statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
normed_comm_group.of_separation [seminormed_comm_group E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
normed_comm_group E | { ..‹seminormed_comm_group E›, ..normed_group.of_separation h } | def | normed_comm_group.of_separation | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group",
"normed_group.of_separation",
"seminormed_comm_group"
] | Construct a `normed_comm_group` from a `seminormed_comm_group` satisfying
`∀ x, ‖x‖ = 0 → x = 1`. This avoids having to go back to the `(pseudo_)metric_space` level when
declaring a `normed_comm_group` instance as a special case of a more general `seminormed_comm_group`
instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_group.of_mul_dist [has_norm E] [group E] [pseudo_metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
seminormed_group E | { dist_eq := λ x y, begin
rw h₁, apply le_antisymm,
{ simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _ },
{ simpa only [div_mul_cancel', one_mul] using h₂ (x/y) 1 y }
end } | def | seminormed_group.of_mul_dist | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_eq_mul_inv",
"div_mul_cancel'",
"group",
"has_norm",
"mul_right_inv",
"one_mul",
"pseudo_metric_space",
"seminormed_group"
] | Construct a seminormed group from a multiplication-invariant distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_group.of_mul_dist' [has_norm E] [group E] [pseudo_metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
seminormed_group E | { dist_eq := λ x y, begin
rw h₁, apply le_antisymm,
{ simpa only [div_mul_cancel', one_mul] using h₂ (x/y) 1 y },
{ simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _ }
end } | def | seminormed_group.of_mul_dist' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_eq_mul_inv",
"div_mul_cancel'",
"group",
"has_norm",
"mul_right_inv",
"one_mul",
"pseudo_metric_space",
"seminormed_group"
] | Construct a seminormed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group.of_mul_dist [has_norm E] [comm_group E] [pseudo_metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
seminormed_comm_group E | { ..seminormed_group.of_mul_dist h₁ h₂ } | def | seminormed_comm_group.of_mul_dist | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"pseudo_metric_space",
"seminormed_comm_group",
"seminormed_group.of_mul_dist"
] | Construct a seminormed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group.of_mul_dist' [has_norm E] [comm_group E] [pseudo_metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
seminormed_comm_group E | { ..seminormed_group.of_mul_dist' h₁ h₂ } | def | seminormed_comm_group.of_mul_dist' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"pseudo_metric_space",
"seminormed_comm_group",
"seminormed_group.of_mul_dist'"
] | Construct a seminormed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group.of_mul_dist [has_norm E] [group E] [metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
normed_group E | { ..seminormed_group.of_mul_dist h₁ h₂ } | def | normed_group.of_mul_dist | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group",
"has_norm",
"metric_space",
"normed_group",
"seminormed_group.of_mul_dist"
] | Construct a normed group from a multiplication-invariant distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group.of_mul_dist' [has_norm E] [group E] [metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
normed_group E | { ..seminormed_group.of_mul_dist' h₁ h₂ } | def | normed_group.of_mul_dist' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group",
"has_norm",
"metric_space",
"normed_group",
"seminormed_group.of_mul_dist'"
] | Construct a normed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_group.of_mul_dist [has_norm E] [comm_group E] [metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
normed_comm_group E | { ..normed_group.of_mul_dist h₁ h₂ } | def | normed_comm_group.of_mul_dist | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"metric_space",
"normed_comm_group",
"normed_group.of_mul_dist"
] | Construct a normed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_group.of_mul_dist' [has_norm E] [comm_group E] [metric_space E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
normed_comm_group E | { ..normed_group.of_mul_dist' h₁ h₂ } | def | normed_comm_group.of_mul_dist' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"has_norm",
"metric_space",
"normed_comm_group",
"normed_group.of_mul_dist'"
] | Construct a normed group from a multiplication-invariant pseudodistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_seminorm.to_seminormed_group [group E] (f : group_seminorm E) : seminormed_group E | { dist := λ x y, f (x / y),
norm := f,
dist_eq := λ x y, rfl,
dist_self := λ x, by simp only [div_self', map_one_eq_zero],
dist_triangle := le_map_div_add_map_div f,
dist_comm := map_div_rev f } | def | group_seminorm.to_seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"div_self'",
"group",
"group_seminorm",
"le_map_div_add_map_div",
"map_div_rev",
"seminormed_group"
] | Construct a seminormed group from a seminorm, i.e., registering the pseudodistance and the
pseudometric space structure from the seminorm properties. Note that in most cases this instance
creates bad definitional equalities (e.g., it does not take into account a possibly existing
`uniform_space` instance on `E`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_seminorm.to_seminormed_comm_group [comm_group E] (f : group_seminorm E) :
seminormed_comm_group E | { ..f.to_seminormed_group } | def | group_seminorm.to_seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"group_seminorm",
"seminormed_comm_group"
] | Construct a seminormed group from a seminorm, i.e., registering the pseudodistance and the
pseudometric space structure from the seminorm properties. Note that in most cases this instance
creates bad definitional equalities (e.g., it does not take into account a possibly existing
`uniform_space` instance on `E`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_norm.to_normed_group [group E] (f : group_norm E) : normed_group E | { eq_of_dist_eq_zero := λ x y h, div_eq_one.1 $ eq_one_of_map_eq_zero f h,
..f.to_group_seminorm.to_seminormed_group } | def | group_norm.to_normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"eq_of_dist_eq_zero",
"group",
"group_norm",
"normed_group"
] | Construct a normed group from a norm, i.e., registering the distance and the metric space
structure from the norm properties. Note that in most cases this instance creates bad definitional
equalities (e.g., it does not take into account a possibly existing `uniform_space` instance on
`E`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_norm.to_normed_comm_group [comm_group E] (f : group_norm E) : normed_comm_group E | { ..f.to_normed_group } | def | group_norm.to_normed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"comm_group",
"group_norm",
"normed_comm_group"
] | Construct a normed group from a norm, i.e., registering the distance and the metric space
structure from the norm properties. Note that in most cases this instance creates bad definitional
equalities (e.g., it does not take into account a possibly existing `uniform_space` instance on
`E`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit.norm_eq_zero (r : punit) : ‖r‖ = 0 | rfl | lemma | punit.norm_eq_zero | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ | seminormed_group.dist_eq _ _ | lemma | dist_eq_norm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ | by rw [dist_comm, dist_eq_norm_div] | lemma | dist_eq_norm_div' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_comm",
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_group.to_has_isometric_smul_right : has_isometric_smul Eᵐᵒᵖ E | ⟨λ a, isometry.of_dist_eq $ λ b c, by simp [dist_eq_norm_div]⟩ | instance | normed_group.to_has_isometric_smul_right | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_one_right (a : E) : dist a 1 = ‖a‖ | by rw [dist_eq_norm_div, div_one] | lemma | dist_one_right | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_one_left : dist (1 : E) = norm | funext $ λ a, by rw [dist_comm, dist_one_right] | lemma | dist_one_left | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_comm",
"dist_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry.norm_map_of_map_one {f : E → F} (hi : isometry f) (h₁ : f 1 = 1) (x : E) :
‖f x‖ = ‖x‖ | by rw [←dist_one_right, ←h₁, hi.dist_eq, dist_one_right] | lemma | isometry.norm_map_of_map_one | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_one_right",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_cocompact_at_top' [proper_space E] : tendsto norm (cocompact E) at_top | by simpa only [dist_one_right] using tendsto_dist_right_cocompact_at_top (1 : E) | lemma | tendsto_norm_cocompact_at_top' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_one_right",
"proper_space",
"tendsto_dist_right_cocompact_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_rev (a b : E) : ‖a / b‖ = ‖b / a‖ | by simpa only [dist_eq_norm_div] using dist_comm a b | lemma | norm_div_rev | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_comm",
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inv' (a : E) : ‖a⁻¹‖ = ‖a‖ | by simpa using norm_div_rev 1 a | lemma | norm_inv' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_div_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_self_right (a b : E) : dist b (a * b) = ‖a‖ | by rw [←dist_one_left, ←dist_mul_right 1 a b, one_mul] | lemma | dist_mul_self_right | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_self_left (a b : E) : dist (a * b) b = ‖a‖ | by rw [dist_comm, dist_mul_self_right] | lemma | dist_mul_self_left | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_comm",
"dist_mul_self_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_eq_dist_mul_left (a b c : E) :
dist (a / b) c = dist a (c * b) | by rw [←dist_mul_right _ _ b, div_mul_cancel'] | lemma | dist_div_eq_dist_mul_left | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_mul_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_eq_dist_mul_right (a b c : E) :
dist a (b / c) = dist (a * c) b | by rw [←dist_mul_right _ _ c, div_mul_cancel'] | lemma | dist_div_eq_dist_mul_right | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_mul_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_inv_cobounded :
tendsto (has_inv.inv : E → E) (comap norm at_top) (comap norm at_top) | by simpa only [norm_inv', tendsto_comap_iff, (∘)] using tendsto_comap | lemma | filter.tendsto_inv_cobounded | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_inv'"
] | In a (semi)normed group, inversion `x ↦ x⁻¹` tends to infinity at infinity. TODO: use
`bornology.cobounded` instead of `filter.comap has_norm.norm filter.at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mul_le' (a b : E) : ‖a * b‖ ≤ ‖a‖ + ‖b‖ | by simpa [dist_eq_norm_div] using dist_triangle a 1 b⁻¹ | lemma | norm_mul_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"dist_triangle"
] | **Triangle inequality** for the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mul_le_of_le (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂) : ‖a₁ * a₂‖ ≤ r₁ + r₂ | (norm_mul_le' a₁ a₂).trans $ add_le_add h₁ h₂ | lemma | norm_mul_le_of_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_mul_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mul₃_le (a b c : E) : ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ | norm_mul_le_of_le (norm_mul_le' _ _) le_rfl | lemma | norm_mul₃_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"le_rfl",
"norm_mul_le'",
"norm_mul_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_nonneg' (a : E) : 0 ≤ ‖a‖ | by { rw [←dist_one_right], exact dist_nonneg } | lemma | norm_nonneg' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.tactic.positivity_norm : expr → tactic strictness | | `(‖%%a‖) := nonnegative <$> mk_app ``norm_nonneg [a] <|> nonnegative <$> mk_app ``norm_nonneg' [a]
| _ := failed | def | tactic.positivity_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_nonneg'"
] | Extension for the `positivity` tactic: norms are nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_one' : ‖(1 : E)‖ = 0 | by rw [←dist_one_right, dist_self] | lemma | norm_one' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_of_norm_ne_zero : ‖a‖ ≠ 0 → a ≠ 1 | mt $ by { rintro rfl, exact norm_one' } | lemma | ne_one_of_norm_ne_zero | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_subsingleton' [subsingleton E] (a : E) : ‖a‖ = 0 | by rw [subsingleton.elim a 1, norm_one'] | lemma | norm_of_subsingleton' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖^2 | by positivity | lemma | zero_lt_one_add_norm_sq' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_le (a b : E) : ‖a / b‖ ≤ ‖a‖ + ‖b‖ | by simpa [dist_eq_norm_div] using dist_triangle a 1 b | lemma | norm_div_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_le_of_le {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂) :
‖a₁ / a₂‖ ≤ r₁ + r₂ | (norm_div_le a₁ a₂).trans $ add_le_add H₁ H₂ | lemma | norm_div_le_of_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_norm_add_norm' (a b : E) :
dist a b ≤ ‖a‖ + ‖b‖ | by { rw dist_eq_norm_div, apply norm_div_le } | lemma | dist_le_norm_add_norm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"norm_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_sub_norm_le' (a b : E) : |‖a‖ - ‖b‖| ≤ ‖a / b‖ | by simpa [dist_eq_norm_div] using abs_dist_sub_le a b 1 | lemma | abs_norm_sub_norm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"abs_dist_sub_le",
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sub_norm_le' (a b : E) : ‖a‖ - ‖b‖ ≤ ‖a / b‖ | (le_abs_self _).trans (abs_norm_sub_norm_le' a b) | lemma | norm_sub_norm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"abs_norm_sub_norm_le'",
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_norm_norm_le' (a b : E) : dist ‖a‖ ‖b‖ ≤ ‖a / b‖ | abs_norm_sub_norm_le' a b | lemma | dist_norm_norm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"abs_norm_sub_norm_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ | by { rw add_comm, refine (norm_mul_le' _ _).trans_eq' _, rw div_mul_cancel' } | lemma | norm_le_norm_add_norm_div' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_mul_cancel'",
"norm_mul_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_norm_add_norm_div (u v : E) : ‖v‖ ≤ ‖u‖ + ‖u / v‖ | by { rw norm_div_rev, exact norm_le_norm_add_norm_div' v u } | lemma | norm_le_norm_add_norm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_div_rev",
"norm_le_norm_add_norm_div'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_mul_norm_add (u v : E) : ‖u‖ ≤ ‖u * v‖ + ‖v‖ | calc ‖u‖ = ‖u * v / v‖ : by rw mul_div_cancel''
... ≤ ‖u * v‖ + ‖v‖ : norm_div_le _ _ | lemma | norm_le_mul_norm_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"mul_div_cancel''",
"norm_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq' (y : E) (ε : ℝ) : ball y ε = {x | ‖x / y‖ < ε} | set.ext $ λ a, by simp [dist_eq_norm_div] | lemma | ball_eq' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_one_eq (r : ℝ) : ball (1 : E) r = {x | ‖x‖ < r} | set.ext $ assume a, by simp | lemma | ball_one_eq | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_iff_norm'' : b ∈ ball a r ↔ ‖b / a‖ < r | by rw [mem_ball, dist_eq_norm_div] | lemma | mem_ball_iff_norm'' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_iff_norm''' : b ∈ ball a r ↔ ‖a / b‖ < r | by rw [mem_ball', dist_eq_norm_div] | lemma | mem_ball_iff_norm''' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r | by rw [mem_ball, dist_one_right] | lemma | mem_ball_one_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_iff_norm'' : b ∈ closed_ball a r ↔ ‖b / a‖ ≤ r | by rw [mem_closed_ball, dist_eq_norm_div] | lemma | mem_closed_ball_iff_norm'' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_one_iff : a ∈ closed_ball (1 : E) r ↔ ‖a‖ ≤ r | by rw [mem_closed_ball, dist_one_right] | lemma | mem_closed_ball_one_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_ball_iff_norm''' : b ∈ closed_ball a r ↔ ‖a / b‖ ≤ r | by rw [mem_closed_ball', dist_eq_norm_div] | lemma | mem_closed_ball_iff_norm''' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_of_mem_closed_ball' (h : b ∈ closed_ball a r) : ‖b‖ ≤ ‖a‖ + r | (norm_le_norm_add_norm_div' _ _).trans $ add_le_add_left (by rwa ←dist_eq_norm_div) _ | lemma | norm_le_of_mem_closed_ball' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_norm_add_norm_div'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_norm_add_const_of_dist_le' : dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r | norm_le_of_mem_closed_ball' | lemma | norm_le_norm_add_const_of_dist_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_of_mem_closed_ball'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_of_mem_ball' (h : b ∈ ball a r) : ‖b‖ < ‖a‖ + r | (norm_le_norm_add_norm_div' _ _).trans_lt $ add_lt_add_left (by rwa ←dist_eq_norm_div) _ | lemma | norm_lt_of_mem_ball' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_norm_add_norm_div'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_sub_norm_div_le_norm_div (u v w : E) : ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖ | by simpa only [div_div_div_cancel_right'] using norm_sub_norm_le' (u / w) (v / w) | lemma | norm_div_sub_norm_div_le_norm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_div_div_cancel_right'",
"norm_sub_norm_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_iff_forall_norm_le' : bounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C | by simpa only [set.subset_def, mem_closed_ball_one_iff] using bounded_iff_subset_ball (1 : E) | lemma | bounded_iff_forall_norm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"mem_closed_ball_one_iff",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.exists_pos_norm_le' (hs : metric.bounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R | let ⟨R₀, hR₀⟩ := hs.exists_norm_le' in
⟨max R₀ 1, by positivity, λ x hx, (hR₀ x hx).trans $ le_max_left _ _⟩ | lemma | metric.bounded.exists_pos_norm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"metric.bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sphere_iff_norm' : b ∈ sphere a r ↔ ‖b / a‖ = r | by simp [dist_eq_norm_div] | lemma | mem_sphere_iff_norm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r | by simp [dist_eq_norm_div] | lemma | mem_sphere_one_iff_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_of_mem_sphere' (x : sphere (1:E) r) : ‖(x : E)‖ = r | mem_sphere_one_iff_norm.mp x.2 | lemma | norm_eq_of_mem_sphere' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_of_mem_sphere (hr : r ≠ 0) (x : sphere (1 : E) r) : (x : E) ≠ 1 | ne_one_of_norm_ne_zero $ by rwa norm_eq_of_mem_sphere' x | lemma | ne_one_of_mem_sphere | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"ne_one_of_norm_ne_zero",
"norm_eq_of_mem_sphere'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_of_mem_unit_sphere (x : sphere (1 : E) 1) : (x:E) ≠ 1 | ne_one_of_mem_sphere one_ne_zero _ | lemma | ne_one_of_mem_unit_sphere | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"ne_one_of_mem_sphere",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_group_seminorm : group_seminorm E | ⟨norm, norm_one', norm_mul_le', norm_inv'⟩ | def | norm_group_seminorm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group_seminorm",
"norm_mul_le'",
"norm_one'"
] | The norm of a seminormed group as a group seminorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm_group_seminorm : ⇑(norm_group_seminorm E) = norm | rfl | lemma | coe_norm_group_seminorm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.tendsto_nhds_one {f : α → E} {l : filter α} :
tendsto f l (𝓝 1) ↔ ∀ ε > 0, ∀ᶠ x in l, ‖ f x ‖ < ε | metric.tendsto_nhds.trans $ by simp only [dist_one_right] | lemma | normed_comm_group.tendsto_nhds_one | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_one_right",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.tendsto_nhds_nhds {f : E → F} {x : E} {y : F} :
tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ x', ‖x' / x‖ < δ → ‖f x' / y‖ < ε | by simp_rw [metric.tendsto_nhds_nhds, dist_eq_norm_div] | lemma | normed_comm_group.tendsto_nhds_nhds | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"metric.tendsto_nhds_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.cauchy_seq_iff [nonempty α] [semilattice_sup α] {u : α → E} :
cauchy_seq u ↔ ∀ ε > 0, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → ‖u m / u n‖ < ε | by simp [metric.cauchy_seq_iff, dist_eq_norm_div] | lemma | normed_comm_group.cauchy_seq_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"cauchy_seq",
"dist_eq_norm_div",
"metric.cauchy_seq_iff",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.nhds_basis_norm_lt (x : E) :
(𝓝 x).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {y | ‖y / x‖ < ε}) | by { simp_rw ← ball_eq', exact metric.nhds_basis_ball } | lemma | normed_comm_group.nhds_basis_norm_lt | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"ball_eq'",
"metric.nhds_basis_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.nhds_one_basis_norm_lt :
(𝓝 (1 : E)).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {y | ‖y‖ < ε}) | by { convert normed_comm_group.nhds_basis_norm_lt (1 : E), simp } | lemma | normed_comm_group.nhds_one_basis_norm_lt | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group.nhds_basis_norm_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group.uniformity_basis_dist :
(𝓤 E).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {p : E × E | ‖p.fst / p.snd‖ < ε}) | by { convert metric.uniformity_basis_dist, simp [dist_eq_norm_div] } | lemma | normed_comm_group.uniformity_basis_dist | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"metric.uniformity_basis_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_class.lipschitz_of_bound [monoid_hom_class 𝓕 E F] (f : 𝓕) (C : ℝ)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : lipschitz_with (real.to_nnreal C) f | lipschitz_with.of_dist_le' $ λ x y, by simpa only [dist_eq_norm_div, map_div] using h (x / y) | lemma | monoid_hom_class.lipschitz_of_bound | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"lipschitz_with",
"lipschitz_with.of_dist_le'",
"map_div",
"monoid_hom_class",
"real.to_nnreal"
] | A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant `C` such that
for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `normed_space.operator_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_on_with_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
lipschitz_on_with C f s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ‖f x / f y‖ ≤ C * ‖x / y‖ | by simp only [lipschitz_on_with_iff_dist_le_mul, dist_eq_norm_div] | lemma | lipschitz_on_with_iff_norm_div_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"lipschitz_on_with",
"lipschitz_on_with_iff_dist_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_on_with.norm_div_le_of_le {f : E → F} {C : ℝ≥0}
(h : lipschitz_on_with C f s) (ha : a ∈ s) (hb : b ∈ s) (hr : ‖a / b‖ ≤ r) :
‖f a / f b‖ ≤ C * r | (h.norm_div_le ha hb).trans $ mul_le_mul_of_nonneg_left hr C.2 | lemma | lipschitz_on_with.norm_div_le_of_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"lipschitz_on_with",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
lipschitz_with C f ↔ ∀ x y, ‖f x / f y‖ ≤ C * ‖x / y‖ | by simp only [lipschitz_with_iff_dist_le_mul, dist_eq_norm_div] | lemma | lipschitz_with_iff_norm_div_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"lipschitz_with",
"lipschitz_with_iff_dist_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : lipschitz_with C f)
(hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r | (h.norm_div_le _ _).trans $ mul_le_mul_of_nonneg_left hr C.2 | lemma | lipschitz_with.norm_div_le_of_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"lipschitz_with",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_class.continuous_of_bound [monoid_hom_class 𝓕 E F] (f : 𝓕) (C : ℝ)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : continuous f | (monoid_hom_class.lipschitz_of_bound f C h).continuous | lemma | monoid_hom_class.continuous_of_bound | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"continuous",
"monoid_hom_class",
"monoid_hom_class.lipschitz_of_bound"
] | A homomorphism `f` of seminormed groups is continuous, if there exists a constant `C` such that
for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom_class.uniform_continuous_of_bound [monoid_hom_class 𝓕 E F]
(f : 𝓕) (C : ℝ) (h : ∀x, ‖f x‖ ≤ C * ‖x‖) : uniform_continuous f | (monoid_hom_class.lipschitz_of_bound f C h).uniform_continuous | lemma | monoid_hom_class.uniform_continuous_of_bound | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"monoid_hom_class",
"monoid_hom_class.lipschitz_of_bound",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_bound_of_continuous_on' [topological_space α] {s : set α}
(hs : is_compact s) {f : α → E} (hf : continuous_on f s) :
∃ C, ∀ x ∈ s, ‖f x‖ ≤ C | (bounded_iff_forall_norm_le'.1 (hs.image_of_continuous_on hf).bounded).imp $ λ C hC x hx,
hC _ $ set.mem_image_of_mem _ hx | lemma | is_compact.exists_bound_of_continuous_on' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"continuous_on",
"is_compact",
"set.mem_image_of_mem",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_class.isometry_iff_norm [monoid_hom_class 𝓕 E F] (f : 𝓕) :
isometry f ↔ ∀ x, ‖f x‖ = ‖x‖ | begin
simp only [isometry_iff_dist_eq, dist_eq_norm_div, ←map_div],
refine ⟨λ h x, _, λ h x y, h _⟩,
simpa using h x 1,
end | lemma | monoid_hom_class.isometry_iff_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"isometry",
"isometry_iff_dist_eq",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group.to_has_nnnorm : has_nnnorm E | ⟨λ a, ⟨‖a‖, norm_nonneg' a⟩⟩ | instance | seminormed_group.to_has_nnnorm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"has_nnnorm",
"norm_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nnnorm' (a : E) : (‖a‖₊ : ℝ) = ‖a‖ | rfl | lemma | coe_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_nnnorm' : (coe : ℝ≥0 → ℝ) ∘ (nnnorm : E → ℝ≥0) = norm | rfl | lemma | coe_comp_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_nnreal' : ‖a‖.to_nnreal = ‖a‖₊ | @real.to_nnreal_coe ‖a‖₊ | lemma | norm_to_nnreal' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"real.to_nnreal_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq_nnnorm_div (a b : E) : nndist a b = ‖a / b‖₊ | nnreal.eq $ dist_eq_norm_div _ _ | lemma | nndist_eq_nnnorm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_one' : ‖(1 : E)‖₊ = 0 | nnreal.eq norm_one' | lemma | nnnorm_one' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"nnreal.eq",
"norm_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_of_nnnorm_ne_zero {a : E} : ‖a‖₊ ≠ 0 → a ≠ 1 | mt $ by { rintro rfl, exact nnnorm_one' } | lemma | ne_one_of_nnnorm_ne_zero | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"nnnorm_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_mul_le' (a b : E) : ‖a * b‖₊ ≤ ‖a‖₊ + ‖b‖₊ | nnreal.coe_le_coe.1 $ norm_mul_le' a b | lemma | nnnorm_mul_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_mul_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_inv' (a : E) : ‖a⁻¹‖₊ = ‖a‖₊ | nnreal.eq $ norm_inv' a | lemma | nnnorm_inv' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"nnreal.eq",
"norm_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_div_le (a b : E) : ‖a / b‖₊ ≤ ‖a‖₊ + ‖b‖₊ | nnreal.coe_le_coe.1 $ norm_div_le _ _ | lemma | nnnorm_div_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_nnnorm_nnnorm_le' (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a / b‖₊ | nnreal.coe_le_coe.1 $ dist_norm_norm_le' a b | lemma | nndist_nnnorm_nnnorm_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_norm_norm_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_le_nnnorm_add_nnnorm_div (a b : E) : ‖b‖₊ ≤ ‖a‖₊ + ‖a / b‖₊ | norm_le_norm_add_norm_div _ _ | lemma | nnnorm_le_nnnorm_add_nnnorm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_norm_add_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_le_nnnorm_add_nnnorm_div' (a b : E) : ‖a‖₊ ≤ ‖b‖₊ + ‖a / b‖₊ | norm_le_norm_add_norm_div' _ _ | lemma | nnnorm_le_nnnorm_add_nnnorm_div' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_norm_add_norm_div'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_le_mul_nnnorm_add (a b : E) : ‖a‖₊ ≤ ‖a * b‖₊ + ‖b‖₊ | norm_le_mul_norm_add _ _ | lemma | nnnorm_le_mul_nnnorm_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_mul_norm_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_norm_eq_coe_nnnorm' (a : E) : ennreal.of_real ‖a‖ = ‖a‖₊ | ennreal.of_real_eq_coe_nnreal _ | lemma | of_real_norm_eq_coe_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"ennreal.of_real",
"ennreal.of_real_eq_coe_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_coe_nnnorm_div (a b : E) : edist a b = ‖a / b‖₊ | by rw [edist_dist, dist_eq_norm_div, of_real_norm_eq_coe_nnnorm'] | lemma | edist_eq_coe_nnnorm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_eq_norm_div",
"edist_dist",
"of_real_norm_eq_coe_nnnorm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_coe_nnnorm' (x : E) : edist x 1 = (‖x‖₊ : ℝ≥0∞) | by rw [edist_eq_coe_nnnorm_div, div_one] | lemma | edist_eq_coe_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_one",
"edist_eq_coe_nnnorm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_emetric_ball_one_iff {r : ℝ≥0∞} : a ∈ emetric.ball (1 : E) r ↔ ↑‖a‖₊ < r | by rw [emetric.mem_ball, edist_eq_coe_nnnorm'] | lemma | mem_emetric_ball_one_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"edist_eq_coe_nnnorm'",
"emetric.ball",
"emetric.mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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