mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
A B : C
f : A ⟶ B
X : C
inst✝¹ : HasBinaryCoproduct A X
inst✝ : HasBinaryCoproduct B X
⊢ coprod.inl ≫ coprod.map f (𝟙 X) = f ≫ coprod.inl

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹⁸ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝¹⁷ : CommSemiring A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : IsLocalization S A
M : Type u_3
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
M' : Type u_4
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : Module A M'
inst✝⁹ : IsScalarTower R A M'
f : M →ₗ[R] M'
T : Type u_5
B : Type u_6
inst✝⁸ : CommSemiring T
inst✝⁷ : CommSemiring B
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra T B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R T B
inst✝¹ : IsScalarTower R A B
α : Type u_7
inst✝ : IsLocalizedModule S f
⊢ IsLocalizedModule S (Finsupp.mapRange.linearMap f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommSemiring R
S : Submonoid R
A : Type u_2
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
inst✝⁷ : IsLocalization S A
M : Type u_3
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
M' : Type u_4
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M'
inst✝² : Module A M'
inst✝¹ : IsScalarTower R A M'
f : M →ₗ[R] M'
α : Type u_7
v : α → M
inst✝ : IsLocalizedModule S f
⊢ (map S (mapRange.linearMap (Algebra.linearMap R A)) f) (linearCombination R v) ∘ₗ
      mapRange.linearMap (Algebra.linearMap R A) =
    ↑R (linearCombination A (⇑f ∘ v)) ∘ₗ mapRange.linearMap (Algebra.linearMap R A)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝²¹ : CommSemiring R
S✝ : Submonoid R
A : Type u_2
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
inst✝¹⁸ : IsLocalization S✝ A
M : Type u_3
inst✝¹⁷ : AddCommMonoid M
inst✝¹⁶ : Module R M
M' : Type u_4
inst✝¹⁵ : AddCommMonoid M'
inst✝¹⁴ : Module R M'
inst✝¹³ : Module A M'
inst✝¹² : IsScalarTower R A M'
f : M →ₗ[R] M'
T : Type u_5
B : Type u_6
inst✝¹¹ : CommSemiring T
inst✝¹⁰ : CommSemiring B
inst✝⁹ : Algebra R T
inst✝⁸ : Algebra T B
inst✝⁷ : Algebra R B
inst✝⁶ : Algebra A B
inst✝⁵ : IsScalarTower R T B
inst✝⁴ : IsScalarTower R A B
S : Submonoid A
N : Type u_7
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
g : M →ₗ[A] M'
h : IsLocalizedModule S g
Aₚ : Type u_2 := Localization S
this✝ : Module Aₚ M' := AddEquiv.module Aₚ (iso S g).symm.toAddEquiv
this : IsScalarTower A Aₚ M'
r : R
x : M'
⊢ (algebraMap R Aₚ) r • x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝²¹ : CommSemiring R
S✝ : Submonoid R
A : Type u_2
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
inst✝¹⁸ : IsLocalization S✝ A
M : Type u_3
inst✝¹⁷ : AddCommMonoid M
inst✝¹⁶ : Module R M
M' : Type u_4
inst✝¹⁵ : AddCommMonoid M'
inst✝¹⁴ : Module R M'
inst✝¹³ : Module A M'
inst✝¹² : IsScalarTower R A M'
f : M →ₗ[R] M'
T : Type u_5
B : Type u_6
inst✝¹¹ : CommSemiring T
inst✝¹⁰ : CommSemiring B
inst✝⁹ : Algebra R T
inst✝⁸ : Algebra T B
inst✝⁷ : Algebra R B
inst✝⁶ : Algebra A B
inst✝⁵ : IsScalarTower R T B
inst✝⁴ : IsScalarTower R A B
S : Submonoid A
N : Type u_7
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
g : M →ₗ[A] M'
h : IsLocalizedModule S g
⊢ IsLocalizedModule S ((AlgebraTensorModule.rTensor R N) g)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_1
inst✝¹⁴ : CommSemiring R
A : Type u_2
inst✝¹³ : CommSemiring A
inst✝¹² : Algebra R A
M : Type u_3
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
M' : Type u_4
inst✝⁹ : AddCommMonoid M'
inst✝⁸ : Module R M'
inst✝⁷ : Module A M'
inst✝⁶ : IsScalarTower R A M'
S : Submonoid A
N : Type u_7
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module A M
inst✝² : IsScalarTower R A M
P : Type u_8
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
f : N →ₗ[R] P
g : M →ₗ[A] M'
h : IsLocalizedModule S g
⊢ (map S ((AlgebraTensorModule.rTensor R N) g) ((AlgebraTensorModule.rTensor R P) g))
      ((AlgebraTensorModule.lTensor A M) f) =
    (AlgebraTensorModule.lTensor A M') f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⇑(OreLocalization.oreEqv S M) = r S M

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x✝ : M × ↥S
m : M
s : ↥S
⊢ 1 • (m, s).2 • (m, s).1 = 1 • (m, s).2 • (m, s).1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R✝ : Type u_1
inst✝²³ : CommSemiring R✝
S✝ : Submonoid R✝
A✝ : Type u_2
inst✝²² : CommSemiring A✝
inst✝²¹ : Algebra R✝ A✝
inst✝²⁰ : IsLocalization S✝ A✝
M✝ : Type u_3
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : Module R✝ M✝
M' : Type u_4
inst✝¹⁷ : AddCommMonoid M'
inst✝¹⁶ : Module R✝ M'
inst✝¹⁵ : Module A✝ M'
inst✝¹⁴ : IsScalarTower R✝ A✝ M'
f : M✝ →ₗ[R✝] M'
T : Type u_5
B : Type u_6
inst✝¹³ : CommSemiring T
inst✝¹² : CommSemiring B
inst✝¹¹ : Algebra R✝ T
inst✝¹⁰ : Algebra T B
inst✝⁹ : Algebra R✝ B
inst✝⁸ : Algebra A✝ B
inst✝⁷ : IsScalarTower R✝ T B
inst✝⁶ : IsScalarTower R✝ A✝ B
R : Type u_7
S : Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
r : R
A : Type u_9
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
M : Submonoid R
inst✝ : IsLocalization M A
⊢ IsLocalization (Algebra.algebraMapSubmonoid S M) (S ⊗[R] A)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
A B : C
f : A ⟶ B
X : C
inst✝¹ : HasBinaryCoproduct A X
inst✝ : HasBinaryCoproduct B X
s : PushoutCocone coprod.inl f
⊢ { l //
    (PushoutCocone.mk (coprod.map f (𝟙 X)) coprod.inl ⋯).inl ≫ l = s.inl ∧
      (PushoutCocone.mk (coprod.map f (𝟙 X)) coprod.inl ⋯).inr ≫ l = s.inr ∧
        ∀ {m : (PushoutCocone.mk (coprod.map f (𝟙 X)) coprod.inl ⋯).pt ⟶ s.pt},
          (PushoutCocone.mk (coprod.map f (𝟙 X)) coprod.inl ⋯).inl ≫ m = s.inl →
            (PushoutCocone.mk (coprod.map f (𝟙 X)) coprod.inl ⋯).inr ≫ m = s.inr → m = l }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
c : BinaryCofan X Y
hc : IsColimit c
I : C
hI : IsInitial I
⊢ ∀
    (s :
      PushoutCocone (hI.to ((pair X Y).obj { as := WalkingPair.right }))
        (hI.to ((pair X Y).obj { as := WalkingPair.left }))),
    c.inr ≫ BinaryCofan.IsColimit.desc hc s.inr s.inl = s.inl

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
c : BinaryCofan X Y
hc : IsColimit c
I : C
hI : IsInitial I
⊢ ∀
    (s :
      PushoutCocone (hI.to ((pair X Y).obj { as := WalkingPair.right }))
        (hI.to ((pair X Y).obj { as := WalkingPair.left }))),
    c.inl ≫ BinaryCofan.IsColimit.desc hc s.inr s.inl = s.inr

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R✝ : Type u_1
inst✝²³ : CommSemiring R✝
S✝ : Submonoid R✝
A✝ : Type u_2
inst✝²² : CommSemiring A✝
inst✝²¹ : Algebra R✝ A✝
inst✝²⁰ : IsLocalization S✝ A✝
M✝ : Type u_3
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : Module R✝ M✝
M' : Type u_4
inst✝¹⁷ : AddCommMonoid M'
inst✝¹⁶ : Module R✝ M'
inst✝¹⁵ : Module A✝ M'
inst✝¹⁴ : IsScalarTower R✝ A✝ M'
f : M✝ →ₗ[R✝] M'
T : Type u_5
B : Type u_6
inst✝¹³ : CommSemiring T
inst✝¹² : CommSemiring B
inst✝¹¹ : Algebra R✝ T
inst✝¹⁰ : Algebra T B
inst✝⁹ : Algebra R✝ B
inst✝⁸ : Algebra A✝ B
inst✝⁷ : IsScalarTower R✝ T B
inst✝⁶ : IsScalarTower R✝ A✝ B
R : Type u_7
S : Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
r : R
A : Type u_9
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
M : Submonoid R
inst✝ : IsLocalization M A
⊢ IsLocalization (Algebra.algebraMapSubmonoid S M) (A ⊗[R] S)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X₁ X₂ X₃ Y₁ : C
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
i₁ : X₁ ⟶ Y₁
t₁ : PushoutCocone i₁ f₁
i₂ : X₂ ⟶ t₁.pt
t₂ : PushoutCocone «i₂» f₂
hi₂ : «i₂» = t₁.inr
H : IsColimit t₁
H' : IsColimit t₂
⊢ IsColimit (t₁.pasteHoriz t₂ hi₂)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x✝⁴ x✝³ x✝² : M × ↥S
m1 : M
s1 : ↥S
m2 : M
s2 : ↥S
x✝¹ : r S M (m1, s1) (m2, s2)
m3 : M
s3 : ↥S
x✝ : r S M (m2, s2) (m3, s3)
u1 : ↥S
hu1 : u1 • (m2, s2).2 • (m1, s1).1 = u1 • (m1, s1).2 • (m2, s2).1
u2 : ↥S
hu2 : u2 • (m3, s3).2 • (m2, s2).1 = u2 • (m2, s2).2 • (m3, s3).1
⊢ r S M (m1, s1) (m3, s3)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
c : BinaryCofan X Y
hc : IsColimit c
I : C
hI : IsInitial I
s :
  PushoutCocone (hI.to ((pair X Y).obj { as := WalkingPair.right })) (hI.to ((pair X Y).obj { as := WalkingPair.left }))
m : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right } ⟶ s.pt
h₁ : c.inr ≫ m = s.inl
h₂ : c.inl ≫ m = s.inr
⊢ m = BinaryCofan.IsColimit.desc hc s.inr s.inl

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u_3
inst✝ : CommSemiring R
S : Submonoid R
⊢ Localization S = LocalizedModule S R

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
w : f ≫ inl = g ≫ inr
hom_ext : ∀ ⦃T : C⦄ ⦃φ φ' : P ⟶ T⦄, inl ≫ φ = inl ≫ φ' → inr ≫ φ = inr ≫ φ' → φ = φ'
exists_desc : ∀ ⦃T : C⦄ (a : X ⟶ T) (b : Y ⟶ T), f ≫ a = g ≫ b → ∃ l, inl ≫ l = a ∧ inr ≫ l = b
l : ∀ (s : PushoutCocone f g), ∃ l, inl ≫ l = s.inl ∧ inr ≫ l = s.inr :=
  fun s ↦ exists_desc s.inl s.inr (PushoutCocone.condition s)
⊢ Nonempty (IsColimit (PushoutCocone.mk inl inr w))

Tactic:
exact ⟨PushoutCocone.IsColimit.mk _
      (fun s ↦ (l s).choose)
      (fun s ↦ (l s).choose_spec.1)
      (fun s ↦ (l s).choose_spec.2)
      (fun s m h₁ h₂ ↦ hom_ext
        (h₁.trans (l s).choose_spec.1.symm)
        (h₂.trans (l s).choose_spec.2.symm))⟩

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m m' : M
s s' : ↥S
⊢ mk m s = mk m' s' ↔ ∃ u, u • s' • m = u • s • m'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
w : f ≫ inl = g ≫ inr
hom_ext : ∀ ⦃T : C⦄ ⦃φ φ' : P ⟶ T⦄, inl ≫ φ = inl ≫ φ' → inr ≫ φ = inr ≫ φ' → φ = φ'
exists_desc : ∀ ⦃T : C⦄ (a : X ⟶ T) (b : Y ⟶ T), f ≫ a = g ≫ b → ∃ l, inl ≫ l = a ∧ inr ≫ l = b
⊢ Nonempty (IsColimit (PushoutCocone.mk inl inr w))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
β : LocalizedModule S M → Prop
h : ∀ (m : M) (s : ↥S), β (mk m s)
⊢ ∀ (x : LocalizedModule S M), β x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X₁ X₂ X₃ Y₁ : C
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
i₁ : X₁ ⟶ Y₁
t₁ : PushoutCocone i₁ f₁
i₂ : X₂ ⟶ t₁.pt
t₂ : PushoutCocone «i₂» f₂
hi₂ : «i₂» = t₁.inr
H : IsColimit t₁
H' : IsColimit (t₁.pasteHoriz t₂ hi₂)
s : PushoutCocone «i₂» f₂
⊢ i₁ ≫ t₁.inl ≫ s.inl = (f₁ ≫ f₂) ≫ s.inr

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_7
S : Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_9
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
M : Submonoid R
inst✝ : IsLocalization M A
s : S
x : R
y : ↥M
⊢ s ⊗ₜ[R] mk' A x y = mk' (S ⊗[R] A) ((algebraMap R S) x * s) ⟨(algebraMap R S) ↑y, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
β : LocalizedModule S M → LocalizedModule S M → Prop
h : ∀ (m m' : M) (s s' : ↥S), β (mk m s) (mk m' s')
⊢ ∀ (x y : LocalizedModule S M), β x y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_7
S : Type u_8
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
A : Type u_9
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
M : Submonoid R
inst✝ : IsLocalization M A
s : S
x : R
y : ↥M
⊢ mk' A x y ⊗ₜ[R] s = mk' (A ⊗[R] S) ((algebraMap R S) x * s) ⟨(algebraMap R S) ↑y, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝⁸ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
T : Type u_1
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : IsLocalization S T
T' : Type u_2
inst✝² : CommSemiring T'
inst✝¹ : Algebra R T'
inst✝ : IsLocalization S T'
α : Type u_3
x : LocalizedModule S M
f : M × ↥S → α
wd : ∀ (p p' : M × ↥S), p ≈ p' → f p = f p'
⊢ ∀ (a b : M × ↥S), a ≈ b → f a = f b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type u_3
f : M × ↥S → α
wd : ∀ (p p' : M × ↥S), p ≈ p' → f p = f p'
m : M
s : ↥S
⊢ (mk m s).liftOn f wd = f (m, s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝⁸ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
T : Type u_1
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : IsLocalization S T
T' : Type u_2
inst✝² : CommSemiring T'
inst✝¹ : Algebra R T'
inst✝ : IsLocalization S T'
α : Type u_3
x y : LocalizedModule S M
f : M × ↥S → M × ↥S → α
wd : ∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q'
⊢ ∀ (a₁ b₁ a₂ b₂ : M × ↥S), a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ = f a₂ b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type u_3
f : M × ↥S → M × ↥S → α
wd : ∀ (p q p' q' : M × ↥S), p ≈ p' → q ≈ q' → f p q = f p' q'
m m' : M
s s' : ↥S
⊢ (mk m s).liftOn₂ (mk m' s') f wd = f (m, s) (m', s')

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : 0 ∈ S
m✝ m'✝ : M
s✝ s'✝ : ↥S
⊢ ⟨0, h⟩ • s'✝ • m✝ = ⟨0, h⟩ • s✝ • m'✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : 0 ∈ S
⊢ Subsingleton (LocalizedModule S M)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : ↥S
⊢ mk 0 s = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m1 m2 : M
s1 s2 : ↥S
⊢ mk m1 s1 + mk m2 s2 = mk (s2 • m1 + s1 • m2) (s1 * s2)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m : M
s : ↥S
⊢ mk (-m) s = -mk m s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_10
inst✝² : CommRing R
M : Submonoid R
S : Type u_12
inst✝¹ : CommRing S
inst✝ : Algebra R S
Rₘ : Type u_10 := Localization M
Sₘ : Type u_12 := Localization (Algebra.algebraMapSubmonoid S M)
x✝ : R
y : ↥M
this : Algebra Rₘ (S ⊗[R] Rₘ) := Algebra.TensorProduct.rightAlgebra
h1 : 1 ⊗ₜ[R] IsLocalization.mk' Rₘ x✝ y = (algebraMap Rₘ (S ⊗[R] Rₘ)) (IsLocalization.mk' Rₘ x✝ y)
x : R
⊢ (algebraMap S Sₘ) ((algebraMap R S) x) = (algebraMap Rₘ Sₘ) ((algebraMap R Rₘ) x)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
m₁ m₂ : LocalizedModule S A
⊢ ∀ (p q p' q' : A × ↥S), p ≈ p' → q ≈ q' → mk (p.1 * q.1) (q.2 * p.2) = mk (p'.1 * q'.1) (q'.2 * p'.2)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y Z : C
f f' : X ⟶ Y
g g' : Y ⟶ Z
H : IsPullback f f g g'
⊢ IsLimit (Fork.ofι f ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
⊢ ∀ (a b c : LocalizedModule S A), a * b * c = a * (b * c)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_10
inst✝² : CommRing R
M : Submonoid R
S : Type u_12
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : Localization M
⊢ (tensorLeftAlgEquiv M S) (1 ⊗ₜ[R] x) =
    (algebraMap (Localization M) (Localization (Algebra.algebraMapSubmonoid S M))) x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
a✝ : LocalizedModule S A
a : A
s : ↥S
⊢ 1 • s • ((1, 1).1 * (a, s).1) = 1 • ((a, s).2 * (1, 1).2) • a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
⊢ ∀ (a : LocalizedModule S A), 1 * a = a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
a✝ : LocalizedModule S A
a : A
s : ↥S
⊢ 1 • s • ((a, s).1 * (1, 1).1) = 1 • ((1, 1).2 * (a, s).2) • a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
⊢ ∀ (a : LocalizedModule S A), a * 1 = a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
S : Submonoid R
⊢ OreLocalization.instMonoid = instMonoid

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
S : Submonoid R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
a₁ a₂ : A
s₁ s₂ : ↥S
⊢ mk a₁ s₁ * mk a₂ s₂ = mk (a₁ * a₂) (s₁ * s₂)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.BaseChange

Goal state before:
pp: R : Type u_7
S : Type u_8
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Algebra R S
A : Type u_9
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
M : Submonoid A
B : Type u_10
inst✝⁶ : CommSemiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : IsLocalization M B
inst✝¹ : Algebra (S ⊗[R] A) (S ⊗[R] B)
inst✝ : IsScalarTower S (S ⊗[R] A) (S ⊗[R] B)
H :
  (algebraMap (S ⊗[R] A) (S ⊗[R] B)).comp Algebra.TensorProduct.includeRight.toRingHom =
    Algebra.TensorProduct.includeRight.comp (algebraMap A B)
this✝ : Algebra A (S ⊗[R] B) := Algebra.compHom (S ⊗[R] B) (algebraMap A B)
this : IsScalarTower A (S ⊗[R] A) (S ⊗[R] B)
⊢ algebraMap R (S ⊗[R] B) = (algebraMap A (S ⊗[R] B)).comp (algebraMap R A)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
n : ℕ
a : A
s : ↥S
⊢ mk a s ^ n = mk (a ^ n) (s ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
a✝ : LocalizedModule S A
a : A
s : ↥S
⊢ 1 • 1 • ((0, 1).1 * (a, s).1) = 1 • ((a, s).2 * (0, 1).2) • 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Basic

Goal state before:
pp: R : Type u
inst✝¹⁰ : CommSemiring R
S✝ : Submonoid R
M : Type v
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
T : Type u_1
inst✝⁷ : CommSemiring T
inst✝⁶ : Algebra R T
inst✝⁵ : IsLocalization S✝ T
T' : Type u_2
inst✝⁴ : CommSemiring T'
inst✝³ : Algebra R T'
inst✝² : IsLocalization S✝ T'
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Submonoid R
⊢ ∀ (a : LocalizedModule S A), 0 * a = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y Z : C
f f' : X ⟶ Y
g g' : Y ⟶ Z
H : IsPushout f f' g g
⊢ IsColimit (Cofork.ofπ g ⋯)

Tactic:
by

Goal state after: