statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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|---|---|---|---|---|---|---|---|---|---|---|
ring_equiv_iso_Ring_iso {X Y : Type u} [ring X] [ring Y] :
(X ≃+* Y) ≅ (Ring.of X ≅ Ring.of Y) | { hom := λ e, e.to_Ring_iso,
inv := λ i, i.Ring_iso_to_ring_equiv, } | def | ring_equiv_iso_Ring_iso | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring.of",
"ring"
] | Ring equivalences between `ring`s are the same as (isomorphic to) isomorphisms in `Ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv_iso_CommRing_iso {X Y : Type u} [comm_ring X] [comm_ring Y] :
(X ≃+* Y) ≅ (CommRing.of X ≅ CommRing.of Y) | { hom := λ e, e.to_CommRing_iso,
inv := λ i, i.CommRing_iso_to_ring_equiv, } | def | ring_equiv_iso_CommRing_iso | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing.of",
"comm_ring"
] | Ring equivalences between `comm_ring`s are the same as (isomorphic to) isomorphisms
in `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ring.forget_reflects_isos : reflects_isomorphisms (forget Ring.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget Ring).map f),
let e : X ≃+* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Ring_iso).1⟩,
end } | instance | Ring.forget_reflects_isos | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommRing.forget_reflects_isos : reflects_isomorphisms (forget CommRing.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget CommRing).map f),
let e : X ≃+* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_CommRing_iso).1⟩,
end } | instance | CommRing.forget_reflects_isos | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommRing.comp_eq_ring_hom_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
f ≫ g = g.comp f | rfl | lemma | CommRing.comp_eq_ring_hom_comp | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommRing.ring_hom_comp_eq_comp {R S T : Type*} [comm_ring R] [comm_ring S]
[comm_ring T] (f : R →+* S) (g : S →+* T) :
g.comp f = CommRing.of_hom f ≫ CommRing.of_hom g | rfl | lemma | CommRing.ring_hom_comp_eq_comp | algebra.category.Ring | src/algebra/category/Ring/basic.lean | [
"algebra.category.Group.basic",
"category_theory.concrete_category.reflects_isomorphisms",
"category_theory.elementwise",
"algebra.ring.equiv"
] | [
"CommRing.of_hom",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prequotient
-- There's always `of`
| of : Π (j : J) (x : F.obj j), prequotient
-- Then one generator for each operation
| zero : prequotient
| one : prequotient
| neg : prequotient → prequotient
| add : prequotient → prequotient → prequotient
| mul : prequotient → prequotient → prequotient | inductive | CommRing.colimits.prequotient | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | An inductive type representing all commutative ring expressions (without relations)
on a collection of types indexed by the objects of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relation : prequotient F → prequotient F → Prop
-- Make it an equivalence relation:
| refl : Π (x), relation x x
| symm : Π (x y) (h : relation x y), relation y x
| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z
-- There's always a `map` relation
| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), r... | inductive | CommRing.colimits.relation | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"left_distrib",
"mul_assoc",
"mul_comm",
"mul_one",
"one_mul",
"right_distrib"
] | The relation on `prequotient` saying when two expressions are equal
because of the commutative ring laws, or
because one element is mapped to another by a morphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_mul (x y) :
quot.mk setoid.r (mul x y) = ((quot.mk setoid.r x) * (quot.mk setoid.r y) : colimit_type F) | rfl | lemma | CommRing.colimits.quot_mul | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : CommRing | CommRing.of (colimit_type F) | def | CommRing.colimits.colimit | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"CommRing",
"CommRing.of"
] | The bundled commutative ring giving the colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit F | { to_fun := cocone_fun F j,
map_one' := by apply quot.sound; apply relation.one,
map_mul' := by intros; apply quot.sound; apply relation.mul,
map_zero' := by apply quot.sound; apply relation.zero,
map_add' := by intros; apply quot.sound; apply relation.add } | def | CommRing.colimits.cocone_morphism | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The ring homomorphism from a given commutative ring in the diagram to the colimit commutative
ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fun_lift (s : cocone F) : prequotient F → s.X | | (of j x) := (s.ι.app j) x
| zero := 0
| one := 1
| (neg x) := -(desc_fun_lift x)
| (add x y) := desc_fun_lift x + desc_fun_lift y
| (mul x y) := desc_fun_lift x * desc_fun_lift y | def | CommRing.colimits.desc_fun_lift | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The function from the free commutative ring on the diagram to the cone point of any other
cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fun (s : cocone F) : colimit_type F → s.X | begin
fapply quot.lift,
{ exact desc_fun_lift F s },
{ intros x y r,
induction r; try { dsimp },
-- refl
{ refl },
-- symm
{ exact r_ih.symm },
-- trans
{ exact eq.trans r_ih_h r_ih_k },
-- map
{ simp, },
-- zero
{ simp, },
-- one
{ simp, },
-- neg
{ sim... | def | CommRing.colimits.desc_fun | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"left_distrib",
"mul_assoc",
"mul_comm",
"mul_one",
"one_mul",
"right_distrib"
] | The function from the colimit commutative ring to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_morphism (s : cocone F) : colimit F ⟶ s.X | { to_fun := desc_fun F s,
map_one' := rfl,
map_zero' := rfl,
map_add' := λ x y, by { induction x; induction y; refl },
map_mul' := λ x y, by { induction x; induction y; refl }, } | def | CommRing.colimits.desc_morphism | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | The ring homomorphism from the colimit commutative ring to the cone point of any other
cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_is_colimit : is_colimit (colimit_cocone F) | { desc := λ s, desc_morphism F s,
uniq' := λ s m w,
begin
ext,
induction x,
induction x,
{ have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,
erw w',
refl, },
{ simp, },
{ simp, },
{ simp *, },
{ simp *, },
{ simp *, },
re... | def | CommRing.colimits.colimit_is_colimit | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [] | Evidence that the proposed colimit is the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_CommRing : has_colimits CommRing | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F,
is_colimit := colimit_is_colimit F } } } | instance | CommRing.colimits.has_colimits_CommRing | algebra.category.Ring | src/algebra/category/Ring/colimits.lean | [
"algebra.category.Ring.basic",
"category_theory.limits.has_limits",
"category_theory.concrete_category.elementwise"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone : limits.pushout_cocone f g | begin
letI := ring_hom.to_algebra f,
letI := ring_hom.to_algebra g,
apply limits.pushout_cocone.mk,
show CommRing, from CommRing.of (A ⊗[R] B),
show A ⟶ _, from algebra.tensor_product.include_left.to_ring_hom,
show B ⟶ _, from algebra.tensor_product.include_right.to_ring_hom,
ext r,
transitivity algeb... | def | CommRing.pushout_cocone | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing",
"CommRing.of",
"algebra_map",
"ring_hom.to_algebra"
] | The explicit cocone with tensor products as the fibered product in `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone_inl : (pushout_cocone f g).inl = (by
{ letI := f.to_algebra, letI := g.to_algebra,
exactI algebra.tensor_product.include_left.to_ring_hom }) | rfl | lemma | CommRing.pushout_cocone_inl | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_inr : (pushout_cocone f g).inr = (by
{ letI := f.to_algebra, letI := g.to_algebra,
exactI algebra.tensor_product.include_right.to_ring_hom }) | rfl | lemma | CommRing.pushout_cocone_inr | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_X : (pushout_cocone f g).X = (by
{ letI := f.to_algebra, letI := g.to_algebra,
exactI CommRing.of (A ⊗[R] B) }) | rfl | lemma | CommRing.pushout_cocone_X | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_is_colimit : limits.is_colimit (pushout_cocone f g) | limits.pushout_cocone.is_colimit_aux' _ (λ s,
begin
letI := ring_hom.to_algebra f,
letI := ring_hom.to_algebra g,
letI := ring_hom.to_algebra (f ≫ s.inl),
let f' : A →ₐ[R] s.X := { commutes' := λ r, by
{ change s.inl.to_fun (f r) = (f ≫ s.inl) r, refl }, ..s.inl },
let g' : B →ₐ[R] s.X := { commutes' :=... | def | CommRing.pushout_cocone_is_colimit | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"algebra.tensor_product.ext",
"algebra.tensor_product.product_map",
"algebra.tensor_product.product_map_left_apply",
"algebra.tensor_product.product_map_right_apply",
"ring_hom.to_algebra"
] | Verify that the `pushout_cocone` is indeed the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_is_terminal : is_terminal (CommRing.of.{u} punit) | begin
apply_with is_terminal.of_unique { instances := ff },
tidy
end | def | CommRing.punit_is_terminal | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [] | The trivial ring is the (strict) terminal object of `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommRing_has_strict_terminal_objects : has_strict_terminal_objects CommRing.{u} | begin
apply has_strict_terminal_objects_of_terminal_is_strict (CommRing.of punit),
intros X f,
refine ⟨⟨by tidy, by ext, _⟩⟩,
ext,
have e : (0 : X) = 1 := by { rw [← f.map_one, ← f.map_zero], congr },
replace e : 0 * x = 1 * x := congr_arg (λ a, a * x) e,
rw [one_mul, zero_mul, ← f.map_zero] at e,
exact... | instance | CommRing.CommRing_has_strict_terminal_objects | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_of_is_terminal {X : CommRing} (hX : is_terminal X) : subsingleton X | (hX.unique_up_to_iso punit_is_terminal).CommRing_iso_to_ring_equiv.to_equiv
.subsingleton_congr.mpr (show subsingleton punit, by apply_instance) | lemma | CommRing.subsingleton_of_is_terminal | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Z_is_initial : is_initial (CommRing.of ℤ) | begin
apply_with is_initial.of_unique { instances := ff },
exact λ R, ⟨⟨int.cast_ring_hom R⟩, λ a, a.ext_int _⟩,
end | def | CommRing.Z_is_initial | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of"
] | `ℤ` is the initial object of `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_fan : binary_fan A B | binary_fan.mk (CommRing.of_hom $ ring_hom.fst A B) (CommRing.of_hom $ ring_hom.snd A B) | def | CommRing.prod_fan | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"ring_hom.fst",
"ring_hom.snd"
] | The product in `CommRing` is the cartesian product. This is the binary fan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_fan_is_limit : is_limit (prod_fan A B) | { lift := λ c, ring_hom.prod (c.π.app ⟨walking_pair.left⟩) (c.π.app ⟨walking_pair.right⟩),
fac' := λ c j, by { ext, rcases j with ⟨⟨⟩⟩;
simpa only [binary_fan.π_app_left, binary_fan.π_app_right, comp_apply, ring_hom.prod_apply] },
uniq' := λ s m h, by { ext, { simpa using congr_hom (h ⟨walking_pair.left⟩) x },
... | def | CommRing.prod_fan_is_limit | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"lift",
"ring_hom.prod",
"ring_hom.prod_apply"
] | The product in `CommRing` is the cartesian product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_fork : fork f g | fork.of_ι (CommRing.of_hom (ring_hom.eq_locus f g).subtype) (by { ext ⟨x, e⟩, simpa using e }) | def | CommRing.equalizer_fork | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"ring_hom.eq_locus"
] | The equalizer in `CommRing` is the equalizer as sets. This is the equalizer fork. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_fork_is_limit : is_limit (equalizer_fork f g) | begin
fapply fork.is_limit.mk',
intro s,
use s.ι.cod_restrict _ (λ x, (concrete_category.congr_hom s.condition x : _)),
split,
{ ext, refl },
{ intros m hm, ext x, exact concrete_category.congr_hom hm x }
end | def | CommRing.equalizer_fork_is_limit | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [] | The equalizer in `CommRing` is the equalizer as sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_ι_is_local_ring_hom (F : walking_parallel_pair ⥤ CommRing.{u}) :
is_local_ring_hom (limit.π F walking_parallel_pair.zero) | begin
have := lim_map_π (diagram_iso_parallel_pair F).hom walking_parallel_pair.zero,
rw ← is_iso.comp_inv_eq at this,
rw ← this,
rw ← limit.iso_limit_cone_hom_π ⟨_, equalizer_fork_is_limit
(F.map walking_parallel_pair_hom.left) (F.map walking_parallel_pair_hom.right)⟩
walking_parallel_pair.zero,
chan... | instance | CommRing.equalizer_ι_is_local_ring_hom | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"is_local_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_ι_is_local_ring_hom' (F : walking_parallel_pairᵒᵖ ⥤ CommRing.{u}) :
is_local_ring_hom (limit.π F (opposite.op walking_parallel_pair.one)) | begin
have : _ = limit.π F (walking_parallel_pair_op_equiv.functor.obj _) :=
(limit.iso_limit_cone_inv_π ⟨_, is_limit.whisker_equivalence (limit.is_limit F)
walking_parallel_pair_op_equiv⟩ walking_parallel_pair.zero : _),
erw ← this,
apply_instance
end | instance | CommRing.equalizer_ι_is_local_ring_hom' | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"is_local_ring_hom",
"opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_cone {A B C : CommRing.{u}} (f : A ⟶ C) (g : B ⟶ C) : pullback_cone f g | pullback_cone.mk
(CommRing.of_hom $ (ring_hom.fst A B).comp
(ring_hom.eq_locus (f.comp (ring_hom.fst A B)) (g.comp (ring_hom.snd A B))).subtype)
(CommRing.of_hom $ (ring_hom.snd A B).comp
(ring_hom.eq_locus (f.comp (ring_hom.fst A B)) (g.comp (ring_hom.snd A B))).subtype)
(by { ext ⟨x, e⟩, simpa [CommRing... | def | CommRing.pullback_cone | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"ring_hom.eq_locus",
"ring_hom.fst",
"ring_hom.snd"
] | In the category of `CommRing`, the pullback of `f : A ⟶ C` and `g : B ⟶ C` is the `eq_locus` of
the two maps `A × B ⟶ C`. This is the constructed pullback cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_is_limit {A B C : CommRing.{u}} (f : A ⟶ C) (g : B ⟶ C) :
is_limit (pullback_cone f g) | begin
fapply pullback_cone.is_limit.mk,
{ intro s,
apply (s.fst.prod s.snd).cod_restrict,
intro x, exact congr_arg (λ f : s.X →+* C, f x) s.condition },
{ intro s, ext x, refl },
{ intro s, ext x, refl },
{ intros s m e₁ e₂, ext,
{ exact (congr_arg (λ f : s.X →+* A, f x) e₁ : _) },
{ exact (co... | def | CommRing.pullback_cone_is_limit | algebra.category.Ring | src/algebra/category/Ring/constructions.lean | [
"category_theory.limits.shapes.pullbacks",
"ring_theory.tensor_product",
"algebra.category.Ring.limits",
"algebra.category.Ring.instances",
"category_theory.limits.shapes.strict_initial",
"ring_theory.subring.basic"
] | [] | The constructed pullback cone is indeed the limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiring_obj (j : J) :
semiring (((F ⋙ forget₂ SemiRing Mon.{max v u}) ⋙ forget Mon).obj j) | show semiring (F.obj j), by apply_instance | instance | SemiRing.filtered_colimits.semiring_obj | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon",
"SemiRing",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
R : Mon | Mon.filtered_colimits.colimit (F ⋙ forget₂ SemiRing Mon.{max v u}) | abbreviation | SemiRing.filtered_colimits.R | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon",
"Mon.filtered_colimits.colimit",
"SemiRing"
] | The colimit of `F ⋙ forget₂ SemiRing Mon` in the category `Mon`.
In the following, we will show that this has the structure of a semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_semiring : semiring R | { mul_zero := λ x, begin
apply quot.induction_on x, clear x, intro x,
cases x with j x,
erw [colimit_zero_eq _ j, colimit_mul_mk_eq _ ⟨j, _⟩ ⟨j, _⟩ j (𝟙 j) (𝟙 j)],
rw [category_theory.functor.map_id, id_apply, id_apply, mul_zero x],
refl,
end,
zero_mul := λ x, begin
apply quot.induction_on... | instance | SemiRing.filtered_colimits.colimit_semiring | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"SemiRing",
"left_distrib",
"mul_zero",
"quot.induction_on₃",
"right_distrib",
"semiring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : SemiRing | SemiRing.of R | def | SemiRing.filtered_colimits.colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"SemiRing",
"SemiRing.of"
] | The bundled semiring giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι :=
{ app := λ j,
{ ..(Mon.filtered_colimits.colimit_cocone (F ⋙ forget₂ SemiRing Mon.{max v u})).ι.app j,
..(AddCommMon.filtered_colimits.colimit_cocone
(F ⋙ forget₂ SemiRing AddCommMon.{max v u})).ι.app j },
naturality' := λ j j' f,
(ring_hom.coe_inj ((types.colimit_co... | def | SemiRing.filtered_colimits.colimit_cocone | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon.filtered_colimits.colimit_cocone",
"SemiRing",
"ring_hom.coe_inj"
] | The cocone over the proposed colimit semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t,
{ .. (Mon.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ SemiRing Mon.{max v u})).desc ((forget₂ SemiRing Mon.{max v u}).map_cocone t),
.. (AddCommMon.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ SemiRing AddCommMon.{max v u})).desc
((forget₂ SemiRing AddCommMon.... | def | SemiRing.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon.filtered_colimits.colimit_cocone_is_colimit",
"SemiRing",
"ring_hom.coe_inj",
"ring_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `SemiRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ SemiRing Mon.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F)
(Mon.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ SemiRing Mon.{u})) } } | instance | SemiRing.filtered_colimits.forget₂_Mon_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon.filtered_colimits.colimit_cocone_is_colimit",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget SemiRing.{u}) | limits.comp_preserves_filtered_colimits (forget₂ SemiRing Mon) (forget Mon.{u}) | instance | SemiRing.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Mon",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
R : SemiRing | SemiRing.filtered_colimits.colimit (F ⋙ forget₂ CommSemiRing SemiRing.{max v u}) | abbreviation | CommSemiRing.filtered_colimits.R | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"SemiRing",
"SemiRing.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ CommSemiRing SemiRing` in the category `SemiRing`.
In the following, we will show that this has the structure of a _commutative_ semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_comm_semiring : comm_semiring R | { ..R.semiring,
..CommMon.filtered_colimits.colimit_comm_monoid (F ⋙ forget₂ CommSemiRing CommMon.{max v u}) } | instance | CommSemiRing.filtered_colimits.colimit_comm_semiring | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommMon.filtered_colimits.colimit_comm_monoid",
"CommSemiRing",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : CommSemiRing | CommSemiRing.of R | def | CommSemiRing.filtered_colimits.colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"CommSemiRing.of"
] | The bundled commutative semiring giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι :=
{ ..(SemiRing.filtered_colimits.colimit_cocone
(F ⋙ forget₂ CommSemiRing SemiRing.{max v u})).ι } } | def | CommSemiRing.filtered_colimits.colimit_cocone | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"SemiRing.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit commutative semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t,
(SemiRing.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ CommSemiRing SemiRing.{max v u})).desc
((forget₂ CommSemiRing SemiRing).map_cocone t),
fac' := λ t j, ring_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget CommSemiRing)).fac
((forget CommSemiRing).map_cocone t... | def | CommSemiRing.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"SemiRing",
"SemiRing.filtered_colimits.colimit_cocone_is_colimit",
"ring_hom.coe_inj",
"ring_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `CommSemiRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_SemiRing_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ CommSemiRing SemiRing.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F)
(SemiRing.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ CommSemiRing SemiRing.{u})) } } | instance | CommSemiRing.filtered_colimits.forget₂_SemiRing_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"SemiRing.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget CommSemiRing.{u}) | limits.comp_preserves_filtered_colimits (forget₂ CommSemiRing SemiRing) (forget SemiRing.{u}) | instance | CommSemiRing.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommSemiRing",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
R : SemiRing | SemiRing.filtered_colimits.colimit (F ⋙ forget₂ Ring SemiRing.{max v u}) | abbreviation | Ring.filtered_colimits.R | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"SemiRing",
"SemiRing.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ Ring SemiRing` in the category `SemiRing`.
In the following, we will show that this has the structure of a ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_ring : ring R | { ..R.semiring,
..AddCommGroup.filtered_colimits.colimit_add_comm_group
(F ⋙ forget₂ Ring AddCommGroup.{max v u}) } | instance | Ring.filtered_colimits.colimit_ring | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Ring | Ring.of R | def | Ring.filtered_colimits.colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"Ring.of"
] | The bundled ring giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι := { ..(SemiRing.filtered_colimits.colimit_cocone (F ⋙ forget₂ Ring SemiRing.{max v u})).ι } } | def | Ring.filtered_colimits.colimit_cocone | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"SemiRing.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t,
(SemiRing.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ Ring SemiRing.{max v u})).desc
((forget₂ Ring SemiRing).map_cocone t),
fac' := λ t j, ring_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget Ring)).fac ((forget Ring).map_cocone t) j,
uniq' := λ t m h, ring_hom.coe_in... | def | Ring.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"SemiRing",
"SemiRing.filtered_colimits.colimit_cocone_is_colimit",
"ring_hom.coe_inj",
"ring_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `Ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_SemiRing_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ Ring SemiRing.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F)
(SemiRing.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ Ring SemiRing.{u})) } } | instance | Ring.filtered_colimits.forget₂_SemiRing_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"SemiRing.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget Ring.{u}) | limits.comp_preserves_filtered_colimits (forget₂ Ring SemiRing) (forget SemiRing.{u}) | instance | Ring.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"Ring",
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
R : Ring | Ring.filtered_colimits.colimit (F ⋙ forget₂ CommRing Ring.{max v u}) | abbreviation | CommRing.filtered_colimits.R | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"Ring",
"Ring.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ CommRing Ring` in the category `Ring`.
In the following, we will show that this has the structure of a _commutative_ ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_comm_ring : comm_ring R | { ..R.ring,
..CommSemiRing.filtered_colimits.colimit_comm_semiring
(F ⋙ forget₂ CommRing CommSemiRing.{max v u}) } | instance | CommRing.filtered_colimits.colimit_comm_ring | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"CommSemiRing.filtered_colimits.colimit_comm_semiring",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : CommRing | CommRing.of R | def | CommRing.filtered_colimits.colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"CommRing.of"
] | The bundled commutative ring giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι := { ..(Ring.filtered_colimits.colimit_cocone (F ⋙ forget₂ CommRing Ring.{max v u})).ι } } | def | CommRing.filtered_colimits.colimit_cocone | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"Ring.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit commutative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t,
(Ring.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ CommRing Ring.{max v u})).desc
((forget₂ CommRing Ring).map_cocone t),
fac' := λ t j, ring_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget CommRing)).fac ((forget CommRing).map_cocone t) j,
uniq' := λ t m h, ring_hom.co... | def | CommRing.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"Ring",
"Ring.filtered_colimits.colimit_cocone_is_colimit",
"ring_hom.coe_inj",
"ring_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Ring_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ CommRing Ring.{u}) | { preserves_filtered_colimits := λ J _ _, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit.{u u} F)
(Ring.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ CommRing Ring.{u})) } } | instance | CommRing.filtered_colimits.forget₂_Ring_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"Ring.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget CommRing.{u}) | limits.comp_preserves_filtered_colimits (forget₂ CommRing Ring) (forget Ring.{u}) | instance | CommRing.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Ring | src/algebra/category/Ring/filtered_colimits.lean | [
"algebra.category.Ring.basic",
"algebra.category.Group.filtered_colimits"
] | [
"CommRing",
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_unit_is_iso (R : CommRing) :
is_iso (CommRing.of_hom $ algebra_map R (localization.away (1 : R))) | is_iso.of_iso (is_localization.at_one R (localization.away (1 : R))).to_ring_equiv.to_CommRing_iso | instance | localization_unit_is_iso | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"CommRing.of_hom",
"algebra_map",
"is_localization.at_one",
"localization.away"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_unit_is_iso' (R : CommRing) :
@is_iso CommRing _ R _ (CommRing.of_hom $ algebra_map R (localization.away (1 : R))) | by { cases R, exact localization_unit_is_iso _ } | instance | localization_unit_is_iso' | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"CommRing.of_hom",
"algebra_map",
"localization.away",
"localization_unit_is_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.epi {R : Type*} [comm_ring R] (M : submonoid R) (S : Type*) [comm_ring S]
[algebra R S] [is_localization M S] : epi (CommRing.of_hom $ algebra_map R S) | ⟨λ T f₁ f₂, @is_localization.ring_hom_ext R _ M S _ _ T _ _ _ _⟩ | lemma | is_localization.epi | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing.of_hom",
"algebra",
"algebra_map",
"comm_ring",
"is_localization",
"is_localization.ring_hom_ext",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization.epi {R : Type*} [comm_ring R] (M : submonoid R) : epi
(CommRing.of_hom $ algebra_map R $ localization M) | is_localization.epi M _ | instance | localization.epi | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing.of_hom",
"algebra_map",
"comm_ring",
"is_localization.epi",
"localization",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization.epi' {R : CommRing} (M : submonoid R) : @epi CommRing _ R _
(CommRing.of_hom $ algebra_map R $ localization M : _) | by { cases R, exact is_localization.epi M _ } | instance | localization.epi' | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"CommRing.of_hom",
"algebra_map",
"is_localization.epi",
"localization",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CommRing.is_local_ring_hom_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T)
[is_local_ring_hom g] [is_local_ring_hom f] :
is_local_ring_hom (f ≫ g) | is_local_ring_hom_comp _ _ | instance | CommRing.is_local_ring_hom_comp | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"is_local_ring_hom",
"is_local_ring_hom_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_of_iso {R S : CommRing} (f : R ≅ S) : is_local_ring_hom f.hom | { map_nonunit := λ a ha,
begin
convert f.inv.is_unit_map ha,
rw category_theory.iso.hom_inv_id_apply,
end } | lemma | is_local_ring_hom_of_iso | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"is_local_ring_hom",
"map_nonunit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_of_is_iso {R S : CommRing} (f : R ⟶ S) [is_iso f] :
is_local_ring_hom f | is_local_ring_hom_of_iso (as_iso f) | instance | is_local_ring_hom_of_is_iso | algebra.category.Ring | src/algebra/category/Ring/instances.lean | [
"algebra.category.Ring.basic",
"ring_theory.localization.away.basic",
"ring_theory.ideal.local_ring"
] | [
"CommRing",
"is_local_ring_hom",
"is_local_ring_hom_of_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring_obj (F : J ⥤ SemiRing.{max v u}) (j) :
semiring ((F ⋙ forget SemiRing).obj j) | by { change semiring (F.obj j), apply_instance } | instance | SemiRing.semiring_obj | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_subsemiring (F : J ⥤ SemiRing.{max v u}) :
subsemiring (Π j, F.obj j) | { carrier := (F ⋙ forget SemiRing).sections,
..(AddMon.sections_add_submonoid
(F ⋙ forget₂ SemiRing AddCommMon.{max v u} ⋙ forget₂ AddCommMon AddMon.{max v u})),
..(Mon.sections_submonoid (F ⋙ forget₂ SemiRing Mon.{max v u})) } | def | SemiRing.sections_subsemiring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Mon.sections_submonoid",
"SemiRing",
"subsemiring"
] | The flat sections of a functor into `SemiRing` form a subsemiring of all sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_semiring (F : J ⥤ SemiRing.{max v u}) :
semiring (types.limit_cone (F ⋙ forget SemiRing.{max v u})).X | (sections_subsemiring F).to_semiring | instance | SemiRing.limit_semiring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_π_ring_hom (F : J ⥤ SemiRing.{max v u}) (j) :
(types.limit_cone (F ⋙ forget SemiRing)).X →+* (F ⋙ forget SemiRing).obj j | { to_fun := (types.limit_cone (F ⋙ forget SemiRing)).π.app j,
..AddMon.limit_π_add_monoid_hom
(F ⋙ forget₂ SemiRing AddCommMon.{max v u} ⋙ forget₂ AddCommMon AddMon.{max v u}) j,
..Mon.limit_π_monoid_hom (F ⋙ forget₂ SemiRing Mon.{max v u}) j, } | def | SemiRing.limit_π_ring_hom | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Mon.limit_π_monoid_hom",
"SemiRing"
] | `limit.π (F ⋙ forget SemiRing) j` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ SemiRing.{max v u}) : cone F | { X := SemiRing.of (types.limit_cone (F ⋙ forget _)).X,
π :=
{ app := limit_π_ring_hom F,
naturality' := λ j j' f,
ring_hom.coe_inj ((types.limit_cone (F ⋙ forget _)).π.naturality f) } } | def | SemiRing.has_limits.limit_cone | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing.of",
"ring_hom.coe_inj"
] | Construction of a limit cone in `SemiRing`.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ SemiRing.{max v u}) : is_limit (limit_cone F) | begin
refine is_limit.of_faithful
(forget SemiRing) (types.limit_cone_is_limit _)
(λ s, ⟨_, _, _, _, _⟩) (λ s, rfl); tidy
end | def | SemiRing.has_limits.limit_cone_is_limit | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | Witness that the limit cone in `SemiRing` is a limit cone.
(Internal use only; use the limits API.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v} SemiRing.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk
{ cone := limit_cone F,
is_limit := limit_cone_is_limit F } } } | instance | SemiRing.has_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | The category of rings has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits SemiRing.{u} | SemiRing.has_limits_of_size.{u u} | instance | SemiRing.has_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_AddCommMon_preserves_limits_aux (F : J ⥤ SemiRing.{max v u}) :
is_limit ((forget₂ SemiRing AddCommMon).map_cone (limit_cone F)) | by apply AddCommMon.limit_cone_is_limit (F ⋙ forget₂ SemiRing AddCommMon.{max v u}) | def | SemiRing.forget₂_AddCommMon_preserves_limits_aux | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | An auxiliary declaration to speed up typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommMon_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ SemiRing AddCommMon.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (forget₂_AddCommMon_preserves_limits_aux F) } } | instance | SemiRing.forget₂_AddCommMon_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | The forgetful functor from semirings to additive commutative monoids preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommMon_preserves_limits : preserves_limits (forget₂ SemiRing AddCommMon.{u}) | SemiRing.forget₂_AddCommMon_preserves_limits_of_size.{u u} | instance | SemiRing.forget₂_AddCommMon_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_Mon_preserves_limits_aux (F : J ⥤ SemiRing.{max v u}) :
is_limit ((forget₂ SemiRing Mon).map_cone (limit_cone F)) | by apply Mon.has_limits.limit_cone_is_limit (F ⋙ forget₂ SemiRing Mon.{max v u}) | def | SemiRing.forget₂_Mon_preserves_limits_aux | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Mon",
"Mon.has_limits.limit_cone_is_limit",
"SemiRing"
] | An auxiliary declaration to speed up typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ SemiRing Mon.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (forget₂_Mon_preserves_limits_aux F) } } | instance | SemiRing.forget₂_Mon_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | The forgetful functor from semirings to monoids preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_limits : preserves_limits (forget₂ SemiRing Mon.{u}) | SemiRing.forget₂_Mon_preserves_limits_of_size.{u u} | instance | SemiRing.forget₂_Mon_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"SemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget SemiRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (types.limit_cone_is_limit (F ⋙ forget _)) } } | instance | SemiRing.forget_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | The forgetful functor from semirings to types preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget SemiRing.{u}) | SemiRing.forget_preserves_limits_of_size.{u u} | instance | SemiRing.forget_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring_obj (F : J ⥤ CommSemiRing.{max v u}) (j) :
comm_semiring ((F ⋙ forget CommSemiRing).obj j) | by { change comm_semiring (F.obj j), apply_instance } | instance | CommSemiRing.comm_semiring_obj | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_comm_semiring (F : J ⥤ CommSemiRing.{max v u}) :
comm_semiring (types.limit_cone (F ⋙ forget CommSemiRing.{max v u})).X | @subsemiring.to_comm_semiring (Π j, F.obj j) _
(SemiRing.sections_subsemiring (F ⋙ forget₂ CommSemiRing SemiRing.{max v u})) | instance | CommSemiRing.limit_comm_semiring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing",
"SemiRing.sections_subsemiring",
"comm_semiring",
"subsemiring.to_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone (F : J ⥤ CommSemiRing.{max v u}) : cone F | lift_limit (limit.is_limit (F ⋙ (forget₂ CommSemiRing SemiRing.{max v u}))) | def | CommSemiRing.limit_cone | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing"
] | A choice of limit cone for a functor into `CommSemiRing`.
(Generally, you'll just want to use `limit F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ CommSemiRing.{max v u}) : is_limit (limit_cone F) | lifted_limit_is_limit _ | def | CommSemiRing.limit_cone_is_limit | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | The chosen cone is a limit cone.
(Generally, you'll just want to use `limit.cone F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v v} CommSemiRing.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit_of_created F (forget₂ CommSemiRing SemiRing.{max v u}) } } | instance | CommSemiRing.has_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing"
] | The category of rings has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits CommSemiRing.{u} | CommSemiRing.has_limits_of_size.{u u} | instance | CommSemiRing.has_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_SemiRing_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ CommSemiRing SemiRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F, by apply_instance } } | instance | CommSemiRing.forget₂_SemiRing_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing"
] | The forgetful functor from rings to semirings preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_SemiRing_preserves_limits : preserves_limits (forget₂ CommSemiRing SemiRing.{u}) | CommSemiRing.forget₂_SemiRing_preserves_limits_of_size.{u u} | instance | CommSemiRing.forget₂_SemiRing_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget CommSemiRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F,
limits.comp_preserves_limit (forget₂ CommSemiRing SemiRing) (forget SemiRing) } } | instance | CommSemiRing.forget_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommSemiRing",
"SemiRing"
] | The forgetful functor from rings to types preserves all limits. (That is, the underlying
types could have been computed instead as limits in the category of types.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget CommSemiRing.{u}) | CommSemiRing.forget_preserves_limits_of_size.{u u} | instance | CommSemiRing.forget_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_obj (F : J ⥤ Ring.{max v u}) (j) :
ring ((F ⋙ forget Ring).obj j) | by { change ring (F.obj j), apply_instance } | instance | Ring.ring_obj | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_subring (F : J ⥤ Ring.{max v u}) :
subring (Π j, F.obj j) | { carrier := (F ⋙ forget Ring).sections,
.. AddGroup.sections_add_subgroup
(F ⋙ forget₂ Ring AddCommGroup.{max v u} ⋙ forget₂ AddCommGroup AddGroup.{max v u}),
.. SemiRing.sections_subsemiring (F ⋙ forget₂ Ring SemiRing.{max v u}) } | def | Ring.sections_subring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring",
"SemiRing.sections_subsemiring",
"subring"
] | The flat sections of a functor into `Ring` form a subring of all sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_ring (F : J ⥤ Ring.{max v u}) :
ring (types.limit_cone (F ⋙ forget Ring.{max v u})).X | (sections_subring F).to_ring | instance | Ring.limit_ring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone (F : J ⥤ Ring.{max v u}) : cone F | lift_limit (limit.is_limit (F ⋙ (forget₂ Ring SemiRing.{max v u}))) | def | Ring.limit_cone | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | A choice of limit cone for a functor into `Ring`.
(Generally, you'll just want to use `limit F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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