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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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smul_apply (r : R) (p : nonarch_add_group_seminorm E) (x : E) : (r • p) x = r • p x | rfl | lemma | nonarch_add_group_seminorm.smul_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup (r : R) (p q : nonarch_add_group_seminorm E) : r • (p ⊔ q) = r • p ⊔ r • q | have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).coe_nonneg,
ext $ λ x, real.smul_max _ _ | lemma | nonarch_add_group_seminorm.smul_sup | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"mul_max_of_nonneg",
"nonarch_add_group_seminorm",
"smul_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_norm_class : group_norm_class (group_norm E) E ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_one_eq_zero := λ f, f.map_one',
map_mul_le_add := λ f, f.mul_le',
map_inv_eq_map := λ f, f.inv',
eq_one_of_map_eq_zero := λ f, f.eq_one_of_map_eq_zero' } | instance | group_norm.group_norm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_norm",
"group_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (x : E) : (1 : add_group_norm E) x = if x = 0 then 0 else 1 | rfl | lemma | add_group_norm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (x : E) : (1 : group_norm E) x = if x = 1 then 0 else 1 | rfl | lemma | group_norm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonarch_add_group_norm_class :
nonarch_add_group_norm_class (nonarch_add_group_norm E) E | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_add_le_max := λ f, f.add_le_max',
map_zero := λ f, f.map_zero',
map_neg_eq_map' := λ f, f.neg',
eq_zero_of_map_eq_zero := λ f, f.eq_zero_of_map_eq_zero' } | instance | nonarch_add_group_norm.nonarch_add_group_norm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_norm",
"nonarch_add_group_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq E] (x : E) :
(1 : nonarch_add_group_norm E) x = if x = 0 then 0 else 1 | rfl | lemma | nonarch_add_group_norm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SemiNormedGroup : Type (u+1) | bundled seminormed_add_comm_group | def | SemiNormedGroup | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"seminormed_add_comm_group"
] | The category of seminormed abelian groups and bounded group homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom @normed_add_group_hom | ⟨@normed_add_group_hom.to_fun, @normed_add_group_hom.id, @normed_add_group_hom.comp,
@normed_add_group_hom.coe_inj⟩ | instance | SemiNormedGroup.bundled_hom | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"normed_add_group_hom",
"normed_add_group_hom.comp",
"normed_add_group_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (M : Type u) [seminormed_add_comm_group M] : SemiNormedGroup | bundled.of M | def | SemiNormedGroup.of | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup",
"seminormed_add_comm_group"
] | Construct a bundled `SemiNormedGroup` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (V : Type u) [seminormed_add_comm_group V] :
(SemiNormedGroup.of V : Type u) = V | rfl | lemma | SemiNormedGroup.coe_of | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup.of",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id (V : SemiNormedGroup) : ⇑(𝟙 V) = id | rfl | lemma | SemiNormedGroup.coe_id | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp {M N K : SemiNormedGroup} (f : M ⟶ N) (g : N ⟶ K) :
((f ≫ g) : M → K) = g ∘ f | rfl | lemma | SemiNormedGroup.coe_comp | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_unique (V : Type u) [seminormed_add_comm_group V] [i : unique V] :
unique (SemiNormedGroup.of V) | i | instance | SemiNormedGroup.of_unique | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup.of",
"seminormed_add_comm_group",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply {V W : SemiNormedGroup} (x : V) : (0 : V ⟶ W) x = 0 | rfl | lemma | SemiNormedGroup.zero_apply | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero_of_subsingleton (V : SemiNormedGroup) [subsingleton V] :
limits.is_zero V | begin
refine ⟨λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩,
{ ext, have : x = 0 := subsingleton.elim _ _, simp only [this, map_zero], },
{ ext, apply subsingleton.elim }
end | lemma | SemiNormedGroup.is_zero_of_subsingleton | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object : limits.has_zero_object SemiNormedGroup.{u} | ⟨⟨of punit, is_zero_of_subsingleton _⟩⟩ | instance | SemiNormedGroup.has_zero_object | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_isometry_of_norm_noninc {V W : SemiNormedGroup} (i : V ≅ W)
(h1 : i.hom.norm_noninc) (h2 : i.inv.norm_noninc) :
isometry i.hom | begin
apply add_monoid_hom_class.isometry_of_norm,
intro v,
apply le_antisymm (h1 v),
calc ‖v‖ = ‖i.inv (i.hom v)‖ : by rw [iso.hom_inv_id_apply]
... ≤ ‖i.hom v‖ : h2 _,
end | lemma | SemiNormedGroup.iso_isometry_of_norm_noninc | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SemiNormedGroup₁ : Type (u+1) | bundled seminormed_add_comm_group | def | SemiNormedGroup₁ | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"seminormed_add_comm_group"
] | `SemiNormedGroup₁` is a type synonym for `SemiNormedGroup`,
which we shall equip with the category structure consisting only of the norm non-increasing maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext {M N : SemiNormedGroup₁} (f g : M ⟶ N) (w : (f : M → N) = (g : M → N)) :
f = g | subtype.eq (normed_add_group_hom.ext (congr_fun w)) | lemma | SemiNormedGroup₁.hom_ext | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁",
"hom_ext",
"normed_add_group_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (M : Type u) [seminormed_add_comm_group M] : SemiNormedGroup₁ | bundled.of M | def | SemiNormedGroup₁.of | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁",
"seminormed_add_comm_group"
] | Construct a bundled `SemiNormedGroup₁` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom {M N : SemiNormedGroup} (f : M ⟶ N) (i : f.norm_noninc) :
SemiNormedGroup₁.of M ⟶ SemiNormedGroup₁.of N | ⟨f, i⟩ | def | SemiNormedGroup₁.mk_hom | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup",
"SemiNormedGroup₁.of"
] | Promote a morphism in `SemiNormedGroup` to a morphism in `SemiNormedGroup₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom_apply {M N : SemiNormedGroup} (f : M ⟶ N) (i : f.norm_noninc) (x) :
mk_hom f i x = f x | rfl | lemma | SemiNormedGroup₁.mk_hom_apply | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_iso {M N : SemiNormedGroup} (f : M ≅ N) (i : f.hom.norm_noninc) (i' : f.inv.norm_noninc) :
SemiNormedGroup₁.of M ≅ SemiNormedGroup₁.of N | { hom := mk_hom f.hom i,
inv := mk_hom f.inv i',
hom_inv_id' := by { apply subtype.eq, exact f.hom_inv_id, },
inv_hom_id' := by { apply subtype.eq, exact f.inv_hom_id, }, } | def | SemiNormedGroup₁.mk_iso | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup",
"SemiNormedGroup₁.of"
] | Promote an isomorphism in `SemiNormedGroup` to an isomorphism in `SemiNormedGroup₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (V : Type u) [seminormed_add_comm_group V] :
(SemiNormedGroup₁.of V : Type u) = V | rfl | lemma | SemiNormedGroup₁.coe_of | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁.of",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id (V : SemiNormedGroup₁) : ⇑(𝟙 V) = id | rfl | lemma | SemiNormedGroup₁.coe_id | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp {M N K : SemiNormedGroup₁} (f : M ⟶ N) (g : N ⟶ K) :
((f ≫ g) : M → K) = g ∘ f | rfl | lemma | SemiNormedGroup₁.coe_comp | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp' {M N K : SemiNormedGroup₁} (f : M ⟶ N) (g : N ⟶ K) :
((f ≫ g) : normed_add_group_hom M K) = (↑g : normed_add_group_hom N K).comp ↑f | rfl | lemma | SemiNormedGroup₁.coe_comp' | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_unique (V : Type u) [seminormed_add_comm_group V] [i : unique V] :
unique (SemiNormedGroup₁.of V) | i | instance | SemiNormedGroup₁.of_unique | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁.of",
"seminormed_add_comm_group",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply {V W : SemiNormedGroup₁} (x : V) : (0 : V ⟶ W) x = 0 | rfl | lemma | SemiNormedGroup₁.zero_apply | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero_of_subsingleton (V : SemiNormedGroup₁) [subsingleton V] :
limits.is_zero V | begin
refine ⟨λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩,
{ ext, have : x = 0 := subsingleton.elim _ _, simp only [this, map_zero],
exact map_zero f.1 },
{ ext, apply subsingleton.elim }
end | lemma | SemiNormedGroup₁.is_zero_of_subsingleton | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object : limits.has_zero_object SemiNormedGroup₁.{u} | ⟨⟨of punit, is_zero_of_subsingleton _⟩⟩ | instance | SemiNormedGroup₁.has_zero_object | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_isometry {V W : SemiNormedGroup₁} (i : V ≅ W) :
isometry i.hom | begin
change isometry (i.hom : V →+ W),
refine add_monoid_hom_class.isometry_of_norm i.hom _,
intro v,
apply le_antisymm (i.hom.2 v),
calc ‖v‖ = ‖i.inv (i.hom v)‖ : by rw [iso.hom_inv_id_apply]
... ≤ ‖i.hom v‖ : i.inv.2 _,
end | lemma | SemiNormedGroup₁.iso_isometry | analysis.normed.group | src/analysis/normed/group/SemiNormedGroup.lean | [
"analysis.normed.group.hom",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise"
] | [
"SemiNormedGroup₁",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion : SemiNormedGroup.{u} ⥤ SemiNormedGroup.{u} | { obj := λ V, SemiNormedGroup.of (completion V),
map := λ V W f, f.completion,
map_id' := λ V, completion_id,
map_comp' := λ U V W f g, (completion_comp f g).symm } | def | SemiNormedGroup.Completion | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup.of"
] | The completion of a seminormed group, as an endofunctor on `SemiNormedGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Completion_complete_space {V : SemiNormedGroup} : complete_space (Completion.obj V) | completion.complete_space _ | instance | SemiNormedGroup.Completion_complete_space | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion.incl {V : SemiNormedGroup} : V ⟶ Completion.obj V | { to_fun := λ v, (v : completion V),
map_add' := completion.coe_add,
bound' := ⟨1, λ v, by simp⟩ } | def | SemiNormedGroup.Completion.incl | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"bound'"
] | The canonical morphism from a seminormed group `V` to its completion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Completion.norm_incl_eq {V : SemiNormedGroup} {v : V} : ‖Completion.incl v‖ = ‖v‖ | by simp | lemma | SemiNormedGroup.Completion.norm_incl_eq | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion.map_norm_noninc {V W : SemiNormedGroup} {f : V ⟶ W} (hf : f.norm_noninc) :
(Completion.map f).norm_noninc | normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.2 $
(normed_add_group_hom.norm_completion f).le.trans $
normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.1 hf | lemma | SemiNormedGroup.Completion.map_norm_noninc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"normed_add_group_hom.norm_completion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion.map_hom (V W : SemiNormedGroup.{u}) :
(V ⟶ W) →+ (Completion.obj V ⟶ Completion.obj W) | add_monoid_hom.mk' (category_theory.functor.map Completion) $ λ f g,
f.completion_add g | def | SemiNormedGroup.Completion.map_hom | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [] | Given a normed group hom `V ⟶ W`, this defines the associated morphism
from the completion of `V` to the completion of `W`.
The difference from the definition obtained from the functoriality of completion is in that the
map sending a morphism `f` to the associated morphism of completions is itself additive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Completion.map_zero (V W : SemiNormedGroup) : Completion.map (0 : V ⟶ W) = 0 | (Completion.map_hom V W).map_zero | lemma | SemiNormedGroup.Completion.map_zero | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion.lift {V W : SemiNormedGroup} [complete_space W] [separated_space W] (f : V ⟶ W) :
Completion.obj V ⟶ W | { to_fun := f.extension,
map_add' := f.extension.to_add_monoid_hom.map_add',
bound' := f.extension.bound' } | def | SemiNormedGroup.Completion.lift | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"bound'",
"complete_space",
"separated_space"
] | Given a normed group hom `f : V → W` with `W` complete, this provides a lift of `f` to
the completion of `V`. The lemmas `lift_unique` and `lift_comp_incl` provide the api for the
universal property of the completion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Completion.lift_comp_incl {V W : SemiNormedGroup} [complete_space W] [separated_space W]
(f : V ⟶ W) : Completion.incl ≫ (Completion.lift f) = f | by { ext, apply normed_add_group_hom.extension_coe } | lemma | SemiNormedGroup.Completion.lift_comp_incl | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"complete_space",
"normed_add_group_hom.extension_coe",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Completion.lift_unique {V W : SemiNormedGroup} [complete_space W] [separated_space W]
(f : V ⟶ W) (g : Completion.obj V ⟶ W) : Completion.incl ≫ g = f → g = Completion.lift f | λ h, (normed_add_group_hom.extension_unique _ (λ v, ((ext_iff.1 h) v).symm)).symm | lemma | SemiNormedGroup.Completion.lift_unique | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/completion.lean | [
"analysis.normed.group.SemiNormedGroup",
"category_theory.preadditive.additive_functor",
"analysis.normed.group.hom_completion"
] | [
"SemiNormedGroup",
"complete_space",
"normed_add_group_hom.extension_unique",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_cocone {X Y : SemiNormedGroup₁.{u}} (f : X ⟶ Y) : cofork f 0 | cofork.of_π
(@SemiNormedGroup₁.mk_hom
_ (SemiNormedGroup.of (Y ⧸ (normed_add_group_hom.range f.1)))
f.1.range.normed_mk
(normed_add_group_hom.is_quotient_quotient _).norm_le)
begin
ext,
simp only [comp_apply, limits.zero_comp, normed_add_group_hom.zero_apply,
SemiNormedGroup₁.mk_hom_apply,... | def | SemiNormedGroup₁.cokernel_cocone | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"SemiNormedGroup.of",
"SemiNormedGroup₁.mk_hom",
"SemiNormedGroup₁.mk_hom_apply",
"SemiNormedGroup₁.zero_apply",
"normed_add_group_hom.is_quotient_quotient",
"normed_add_group_hom.range",
"normed_add_group_hom.zero_apply"
] | Auxiliary definition for `has_cokernels SemiNormedGroup₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_lift {X Y : SemiNormedGroup₁.{u}} (f : X ⟶ Y) (s : cokernel_cofork f) :
(cokernel_cocone f).X ⟶ s.X | begin
fsplit,
-- The lift itself:
{ apply normed_add_group_hom.lift _ s.π.1,
rintro _ ⟨b, rfl⟩,
change (f ≫ s.π) b = 0,
simp, },
-- The lift has norm at most one:
exact normed_add_group_hom.lift_norm_noninc _ _ _ s.π.2,
end | def | SemiNormedGroup₁.cokernel_lift | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.lift",
"normed_add_group_hom.lift_norm_noninc"
] | Auxiliary definition for `has_cokernels SemiNormedGroup₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fork {V W : SemiNormedGroup.{u}} (f g : V ⟶ W) : fork f g | @fork.of_ι _ _ _ _ _ _ (of (f - g).ker) (normed_add_group_hom.incl (f - g).ker) $
begin
ext v,
have : v.1 ∈ (f - g).ker := v.2,
simpa only [normed_add_group_hom.incl_apply, pi.zero_apply, coe_comp,
normed_add_group_hom.coe_zero, subtype.val_eq_coe, normed_add_group_hom.mem_ker,
normed_add_group_hom.coe_su... | def | SemiNormedGroup.fork | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.coe_sub",
"normed_add_group_hom.coe_zero",
"normed_add_group_hom.incl",
"normed_add_group_hom.mem_ker",
"subtype.val_eq_coe"
] | The equalizer cone for a parallel pair of morphisms of seminormed groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_parallel_pair {V W : SemiNormedGroup.{u}} (f g : V ⟶ W) :
has_limit (parallel_pair f g) | { exists_limit := nonempty.intro
{ cone := fork f g,
is_limit := fork.is_limit.mk _
(λ c, normed_add_group_hom.ker.lift (fork.ι c) _ $
show normed_add_group_hom.comp_hom (f - g) c.ι = 0,
by { rw [add_monoid_hom.map_sub, add_monoid_hom.sub_apply, sub_eq_zero], exact c.condition })
(λ c, nor... | instance | SemiNormedGroup.has_limit_parallel_pair | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.comp_hom",
"normed_add_group_hom.ker.incl_comp_lift",
"normed_add_group_hom.ker.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_cocone {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) : cofork f 0 | @cofork.of_π _ _ _ _ _ _
(SemiNormedGroup.of (Y ⧸ (normed_add_group_hom.range f)))
f.range.normed_mk
begin
ext,
simp only [comp_apply, limits.zero_comp, normed_add_group_hom.zero_apply,
←normed_add_group_hom.mem_ker, f.range.ker_normed_mk, f.mem_range, exists_apply_eq_apply],
end | def | SemiNormedGroup.cokernel_cocone | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"SemiNormedGroup.of",
"exists_apply_eq_apply",
"normed_add_group_hom.range",
"normed_add_group_hom.zero_apply"
] | Auxiliary definition for `has_cokernels SemiNormedGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_lift {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) (s : cokernel_cofork f) :
(cokernel_cocone f).X ⟶ s.X | normed_add_group_hom.lift _ s.π
begin
rintro _ ⟨b, rfl⟩,
change (f ≫ s.π) b = 0,
simp,
end | def | SemiNormedGroup.cokernel_lift | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.lift"
] | Auxiliary definition for `has_cokernels SemiNormedGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cokernel_cocone {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
is_colimit (cokernel_cocone f) | is_colimit_aux _ (cokernel_lift f)
(λ s, begin
ext,
apply normed_add_group_hom.lift_mk f.range,
rintro _ ⟨b, rfl⟩,
change (f ≫ s.π) b = 0,
simp,
end)
(λ s m w, normed_add_group_hom.lift_unique f.range _ _ _ w) | def | SemiNormedGroup.is_colimit_cokernel_cocone | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.lift_mk",
"normed_add_group_hom.lift_unique"
] | Auxiliary definition for `has_cokernels SemiNormedGroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_cokernel {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) : SemiNormedGroup.{u} | (cokernel_cocone f).X | def | SemiNormedGroup.explicit_cokernel | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | An explicit choice of cokernel, which has good properties with respect to the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_cokernel_desc {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) : explicit_cokernel f ⟶ Z | (is_colimit_cokernel_cocone f).desc (cofork.of_π g (by simp [w])) | def | SemiNormedGroup.explicit_cokernel_desc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | Descend to the explicit cokernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_cokernel_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) : Y ⟶ explicit_cokernel f | (cokernel_cocone f).ι.app walking_parallel_pair.one | def | SemiNormedGroup.explicit_cokernel_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | The projection from `Y` to the explicit cokernel of `X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_cokernel_π_surjective {X Y : SemiNormedGroup.{u}} {f : X ⟶ Y} :
function.surjective (explicit_cokernel_π f) | surjective_quot_mk _ | lemma | SemiNormedGroup.explicit_cokernel_π_surjective | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"surjective_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_explicit_cokernel_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
f ≫ explicit_cokernel_π f = 0 | begin
convert (cokernel_cocone f).w walking_parallel_pair_hom.left,
simp,
end | lemma | SemiNormedGroup.comp_explicit_cokernel_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_π_apply_dom_eq_zero {X Y : SemiNormedGroup.{u}} {f : X ⟶ Y} (x : X) :
(explicit_cokernel_π f) (f x) = 0 | show (f ≫ (explicit_cokernel_π f)) x = 0, by { rw [comp_explicit_cokernel_π], refl } | lemma | SemiNormedGroup.explicit_cokernel_π_apply_dom_eq_zero | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_π_desc {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) : explicit_cokernel_π f ≫ explicit_cokernel_desc w = g | (is_colimit_cokernel_cocone f).fac _ _ | lemma | SemiNormedGroup.explicit_cokernel_π_desc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_π_desc_apply {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
{cond : f ≫ g = 0} (x : Y) : explicit_cokernel_desc cond (explicit_cokernel_π f x) = g x | show (explicit_cokernel_π f ≫ explicit_cokernel_desc cond) x = g x, by rw explicit_cokernel_π_desc | lemma | SemiNormedGroup.explicit_cokernel_π_desc_apply | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_unique {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) (e : explicit_cokernel f ⟶ Z) (he : explicit_cokernel_π f ≫ e = g) :
e = explicit_cokernel_desc w | begin
apply (is_colimit_cokernel_cocone f).uniq (cofork.of_π g (by simp [w])),
rintro (_|_),
{ convert w.symm,
simp },
{ exact he }
end | lemma | SemiNormedGroup.explicit_cokernel_desc_unique | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_comp_eq_desc {X Y Z W : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
{h : Z ⟶ W} {cond : f ≫ g = 0} :
explicit_cokernel_desc cond ≫ h = explicit_cokernel_desc (show f ≫ (g ≫ h) = 0,
by rw [← category_theory.category.assoc, cond, limits.zero_comp]) | begin
refine explicit_cokernel_desc_unique _ _ _,
rw [← category_theory.category.assoc, explicit_cokernel_π_desc]
end | lemma | SemiNormedGroup.explicit_cokernel_desc_comp_eq_desc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_zero {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} :
explicit_cokernel_desc (show f ≫ (0 : Y ⟶ Z) = 0, from category_theory.limits.comp_zero) = 0 | eq.symm $ explicit_cokernel_desc_unique _ _ category_theory.limits.comp_zero | lemma | SemiNormedGroup.explicit_cokernel_desc_zero | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"category_theory.limits.comp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_hom_ext {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y}
(e₁ e₂ : explicit_cokernel f ⟶ Z)
(h : explicit_cokernel_π f ≫ e₁ = explicit_cokernel_π f ≫ e₂) : e₁ = e₂ | begin
let g : Y ⟶ Z := explicit_cokernel_π f ≫ e₂,
have w : f ≫ g = 0, by simp,
have : e₂ = explicit_cokernel_desc w,
{ apply explicit_cokernel_desc_unique, refl },
rw this,
apply explicit_cokernel_desc_unique,
exact h,
end | lemma | SemiNormedGroup.explicit_cokernel_hom_ext | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_π.epi {X Y : SemiNormedGroup.{u}} {f : X ⟶ Y} :
epi (explicit_cokernel_π f) | begin
constructor,
intros Z g h H,
ext x,
obtain ⟨x, hx⟩ := explicit_cokernel_π_surjective (explicit_cokernel_π f x),
change (explicit_cokernel_π f ≫ g) _ = _,
rw [H]
end | instance | SemiNormedGroup.explicit_cokernel_π.epi | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quotient_explicit_cokernel_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
normed_add_group_hom.is_quotient (explicit_cokernel_π f) | normed_add_group_hom.is_quotient_quotient _ | lemma | SemiNormedGroup.is_quotient_explicit_cokernel_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.is_quotient",
"normed_add_group_hom.is_quotient_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_noninc_explicit_cokernel_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
(explicit_cokernel_π f).norm_noninc | (is_quotient_explicit_cokernel_π f).norm_le | lemma | SemiNormedGroup.norm_noninc_explicit_cokernel_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_norm_le_of_norm_le {X Y Z : SemiNormedGroup.{u}}
{f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0) (c : ℝ≥0) (h : ‖ g ‖ ≤ c) :
‖ explicit_cokernel_desc w ‖ ≤ c | normed_add_group_hom.lift_norm_le _ _ _ h | lemma | SemiNormedGroup.explicit_cokernel_desc_norm_le_of_norm_le | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"normed_add_group_hom.lift_norm_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_norm_noninc {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
{cond : f ≫ g = 0} (hg : g.norm_noninc) :
(explicit_cokernel_desc cond).norm_noninc | begin
refine normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.2 _,
rw [← nnreal.coe_one],
exact explicit_cokernel_desc_norm_le_of_norm_le cond 1
(normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.1 hg)
end | lemma | SemiNormedGroup.explicit_cokernel_desc_norm_noninc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"nnreal.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_comp_eq_zero {X Y Z W : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
{h : Z ⟶ W} (cond : f ≫ g = 0) (cond2 : g ≫ h = 0) :
explicit_cokernel_desc cond ≫ h = 0 | begin
rw [← cancel_epi (explicit_cokernel_π f), ← category.assoc, explicit_cokernel_π_desc],
simp [cond2]
end | lemma | SemiNormedGroup.explicit_cokernel_desc_comp_eq_zero | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_desc_norm_le {X Y Z : SemiNormedGroup.{u}}
{f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0) : ‖ explicit_cokernel_desc w ‖ ≤ ‖ g ‖ | explicit_cokernel_desc_norm_le_of_norm_le w ‖ g ‖₊ le_rfl | lemma | SemiNormedGroup.explicit_cokernel_desc_norm_le | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_iso {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
explicit_cokernel f ≅ cokernel f | (is_colimit_cokernel_cocone f).cocone_point_unique_up_to_iso (colimit.is_colimit _) | def | SemiNormedGroup.explicit_cokernel_iso | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | The explicit cokernel is isomorphic to the usual cokernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_cokernel_iso_hom_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
explicit_cokernel_π f ≫ (explicit_cokernel_iso f).hom = cokernel.π _ | by simp [explicit_cokernel_π, explicit_cokernel_iso, is_colimit.cocone_point_unique_up_to_iso] | lemma | SemiNormedGroup.explicit_cokernel_iso_hom_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_iso_inv_π {X Y : SemiNormedGroup.{u}} (f : X ⟶ Y) :
cokernel.π f ≫ (explicit_cokernel_iso f).inv = explicit_cokernel_π f | by simp [explicit_cokernel_π, explicit_cokernel_iso] | lemma | SemiNormedGroup.explicit_cokernel_iso_inv_π | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel_iso_hom_desc {X Y Z : SemiNormedGroup.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) :
(explicit_cokernel_iso f).hom ≫ cokernel.desc f g w = explicit_cokernel_desc w | begin
ext1,
simp [explicit_cokernel_desc, explicit_cokernel_π, explicit_cokernel_iso,
is_colimit.cocone_point_unique_up_to_iso],
end | lemma | SemiNormedGroup.explicit_cokernel_iso_hom_desc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
explicit_cokernel.map {A B C D : SemiNormedGroup.{u}} {fab : A ⟶ B}
{fbd : B ⟶ D} {fac : A ⟶ C} {fcd : C ⟶ D} (h : fab ≫ fbd = fac ≫ fcd) :
explicit_cokernel fab ⟶ explicit_cokernel fcd | @explicit_cokernel_desc _ _ _ fab (fbd ≫ explicit_cokernel_π _) $ by simp [reassoc_of h] | def | SemiNormedGroup.explicit_cokernel.map | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | A special case of `category_theory.limits.cokernel.map` adapted to `explicit_cokernel`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
explicit_coker.map_desc {A B C D B' D' : SemiNormedGroup.{u}}
{fab : A ⟶ B} {fbd : B ⟶ D} {fac : A ⟶ C} {fcd : C ⟶ D}
{h : fab ≫ fbd = fac ≫ fcd} {fbb' : B ⟶ B'} {fdd' : D ⟶ D'}
{condb : fab ≫ fbb' = 0} {condd : fcd ≫ fdd' = 0} {g : B' ⟶ D'}
(h' : fbb' ≫ g = fbd ≫ fdd'):
explicit_cokernel_desc condb ≫ g = exp... | begin
delta explicit_cokernel.map,
simp [← cancel_epi (explicit_cokernel_π fab), category.assoc, explicit_cokernel_π_desc, h']
end | lemma | SemiNormedGroup.explicit_coker.map_desc | analysis.normed.group.SemiNormedGroup | src/analysis/normed/group/SemiNormedGroup/kernels.lean | [
"analysis.normed.group.SemiNormedGroup",
"analysis.normed.group.quotient",
"category_theory.limits.shapes.kernels"
] | [] | A special case of `category_theory.limits.cokernel.map_desc` adapted to `explicit_cokernel`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ordered_add_group (α : Type*)
extends ordered_add_comm_group α, has_norm α, metric_space α | (dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | normed_ordered_add_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"has_norm",
"metric_space",
"ordered_add_comm_group"
] | A `normed_ordered_add_group` is an additive group that is both a `normed_add_comm_group` and an
`ordered_add_comm_group`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ordered_group (α : Type*) extends ordered_comm_group α, has_norm α, metric_space α | (dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | normed_ordered_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"has_norm",
"metric_space",
"ordered_comm_group"
] | A `normed_ordered_group` is a group that is both a `normed_comm_group` and an
`ordered_comm_group`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_linear_ordered_add_group (α : Type*)
extends linear_ordered_add_comm_group α, has_norm α, metric_space α | (dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously) | class | normed_linear_ordered_add_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"has_norm",
"linear_ordered_add_comm_group",
"metric_space"
] | A `normed_linear_ordered_add_group` is an additive group that is both a `normed_add_comm_group`
and a `linear_ordered_add_comm_group`. This class is necessary to avoid diamonds caused by both
classes carrying their own group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_linear_ordered_group (α : Type*)
extends linear_ordered_comm_group α, has_norm α, metric_space α | (dist_eq : ∀ x y, dist x y = ‖x / y‖ . obviously) | class | normed_linear_ordered_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"has_norm",
"linear_ordered_comm_group",
"metric_space"
] | A `normed_linear_ordered_group` is a group that is both a `normed_comm_group` and a
`linear_ordered_comm_group`. This class is necessary to avoid diamonds caused by both classes
carrying their own group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_linear_ordered_field (α : Type*)
extends linear_ordered_field α, has_norm α, metric_space α | (dist_eq : ∀ x y, dist x y = ‖x - y‖ . obviously)
(norm_mul' : ∀ x y : α, ‖x * y‖ = ‖x‖ * ‖y‖) | class | normed_linear_ordered_field | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"has_norm",
"linear_ordered_field",
"metric_space"
] | A `normed_linear_ordered_field` is a field that is both a `normed_field` and a
`linear_ordered_field`. This class is necessary to avoid diamonds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ordered_group.to_normed_comm_group [normed_ordered_group α] : normed_comm_group α | ⟨normed_ordered_group.dist_eq⟩ | instance | normed_ordered_group.to_normed_comm_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"normed_comm_group",
"normed_ordered_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_linear_ordered_group.to_normed_ordered_group [normed_linear_ordered_group α] :
normed_ordered_group α | ⟨normed_linear_ordered_group.dist_eq⟩ | instance | normed_linear_ordered_group.to_normed_ordered_group | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"normed_linear_ordered_group",
"normed_ordered_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_linear_ordered_field.to_normed_field (α : Type*)
[normed_linear_ordered_field α] : normed_field α | { dist_eq := normed_linear_ordered_field.dist_eq,
norm_mul' := normed_linear_ordered_field.norm_mul' } | instance | normed_linear_ordered_field.to_normed_field | analysis.normed.order | src/analysis/normed/order/basic.lean | [
"algebra.order.group.type_tags",
"analysis.normed_space.basic"
] | [
"normed_field",
"normed_linear_ordered_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_solid_norm (α : Type*) [normed_add_comm_group α] [lattice α] : Prop | (solid : ∀ ⦃x y : α⦄, |x| ≤ |y| → ‖x‖ ≤ ‖y‖) | class | has_solid_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"lattice",
"normed_add_comm_group"
] | Let `α` be an `add_comm_group` with a `lattice` structure. A norm on `α` is *solid* if, for `a`
and `b` in `α`, with absolute values `|a|` and `|b|` respectively, `|a| ≤ |b|` implies `‖a‖ ≤ ‖b‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_norm_of_abs_le_abs {a b : α} (h : |a| ≤ |b|) : ‖a‖ ≤ ‖b‖ | has_solid_norm.solid h | lemma | norm_le_norm_of_abs_le_abs | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lattice_ordered_add_comm_group.is_solid_ball (r : ℝ) :
lattice_ordered_add_comm_group.is_solid (metric.ball (0 : α) r) | λ _ hx _ hxy, mem_ball_zero_iff.mpr ((has_solid_norm.solid hxy).trans_lt (mem_ball_zero_iff.mp hx)) | lemma | lattice_ordered_add_comm_group.is_solid_ball | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"lattice_ordered_add_comm_group.is_solid",
"metric.ball"
] | If `α` has a solid norm, then the balls centered at the origin of `α` are solid sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_lattice_add_comm_group (α : Type*)
extends normed_add_comm_group α, lattice α, has_solid_norm α | (add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b) | class | normed_lattice_add_comm_group | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"has_solid_norm",
"lattice",
"normed_add_comm_group"
] | Let `α` be a normed commutative group equipped with a partial order covariant with addition, with
respect which `α` forms a lattice. Suppose that `α` is *solid*, that is to say, for `a` and `b` in
`α`, with absolute values `|a|` and `|b|` respectively, `|a| ≤ |b|` implies `‖a‖ ≤ ‖b‖`. Then `α` is
said to be a normed la... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_lattice_add_comm_group_to_ordered_add_comm_group {α : Type*}
[h : normed_lattice_add_comm_group α] : ordered_add_comm_group α | { ..h } | instance | normed_lattice_add_comm_group_to_ordered_add_comm_group | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"normed_lattice_add_comm_group",
"ordered_add_comm_group"
] | A normed lattice ordered group is an ordered additive commutative group | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_solid (a b : α) (h: b⊓-b ≤ a⊓-a) : ‖a‖ ≤ ‖b‖ | begin
apply solid,
rw abs_eq_sup_neg,
nth_rewrite 0 ← neg_neg a,
rw ← neg_inf_eq_sup_neg,
rw abs_eq_sup_neg,
nth_rewrite 0 ← neg_neg b,
rwa [← neg_inf_eq_sup_neg, neg_le_neg_iff, @inf_comm _ _ _ b, @inf_comm _ _ _ a],
end | lemma | dual_solid | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_abs_eq_norm (a : α) : ‖|a|‖ = ‖a‖ | (solid (abs_abs a).le).antisymm (solid (abs_abs a).symm.le) | lemma | norm_abs_eq_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"abs_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inf_sub_inf_le_add_norm (a b c d : α) : ‖a ⊓ b - c ⊓ d‖ ≤ ‖a - c‖ + ‖b - d‖ | begin
rw [← norm_abs_eq_norm (a - c), ← norm_abs_eq_norm (b - d)],
refine le_trans (solid _) (norm_add_le (|a - c|) (|b - d|)),
rw abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d))),
calc |a ⊓ b - c ⊓ d| =
|a ⊓ b - c ⊓ b + (c ⊓ b - c ⊓ d)| : by rw sub_add_sub_cancel
.... | lemma | norm_inf_sub_inf_le_add_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"abs_nonneg",
"abs_of_nonneg",
"inf_comm",
"norm_abs_eq_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sup_sub_sup_le_add_norm (a b c d : α) : ‖a ⊔ b - (c ⊔ d)‖ ≤ ‖a - c‖ + ‖b - d‖ | begin
rw [← norm_abs_eq_norm (a - c), ← norm_abs_eq_norm (b - d)],
refine le_trans (solid _) (norm_add_le (|a - c|) (|b - d|)),
rw abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d))),
calc |a ⊔ b - (c ⊔ d)| =
|a ⊔ b - (c ⊔ b) + (c ⊔ b - (c ⊔ d))| : by rw sub_add_sub_canc... | lemma | norm_sup_sub_sup_le_add_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"abs_nonneg",
"abs_of_nonneg",
"norm_abs_eq_norm",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inf_le_add (x y : α) : ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖ | begin
have h : ‖x ⊓ y - 0 ⊓ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_inf_sub_inf_le_add_norm x y 0 0,
simpa only [inf_idem, sub_zero] using h,
end | lemma | norm_inf_le_add | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"inf_idem",
"norm_inf_sub_inf_le_add_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sup_le_add (x y : α) : ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖ | begin
have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0,
simpa only [sup_idem, sub_zero] using h,
end | lemma | norm_sup_le_add | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"norm_sup_sub_sup_le_add_norm",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_lattice_add_comm_group_has_continuous_inf : has_continuous_inf α | begin
refine ⟨continuous_iff_continuous_at.2 $ λ q, tendsto_iff_norm_tendsto_zero.2 $ _⟩,
have : ∀ p : α × α, ‖p.1 ⊓ p.2 - q.1 ⊓ q.2‖ ≤ ‖p.1 - q.1‖ + ‖p.2 - q.2‖,
from λ _, norm_inf_sub_inf_le_add_norm _ _ _ _,
refine squeeze_zero (λ e, norm_nonneg _) this _,
convert (((continuous_fst.tendsto q).sub tendsto... | instance | normed_lattice_add_comm_group_has_continuous_inf | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"has_continuous_inf",
"norm_inf_sub_inf_le_add_norm",
"squeeze_zero",
"tendsto_const_nhds"
] | Let `α` be a normed lattice ordered group. Then the infimum is jointly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_lattice_add_comm_group_has_continuous_sup {α : Type*}
[normed_lattice_add_comm_group α] :
has_continuous_sup α | order_dual.has_continuous_sup αᵒᵈ | instance | normed_lattice_add_comm_group_has_continuous_sup | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"has_continuous_sup",
"normed_lattice_add_comm_group",
"order_dual.has_continuous_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_lattice_add_comm_group_topological_lattice : topological_lattice α | topological_lattice.mk | instance | normed_lattice_add_comm_group_topological_lattice | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"topological_lattice"
] | Let `α` be a normed lattice ordered group. Then `α` is a topological lattice in the norm topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_abs_sub_abs (a b : α) :
‖ |a| - |b| ‖ ≤ ‖a-b‖ | solid (lattice_ordered_comm_group.abs_abs_sub_abs_le _ _) | lemma | norm_abs_sub_abs | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sup_sub_sup_le_norm (x y z : α) : ‖x ⊔ z - (y ⊔ z)‖ ≤ ‖x - y‖ | solid (abs_sup_sub_sup_le_abs x y z) | lemma | norm_sup_sub_sup_le_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inf_sub_inf_le_norm (x y z : α) : ‖x ⊓ z - (y ⊓ z)‖ ≤ ‖x - y‖ | solid (abs_inf_sub_inf_le_abs x y z) | lemma | norm_inf_sub_inf_le_norm | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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