Goal state after:
m : Type u_1
n : Type u_2
α : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommSemiring α
A : Matrix n n α
inst✝ : Invertible A
⊢ 1 = 1