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lift (s : fan f) :
s.X ⟶ AddCommGroup.of (Π j,f j) | { to_fun := λ x j, s.π.app ⟨j⟩ x,
map_zero' := by { ext, simp },
map_add' := λ x y, by { ext, simp }, } | def | AddCommGroup.has_limit.lift | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [
"lift"
] | The map from an arbitrary cone over a indexed family of abelian groups
to the cartesian product of those groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_limit_cone : limits.limit_cone (discrete.functor f) | { cone :=
{ X := AddCommGroup.of (Π j, f j),
π := discrete.nat_trans (λ j, pi.eval_add_monoid_hom (λ j, f j) j.as), },
is_limit :=
{ lift := lift f,
fac' := λ s j, by { cases j, ext, simp, },
uniq' := λ s m w,
begin
ext x j,
dsimp only [has_limit.lift],
simp only [add_monoid_hom.... | def | AddCommGroup.has_limit.product_limit_cone | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [
"lift"
] | Construct limit data for a product in `AddCommGroup`, using `AddCommGroup.of (Π j, F.obj j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_iso_pi (f : J → AddCommGroup.{u}) :
(⨁ f : AddCommGroup) ≅ AddCommGroup.of (Π j, f j) | is_limit.cone_point_unique_up_to_iso
(biproduct.is_limit f)
(product_limit_cone f).is_limit | def | AddCommGroup.biproduct_iso_pi | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | We verify that the biproduct we've just defined is isomorphic to the AddCommGroup structure
on the dependent function type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_iso_pi_inv_comp_π (f : J → AddCommGroup.{u}) (j : J) :
(biproduct_iso_pi f).inv ≫ biproduct.π f j = pi.eval_add_monoid_hom (λ j, f j) j | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk j) | lemma | AddCommGroup.biproduct_iso_pi_inv_comp_π | algebra.category.Group | src/algebra/category/Group/biproducts.lean | [
"algebra.group.pi",
"algebra.category.Group.preadditive",
"category_theory.preadditive.biproducts",
"algebra.category.Group.limits"
] | [] | 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
| neg : prequotient → prequotient
| add : prequotient → prequotient → prequotient | inductive | AddCommGroup.colimits.prequotient | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | An inductive type representing all group 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 | AddCommGroup.colimits.relation | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The relation on `prequotient` saying when two expressions are equal
because of the abelian group laws, or
because one element is mapped to another by a morphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_setoid : setoid (prequotient F) | { r := relation F, iseqv := ⟨relation.refl, relation.symm, relation.trans⟩ } | def | AddCommGroup.colimits.colimit_setoid | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The setoid corresponding to group expressions modulo abelian group relations and identifications. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_type : Type v | quotient (colimit_setoid F) | def | AddCommGroup.colimits.colimit_type | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The underlying type of the colimit of a diagram in `AddCommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_zero : quot.mk setoid.r zero = (0 : colimit_type F) | rfl | lemma | AddCommGroup.colimits.quot_zero | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_neg (x) :
quot.mk setoid.r (neg x) = (-(quot.mk setoid.r x) : colimit_type F) | rfl | lemma | AddCommGroup.colimits.quot_neg | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_add (x y) :
quot.mk setoid.r (add x y) = ((quot.mk setoid.r x) + (quot.mk setoid.r y) : colimit_type F) | rfl | lemma | AddCommGroup.colimits.quot_add | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : AddCommGroup | AddCommGroup.of (colimit_type F) | def | AddCommGroup.colimits.colimit | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The bundled abelian group giving the colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_fun (j : J) (x : F.obj j) : colimit_type F | quot.mk _ (of j x) | def | AddCommGroup.colimits.cocone_fun | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The function from a given abelian group in the diagram to the colimit abelian group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (j : J) : F.obj j ⟶ colimit F | { to_fun := cocone_fun F j,
map_zero' := by apply quot.sound; apply relation.zero,
map_add' := by intros; apply quot.sound; apply relation.add } | def | AddCommGroup.colimits.cocone_morphism | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The group homomorphism from a given abelian group in the diagram to the colimit abelian
group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_naturality {j j' : J} (f : j ⟶ j') :
F.map f ≫ (cocone_morphism F j') = cocone_morphism F j | begin
ext,
apply quot.sound,
apply relation.map,
end | lemma | AddCommGroup.colimits.cocone_naturality | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [
"relation.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j):
(cocone_morphism F j') (F.map f x) = (cocone_morphism F j) x | by { rw ←cocone_naturality F f, refl } | lemma | AddCommGroup.colimits.cocone_naturality_components | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone : cocone F | { X := colimit F,
ι :=
{ app := cocone_morphism F } }. | def | AddCommGroup.colimits.colimit_cocone | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The cocone over the proposed colimit abelian group. | 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
| (neg x) := -(desc_fun_lift x)
| (add x y) := desc_fun_lift x + desc_fun_lift y | def | AddCommGroup.colimits.desc_fun_lift | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The function from the free abelian group 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, },
-- neg
{ simp, },
-- add
{ sim... | def | AddCommGroup.colimits.desc_fun | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The function from the colimit abelian group 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_zero' := rfl,
map_add' := λ x y, by { induction x; induction y; refl }, } | def | AddCommGroup.colimits.desc_morphism | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | The group homomorphism from the colimit abelian group to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_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 *, },
refl
end }. | def | AddCommGroup.colimits.colimit_cocone_is_colimit | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | Evidence that the proposed colimit is the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_AddCommGroup : has_colimits AddCommGroup | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F,
is_colimit := colimit_cocone_is_colimit F } } } | instance | AddCommGroup.colimits.has_colimits_AddCommGroup | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_iso_quotient {G H : AddCommGroup.{u}} (f : G ⟶ H) :
cokernel f ≅ AddCommGroup.of (H ⧸ (add_monoid_hom.range f)) | { hom := cokernel.desc f (mk' _)
(by { ext, apply quotient.sound, apply left_rel_apply.mpr, fsplit, exact -x,
simp only [add_zero, add_monoid_hom.map_neg], }),
inv := quotient_add_group.lift _ (cokernel.π f)
(by { intros x H_1, cases H_1, induction H_1_h,
simp only [cokernel.condition_appl... | def | AddCommGroup.cokernel_iso_quotient | algebra.category.Group | src/algebra/category/Group/colimits.lean | [
"algebra.category.Group.preadditive",
"group_theory.quotient_group",
"category_theory.limits.shapes.kernels",
"category_theory.concrete_category.elementwise"
] | [
"lift_mk",
"mk'"
] | The categorical cokernel of a morphism in `AddCommGroup`
agrees with the usual group-theoretical quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_eq_bot_of_cancel {f : A →* B} (h : ∀ (u v : f.ker →* A), f.comp u = f.comp v → u = v) :
f.ker = ⊥ | by simpa using _root_.congr_arg range (h f.ker.subtype 1 (by tidy)) | lemma | monoid_hom.ker_eq_bot_of_cancel | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_top_of_cancel {f : A →* B}
(h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v) :
f.range = ⊤ | begin
specialize h 1 (quotient_group.mk' _) _,
{ ext1,
simp only [one_apply, coe_comp, coe_mk', function.comp_app],
rw [show (1 : B ⧸ f.range) = (1 : B), from quotient_group.coe_one _, quotient_group.eq,
inv_one, one_mul],
exact ⟨x, rfl⟩, },
replace h : (quotient_group.mk' _).ker = (1 : B →* B ⧸ ... | lemma | monoid_hom.range_eq_top_of_cancel | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"inv_one",
"one_mul",
"quotient_group.coe_one",
"quotient_group.eq",
"quotient_group.ker_mk",
"quotient_group.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot_of_mono [mono f] : f.ker = ⊥ | monoid_hom.ker_eq_bot_of_cancel $ λ u v,
(@cancel_mono _ _ _ _ _ f _ (show Group.of f.ker ⟶ A, from u) _).1 | lemma | Group.ker_eq_bot_of_mono | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"Group.of",
"monoid_hom.ker_eq_bot_of_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_ker_eq_bot : mono f ↔ f.ker = ⊥ | ⟨λ h, @@ker_eq_bot_of_mono f h,
λ h, concrete_category.mono_of_injective _ $ (monoid_hom.ker_eq_bot_iff f).1 h⟩ | lemma | Group.mono_iff_ker_eq_bot | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"monoid_hom.ker_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective : mono f ↔ function.injective f | iff.trans (mono_iff_ker_eq_bot f) $ monoid_hom.ker_eq_bot_iff f | lemma | Group.mono_iff_injective | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"monoid_hom.ker_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_with_infinity
| from_coset : set.range (function.swap left_coset f.range.carrier) → X_with_infinity
| infinity : X_with_infinity | inductive | Group.surjective_of_epi_auxs.X_with_infinity | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"left_coset",
"set.range"
] | Define `X'` to be the set of all left cosets with an extra point at "infinity". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_smul (b b' : B) (x : X') : (b * b') • x = b • b' • x | match x with
| from_coset y :=
begin
change from_coset _ = from_coset _,
simp only [←subtype.val_eq_coe, left_coset_assoc],
end
| ∞ := rfl
end | lemma | Group.surjective_of_epi_auxs.mul_smul | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"left_coset_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_smul (x : X') : (1 : B) • x = x | match x with
| from_coset y :=
begin
change from_coset _ = from_coset _,
simp only [←subtype.val_eq_coe, one_left_coset, subtype.ext_iff_val],
end
| ∞ := rfl
end | lemma | Group.surjective_of_epi_auxs.one_smul | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"one_left_coset",
"one_smul",
"subtype.ext_iff_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_coset_eq_of_mem_range {b : B} (hb : b ∈ f.range) :
from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩ =
from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩ | begin
congr,
change b *l f.range = f.range,
nth_rewrite 1 [show (f.range : set B) = 1 *l f.range, from (one_left_coset _).symm],
rw [left_coset_eq_iff, mul_one],
exact subgroup.inv_mem _ hb,
end | lemma | Group.surjective_of_epi_auxs.from_coset_eq_of_mem_range | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"left_coset_eq_iff",
"mul_one",
"one_left_coset",
"subgroup.inv_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_coset_ne_of_nin_range {b : B} (hb : b ∉ f.range) :
from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩ ≠
from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩ | begin
intros r,
simp only [subtype.mk_eq_mk] at r,
change b *l f.range = f.range at r,
nth_rewrite 1 [show (f.range : set B) = 1 *l f.range, from (one_left_coset _).symm] at r,
rw [left_coset_eq_iff, mul_one] at r,
exact hb (inv_inv b ▸ (subgroup.inv_mem _ r)),
end | lemma | Group.surjective_of_epi_auxs.from_coset_ne_of_nin_range | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"inv_inv",
"left_coset_eq_iff",
"mul_one",
"one_left_coset",
"subgroup.inv_mem",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tau : SX' | equiv.swap (from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩) ∞ | def | Group.surjective_of_epi_auxs.tau | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.swap",
"one_left_coset"
] | Let `τ` be the permutation on `X'` exchanging `f.range` and the point at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
τ_apply_infinity :
τ ∞ = from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩ | equiv.swap_apply_right _ _ | lemma | Group.surjective_of_epi_auxs.τ_apply_infinity | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.swap_apply_right",
"one_left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
τ_apply_from_coset :
τ (from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩) = ∞ | equiv.swap_apply_left _ _ | lemma | Group.surjective_of_epi_auxs.τ_apply_from_coset | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.swap_apply_left",
"one_left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
τ_apply_from_coset' (x : B) (hx : x ∈ f.range) :
τ (from_coset ⟨x *l f.range.carrier, ⟨x, rfl⟩⟩) = ∞ | (from_coset_eq_of_mem_range _ hx).symm ▸ τ_apply_from_coset _ | lemma | Group.surjective_of_epi_auxs.τ_apply_from_coset' | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
τ_symm_apply_from_coset :
(equiv.symm τ) (from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩) = ∞ | by rw [tau, equiv.symm_swap, equiv.swap_apply_left] | lemma | Group.surjective_of_epi_auxs.τ_symm_apply_from_coset | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.swap_apply_left",
"equiv.symm",
"equiv.symm_swap",
"one_left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
τ_symm_apply_infinity :
(equiv.symm τ) ∞ = from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩ | by rw [tau, equiv.symm_swap, equiv.swap_apply_right] | lemma | Group.surjective_of_epi_auxs.τ_symm_apply_infinity | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.swap_apply_right",
"equiv.symm",
"equiv.symm_swap",
"one_left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
G : B →* SX' | { to_fun := λ β,
{ to_fun := λ x, β • x,
inv_fun := λ x, β⁻¹ • x,
left_inv := λ x, by { dsimp only, rw [←mul_smul, mul_left_inv, one_smul] },
right_inv := λ x, by { dsimp only, rw [←mul_smul, mul_right_inv, one_smul] } },
map_one' := by { ext, simp [one_smul] },
map_mul' := λ b1 b2, by { ext, simp [mu... | def | Group.surjective_of_epi_auxs.G | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"inv_fun",
"mul_left_inv",
"mul_right_inv",
"one_smul"
] | Let `g : B ⟶ S(X')` be defined as such that, for any `β : B`, `g(β)` is the function sending
point at infinity to point at infinity and sending coset `y` to `β *l y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
H : B →* SX' | { to_fun := λ β, ((τ).symm.trans (g β)).trans τ,
map_one' := by { ext, simp },
map_mul' := λ b1 b2, by { ext, simp } } | def | Group.surjective_of_epi_auxs.H | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | Define `h : B ⟶ S(X')` to be `τ g τ⁻¹` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
g_apply_from_coset (x : B) (y : X) : (g x) (from_coset y) = from_coset ⟨x *l y, by tidy⟩ | rfl | lemma | Group.surjective_of_epi_auxs.g_apply_from_coset | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
g_apply_infinity (x : B) : (g x) ∞ = ∞ | rfl | lemma | Group.surjective_of_epi_auxs.g_apply_infinity | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
h_apply_infinity (x : B) (hx : x ∈ f.range) : (h x) ∞ = ∞ | begin
simp only [H, monoid_hom.coe_mk, equiv.to_fun_as_coe, equiv.coe_trans, function.comp_app],
rw [τ_symm_apply_infinity, g_apply_from_coset],
simpa only [←subtype.val_eq_coe] using τ_apply_from_coset' f x hx,
end | lemma | Group.surjective_of_epi_auxs.h_apply_infinity | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.coe_trans",
"equiv.to_fun_as_coe",
"monoid_hom.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
h_apply_from_coset (x : B) :
(h x) (from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩) =
from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩ | by simp [H, τ_symm_apply_from_coset, g_apply_infinity, τ_apply_infinity] | lemma | Group.surjective_of_epi_auxs.h_apply_from_coset | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"one_left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
h_apply_from_coset' (x : B) (b : B) (hb : b ∈ f.range):
(h x) (from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩) =
from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩ | (from_coset_eq_of_mem_range _ hb).symm ▸ h_apply_from_coset f x | lemma | Group.surjective_of_epi_auxs.h_apply_from_coset' | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
h_apply_from_coset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
(h x) (from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩) =
from_coset ⟨(x * b) *l f.range.carrier, ⟨x * b, rfl⟩⟩ | begin
simp only [H, tau, monoid_hom.coe_mk, equiv.to_fun_as_coe, equiv.coe_trans, function.comp_app],
rw [equiv.symm_swap, @equiv.swap_apply_of_ne_of_ne X' _
(from_coset ⟨f.range.carrier, ⟨1, one_left_coset _⟩⟩) ∞
(from_coset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩) (from_coset_ne_of_nin_range _ hb) (by simp)],
... | lemma | Group.surjective_of_epi_auxs.h_apply_from_coset_nin_range | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.coe_trans",
"equiv.swap_apply_of_ne_of_ne",
"equiv.symm_swap",
"equiv.to_fun_as_coe",
"left_coset_assoc",
"monoid_hom.coe_mk",
"mul_left_inv",
"one_left_coset",
"one_mul",
"subgroup.inv_mem",
"subgroup.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
agree : f.range.carrier = {x | h x = g x} | begin
refine set.ext (λ b, ⟨_, λ (hb : h b = g b), classical.by_contradiction (λ r, _)⟩),
{ rintros ⟨a, rfl⟩,
change h (f a) = g (f a),
ext ⟨⟨_, ⟨y, rfl⟩⟩⟩,
{ rw [g_apply_from_coset],
by_cases m : y ∈ f.range,
{ rw [h_apply_from_coset' _ _ _ m, from_coset_eq_of_mem_range _ m],
change... | lemma | Group.surjective_of_epi_auxs.agree | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"fun_like.congr_fun",
"left_coset_assoc",
"one_left_coset",
"set.ext",
"subgroup.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq : f ≫ (show B ⟶ Group.of SX', from g) = f ≫ h | fun_like.ext _ _ $ λ a,
by simp only [comp_apply, show h (f a) = _, from (by simp [←agree] : f a ∈ {b | h b = g b})] | lemma | Group.surjective_of_epi_auxs.comp_eq | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"Group.of",
"comp_eq",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
g_ne_h (x : B) (hx : x ∉ f.range) : g ≠ h | begin
intros r,
replace r := fun_like.congr_fun (fun_like.congr_fun r x)
((from_coset ⟨f.range, ⟨1, one_left_coset _⟩⟩)),
rw [H, g_apply_from_coset, monoid_hom.coe_mk, tau] at r,
simp only [monoid_hom.coe_range, subtype.coe_mk, equiv.symm_swap,
equiv.to_fun_as_coe, equiv.coe_trans, function.comp_app] at... | lemma | Group.surjective_of_epi_auxs.g_ne_h | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"equiv.coe_trans",
"equiv.swap_apply_left",
"equiv.swap_apply_right",
"equiv.symm_swap",
"equiv.to_fun_as_coe",
"fun_like.congr_fun",
"monoid_hom.coe_mk",
"monoid_hom.coe_range",
"one_left_coset",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_of_epi [epi f] : function.surjective f | begin
by_contra r,
push_neg at r,
rcases r with ⟨b, hb⟩,
exact surjective_of_epi_auxs.g_ne_h f b (λ ⟨c, hc⟩, hb _ hc)
((cancel_epi f).1 (surjective_of_epi_auxs.comp_eq f)),
end | lemma | Group.surjective_of_epi | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective : epi f ↔ function.surjective f | ⟨λ h, @@surjective_of_epi f h, concrete_category.epi_of_surjective _⟩ | lemma | Group.epi_iff_surjective | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_range_eq_top : epi f ↔ f.range = ⊤ | iff.trans (epi_iff_surjective _) (subgroup.eq_top_iff' f.range).symm | lemma | Group.epi_iff_range_eq_top | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"subgroup.eq_top_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective : epi f ↔ function.surjective f | begin
have i1 : epi f ↔ epi (Group_AddGroup_equivalence.inverse.map f),
{ refine ⟨_, Group_AddGroup_equivalence.inverse.epi_of_epi_map⟩,
introsI e',
apply Group_AddGroup_equivalence.inverse.map_epi },
rwa Group.epi_iff_surjective at i1,
end | lemma | AddGroup.epi_iff_surjective | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"Group.epi_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_range_eq_top : epi f ↔ f.range = ⊤ | iff.trans (epi_iff_surjective _) (add_subgroup.eq_top_iff' f.range).symm | lemma | AddGroup.epi_iff_range_eq_top | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_Group_preserves_mono : (forget Group).preserves_monomorphisms | { preserves := λ X Y f e, by rwa [mono_iff_injective, ←category_theory.mono_iff_injective] at e } | instance | Group.forget_Group_preserves_mono | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_Group_preserves_epi : (forget Group).preserves_epimorphisms | { preserves := λ X Y f e, by rwa [epi_iff_surjective, ←category_theory.epi_iff_surjective] at e } | instance | Group.forget_Group_preserves_epi | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot_of_mono [mono f] : f.ker = ⊥ | monoid_hom.ker_eq_bot_of_cancel $ λ u v,
(@cancel_mono _ _ _ _ _ f _ (show CommGroup.of f.ker ⟶ A, from u) _).1 | lemma | CommGroup.ker_eq_bot_of_mono | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"CommGroup.of",
"monoid_hom.ker_eq_bot_of_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_top_of_epi [epi f] : f.range = ⊤ | monoid_hom.range_eq_top_of_cancel $ λ u v h,
(@cancel_epi _ _ _ _ _ f _ (show B ⟶ ⟨B ⧸ monoid_hom.range f⟩, from u) v).1 h | lemma | CommGroup.range_eq_top_of_epi | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"monoid_hom.range",
"monoid_hom.range_eq_top_of_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_range_eq_top : epi f ↔ f.range = ⊤ | ⟨λ hf, by exactI range_eq_top_of_epi _,
λ hf, concrete_category.epi_of_surjective _ $ monoid_hom.range_top_iff_surjective.mp hf⟩ | lemma | CommGroup.epi_iff_range_eq_top | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective : epi f ↔ function.surjective f | by rw [epi_iff_range_eq_top, monoid_hom.range_top_iff_surjective] | lemma | CommGroup.epi_iff_surjective | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"monoid_hom.range_top_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_CommGroup_preserves_mono : (forget CommGroup).preserves_monomorphisms | { preserves := λ X Y f e, by rwa [mono_iff_injective, ←category_theory.mono_iff_injective] at e } | instance | CommGroup.forget_CommGroup_preserves_mono | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"CommGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_CommGroup_preserves_epi : (forget CommGroup).preserves_epimorphisms | { preserves := λ X Y f e, by rwa [epi_iff_surjective, ←category_theory.epi_iff_surjective] at e } | instance | CommGroup.forget_CommGroup_preserves_epi | algebra.category.Group | src/algebra/category/Group/epi_mono.lean | [
"algebra.category.Group.equivalence_Group_AddGroup",
"group_theory.quotient_group"
] | [
"CommGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_AddGroup : Group ⥤ AddGroup | { obj := λ X, AddGroup.of (additive X),
map := λ X Y, monoid_hom.to_additive } | def | Group.to_AddGroup | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"Group",
"additive",
"monoid_hom.to_additive"
] | The functor `Group ⥤ AddGroup` by sending `X ↦ additive X` and `f ↦ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_AddCommGroup : CommGroup ⥤ AddCommGroup | { obj := λ X, AddCommGroup.of (additive X),
map := λ X Y, monoid_hom.to_additive } | def | CommGroup.to_AddCommGroup | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"CommGroup",
"additive",
"monoid_hom.to_additive"
] | The functor `CommGroup ⥤ AddCommGroup` by sending `X ↦ additive X` and `f ↦ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Group : AddGroup ⥤ Group | { obj := λ X, Group.of (multiplicative X),
map := λ X Y, add_monoid_hom.to_multiplicative } | def | AddGroup.to_Group | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"Group",
"Group.of",
"add_monoid_hom.to_multiplicative",
"multiplicative"
] | The functor `AddGroup ⥤ Group` by sending `X ↦ multiplicative Y` and `f ↦ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_CommGroup : AddCommGroup ⥤ CommGroup | { obj := λ X, CommGroup.of (multiplicative X),
map := λ X Y, add_monoid_hom.to_multiplicative } | def | AddCommGroup.to_CommGroup | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"CommGroup",
"CommGroup.of",
"add_monoid_hom.to_multiplicative",
"multiplicative"
] | The functor `AddCommGroup ⥤ CommGroup` by sending `X ↦ multiplicative Y` and `f ↦ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Group_AddGroup_equivalence : Group ≌ AddGroup | equivalence.mk Group.to_AddGroup AddGroup.to_Group
(nat_iso.of_components
(λ X, mul_equiv.to_Group_iso (mul_equiv.multiplicative_additive X))
(λ X Y f, rfl))
(nat_iso.of_components
(λ X, add_equiv.to_AddGroup_iso (add_equiv.additive_multiplicative X))
(λ X Y f, rfl)) | def | Group_AddGroup_equivalence | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"AddGroup.to_Group",
"Group",
"Group.to_AddGroup",
"add_equiv.additive_multiplicative",
"mul_equiv.multiplicative_additive",
"mul_equiv.to_Group_iso"
] | The equivalence of categories between `Group` and `AddGroup` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CommGroup_AddCommGroup_equivalence : CommGroup ≌ AddCommGroup | equivalence.mk CommGroup.to_AddCommGroup AddCommGroup.to_CommGroup
(nat_iso.of_components
(λ X, mul_equiv.to_CommGroup_iso (mul_equiv.multiplicative_additive X))
(λ X Y f, rfl))
(nat_iso.of_components
(λ X, add_equiv.to_AddCommGroup_iso (add_equiv.additive_multiplicative X))
(λ X Y f, rfl)) | def | CommGroup_AddCommGroup_equivalence | algebra.category.Group | src/algebra/category/Group/equivalence_Group_AddGroup.lean | [
"algebra.category.Group.basic",
"algebra.hom.equiv.type_tags"
] | [
"AddCommGroup.to_CommGroup",
"CommGroup",
"CommGroup.to_AddCommGroup",
"add_equiv.additive_multiplicative",
"mul_equiv.multiplicative_additive",
"mul_equiv.to_CommGroup_iso"
] | The equivalence of categories between `CommGroup` and `AddCommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
G : Mon | Mon.filtered_colimits.colimit (F ⋙ forget₂ Group Mon.{max v u}) | abbreviation | Group.filtered_colimits.G | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Mon",
"Mon.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ Group Mon` in the category `Mon`.
In the following, we will show that this has the structure of a group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
G.mk : (Σ j, F.obj j) → G | quot.mk (types.quot.rel (F ⋙ forget Group)) | abbreviation | Group.filtered_colimits.G.mk | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group"
] | The canonical projection into the colimit, as a quotient type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
G.mk_eq (x y : Σ j, F.obj j)
(h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) :
G.mk x = G.mk y | quot.eqv_gen_sound (types.filtered_colimit.eqv_gen_quot_rel_of_rel (F ⋙ forget Group) x y h) | lemma | Group.filtered_colimits.G.mk_eq | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_inv_aux (x : Σ j, F.obj j) : G | G.mk ⟨x.1, x.2 ⁻¹⟩ | def | Group.filtered_colimits.colimit_inv_aux | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [] | The "unlifted" version of taking inverses in the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_inv_aux_eq_of_rel (x y : Σ j, F.obj j)
(h : types.filtered_colimit.rel (F ⋙ forget Group) x y) :
colimit_inv_aux x = colimit_inv_aux y | begin
apply G.mk_eq,
obtain ⟨k, f, g, hfg⟩ := h,
use [k, f, g],
rw [monoid_hom.map_inv, monoid_hom.map_inv, inv_inj],
exact hfg,
end | lemma | Group.filtered_colimits.colimit_inv_aux_eq_of_rel | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"inv_inj",
"monoid_hom.map_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_has_inv : has_inv G | { inv := λ x, begin
refine quot.lift (colimit_inv_aux F) _ x,
intros x y h,
apply colimit_inv_aux_eq_of_rel,
apply types.filtered_colimit.rel_of_quot_rel,
exact h,
end } | instance | Group.filtered_colimits.colimit_has_inv | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [] | Taking inverses in the colimit. See also `colimit_inv_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_inv_mk_eq (x : Σ j, F.obj j) : (G.mk x) ⁻¹ = G.mk ⟨x.1, x.2 ⁻¹⟩ | rfl | lemma | Group.filtered_colimits.colimit_inv_mk_eq | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_group : group G | { mul_left_inv := λ x, begin
apply quot.induction_on x, clear x, intro x,
cases x with j x,
erw [colimit_inv_mk_eq,
colimit_mul_mk_eq (F ⋙ forget₂ Group Mon.{max v u}) ⟨j, _⟩ ⟨j, _⟩ j (𝟙 j) (𝟙 j),
colimit_one_eq (F ⋙ forget₂ Group Mon.{max v u}) j],
dsimp,
simp only [category_theory.fu... | instance | Group.filtered_colimits.colimit_group | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"group",
"mul_left_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : Group | Group.of G | def | Group.filtered_colimits.colimit | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Group.of"
] | The bundled group giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι := { ..(Mon.filtered_colimits.colimit_cocone (F ⋙ forget₂ Group Mon.{max v u})).ι } } | def | Group.filtered_colimits.colimit_cocone | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Mon.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t, Mon.filtered_colimits.colimit_desc (F ⋙ forget₂ Group Mon.{max v u})
((forget₂ Group Mon).map_cocone t),
fac' := λ t j, monoid_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget Group)).fac ((forget Group).map_cocone t) j,
uniq' := λ t m h, monoid_hom.coe_inj $
(types.colimit_coco... | def | Group.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Mon",
"Mon.filtered_colimits.colimit_desc",
"monoid_hom.coe_inj",
"monoid_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `Group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Mon_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ Group 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 Group Mon.{u})) } } | instance | Group.filtered_colimits.forget₂_Mon_preserves_filtered_colimits | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Mon.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits : preserves_filtered_colimits (forget Group.{u}) | limits.comp_preserves_filtered_colimits (forget₂ Group Mon) (forget Mon.{u}) | instance | Group.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"Group",
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
G : Group | Group.filtered_colimits.colimit (F ⋙ forget₂ CommGroup Group.{max v u}) | abbreviation | CommGroup.filtered_colimits.G | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"Group",
"Group.filtered_colimits.colimit"
] | The colimit of `F ⋙ forget₂ CommGroup Group` in the category `Group`.
In the following, we will show that this has the structure of a _commutative_ group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_comm_group : comm_group G | { ..G.group,
..CommMon.filtered_colimits.colimit_comm_monoid (F ⋙ forget₂ CommGroup CommMon.{max v u}) } | instance | CommGroup.filtered_colimits.colimit_comm_group | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"CommMon.filtered_colimits.colimit_comm_monoid",
"comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit : CommGroup | CommGroup.of G | def | CommGroup.filtered_colimits.colimit | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"CommGroup.of"
] | The bundled commutative group giving the filtered colimit of a diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone : cocone F | { X := colimit,
ι := { ..(Group.filtered_colimits.colimit_cocone (F ⋙ forget₂ CommGroup Group.{max v u})).ι } } | def | CommGroup.filtered_colimits.colimit_cocone | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"Group.filtered_colimits.colimit_cocone"
] | The cocone over the proposed colimit commutative group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit : is_colimit colimit_cocone | { desc := λ t,
(Group.filtered_colimits.colimit_cocone_is_colimit (F ⋙ forget₂ CommGroup Group.{max v u})).desc
((forget₂ CommGroup Group.{max v u}).map_cocone t),
fac' := λ t j, monoid_hom.coe_inj $
(types.colimit_cocone_is_colimit (F ⋙ forget CommGroup)).fac
((forget CommGroup).map_cocone t) j,
uniq... | def | CommGroup.filtered_colimits.colimit_cocone_is_colimit | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"Group.filtered_colimits.colimit_cocone_is_colimit",
"monoid_hom.coe_inj",
"monoid_hom.congr_fun"
] | The proposed colimit cocone is a colimit in `CommGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Group_preserves_filtered_colimits :
preserves_filtered_colimits (forget₂ CommGroup Group.{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)
(Group.filtered_colimits.colimit_cocone_is_colimit
(F ⋙ forget₂ CommGroup Group.{u})) } } | instance | CommGroup.filtered_colimits.forget₂_Group_preserves_filtered_colimits | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"Group.filtered_colimits.colimit_cocone_is_colimit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_filtered_colimits :
preserves_filtered_colimits (forget CommGroup.{u}) | limits.comp_preserves_filtered_colimits (forget₂ CommGroup Group) (forget Group.{u}) | instance | CommGroup.filtered_colimits.forget_preserves_filtered_colimits | algebra.category.Group | src/algebra/category/Group/filtered_colimits.lean | [
"algebra.category.Group.basic",
"algebra.category.Mon.filtered_colimits"
] | [
"CommGroup",
"Group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image : AddCommGroup | AddCommGroup.of (add_monoid_hom.range f) | def | AddCommGroup.image | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | the image of a morphism in AddCommGroup is just the bundling of `add_monoid_hom.range f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.ι : image f ⟶ H | f.range.subtype | def | AddCommGroup.image.ι | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | the inclusion of `image f` into the target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factor_thru_image : G ⟶ image f | f.range_restrict | def | AddCommGroup.factor_thru_image | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | the corestriction map to the image | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.fac : factor_thru_image f ≫ image.ι f = f | by { ext, refl, } | lemma | AddCommGroup.image.fac | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.lift (F' : mono_factorisation f) : image f ⟶ F'.I | { to_fun :=
(λ x, F'.e (classical.indefinite_description _ x.2).1 : image f → F'.I),
map_zero' :=
begin
haveI := F'.m_mono,
apply injective_of_mono F'.m,
change (F'.e ≫ F'.m) _ = _,
rw [F'.fac, add_monoid_hom.map_zero],
exact (classical.indefinite_description (λ y, f y = 0) _).2,
end,
map_... | def | AddCommGroup.image.lift | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | the universal property for the image factorisation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.lift_fac (F' : mono_factorisation f) : image.lift F' ≫ F'.m = image.ι f | begin
ext x,
change (F'.e ≫ F'.m) _ = _,
rw [F'.fac, (classical.indefinite_description _ x.2).2],
refl,
end | lemma | AddCommGroup.image.lift_fac | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_factorisation : mono_factorisation f | { I := image f,
m := image.ι f,
e := factor_thru_image f } | def | AddCommGroup.mono_factorisation | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | the factorisation of any morphism in AddCommGroup through a mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_image : is_image (mono_factorisation f) | { lift := image.lift,
lift_fac' := image.lift_fac } | def | AddCommGroup.is_image | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [
"lift"
] | the factorisation of any morphism in AddCommGroup through a mono has the universal property of
the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_iso_range {G H : AddCommGroup.{0}} (f : G ⟶ H) :
limits.image f ≅ AddCommGroup.of f.range | is_image.iso_ext (image.is_image f) (is_image f) | def | AddCommGroup.image_iso_range | algebra.category.Group | src/algebra/category/Group/images.lean | [
"algebra.category.Group.abelian",
"category_theory.limits.shapes.images"
] | [] | The categorical image of a morphism in `AddCommGroup`
agrees with the usual group-theoretical range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_of_injective_as_module [injective (⟨A⟩ : Module ℤ)] :
category_theory.injective (⟨A⟩ : AddCommGroup) | { factors := λ X Y g f m,
begin
resetI,
let G : (⟨X⟩ : Module ℤ) ⟶ ⟨A⟩ :=
{ map_smul' := by { intros, rw [ring_hom.id_apply, g.to_fun_eq_coe, map_zsmul], }, ..g },
let F : (⟨X⟩ : Module ℤ) ⟶ ⟨Y⟩ :=
{ map_smul' := by { intros, rw [ring_hom.id_apply, f.to_fun_eq_coe, map_zsmul], }, ..f },
ha... | lemma | AddCommGroup.injective_of_injective_as_module | algebra.category.Group | src/algebra/category/Group/injective.lean | [
"algebra.category.Group.epi_mono",
"algebra.category.Module.epi_mono",
"algebra.module.injective",
"category_theory.preadditive.injective",
"group_theory.divisible",
"ring_theory.principal_ideal_domain"
] | [
"Module",
"Module.coe_comp",
"category_theory.comp_apply",
"category_theory.injective",
"fun_like.congr_fun",
"linear_map.coe_mk",
"linear_map.to_add_monoid_hom_coe",
"map_zsmul",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_as_module_of_injective_as_Ab [injective (⟨A⟩ : AddCommGroup)] :
injective (⟨A⟩ : Module ℤ) | { factors := λ X Y g f m,
begin
resetI,
let G : (⟨X⟩ : AddCommGroup) ⟶ ⟨A⟩ := g.to_add_monoid_hom,
let F : (⟨X⟩ : AddCommGroup) ⟶ ⟨Y⟩ := f.to_add_monoid_hom,
haveI : mono F,
{ rw mono_iff_injective, intros _ _ h, exact ((Module.mono_iff_injective f).mp m) h, },
refine ⟨{map_smul' := _, ..injec... | lemma | AddCommGroup.injective_as_module_of_injective_as_Ab | algebra.category.Group | src/algebra/category/Group/injective.lean | [
"algebra.category.Group.epi_mono",
"algebra.category.Module.epi_mono",
"algebra.module.injective",
"category_theory.preadditive.injective",
"group_theory.divisible",
"ring_theory.principal_ideal_domain"
] | [
"Module",
"Module.mono_iff_injective",
"add_smul",
"fun_like.congr_fun",
"int.induction_on",
"one_smul",
"ring_hom.id_apply",
"sub_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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