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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
monoid_hom_class.lipschitz_of_bound_nnnorm [monoid_hom_class 𝓕 E F] (f : 𝓕)
(C : ℝ≥0) (h : ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊) : lipschitz_with C f | @real.to_nnreal_coe C ▸ monoid_hom_class.lipschitz_of_bound f C h | lemma | monoid_hom_class.lipschitz_of_bound_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"
] | [
"lipschitz_with",
"monoid_hom_class",
"monoid_hom_class.lipschitz_of_bound",
"real.to_nnreal_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_class.antilipschitz_of_bound [monoid_hom_class 𝓕 E F] (f : 𝓕)
{K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
antilipschitz_with K f | antilipschitz_with.of_le_mul_dist $ λ x y, by simpa only [dist_eq_norm_div, map_div] using h (x / y) | lemma | monoid_hom_class.antilipschitz_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"
] | [
"antilipschitz_with",
"dist_eq_norm_div",
"map_div",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.norm_le_mul' {f : E → F}
{K : ℝ≥0} (h : lipschitz_with K f) (hf : f 1 = 1) (x) : ‖f x‖ ≤ K * ‖x‖ | by simpa only [dist_one_right, hf] using h.dist_le_mul x 1 | lemma | lipschitz_with.norm_le_mul' | 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",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with.nnorm_le_mul' {f : E → F}
{K : ℝ≥0} (h : lipschitz_with K f) (hf : f 1 = 1) (x) : ‖f x‖₊ ≤ K * ‖x‖₊ | h.norm_le_mul' hf x | lemma | lipschitz_with.nnorm_le_mul' | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with.le_mul_norm' {f : E → F}
{K : ℝ≥0} (h : antilipschitz_with K f) (hf : f 1 = 1) (x) : ‖x‖ ≤ K * ‖f x‖ | by simpa only [dist_one_right, hf] using h.le_mul_dist x 1 | lemma | antilipschitz_with.le_mul_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"
] | [
"antilipschitz_with",
"dist_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with.le_mul_nnnorm' {f : E → F}
{K : ℝ≥0} (h : antilipschitz_with K f) (hf : f 1 = 1) (x) : ‖x‖₊ ≤ K * ‖f x‖₊ | h.le_mul_norm' hf x | lemma | antilipschitz_with.le_mul_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"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom_class.bound_of_antilipschitz [one_hom_class 𝓕 E F] (f : 𝓕)
{K : ℝ≥0} (h : antilipschitz_with K f) (x) : ‖x‖ ≤ K * ‖f x‖ | h.le_mul_nnnorm' (map_one f) x | lemma | one_hom_class.bound_of_antilipschitz | 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"
] | [
"antilipschitz_with",
"map_one",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_norm_tendsto_one {f : α → E} {a : filter α} {b : E} :
tendsto f a (𝓝 b) ↔ tendsto (λ e, ‖f e / b‖) a (𝓝 0) | by { convert tendsto_iff_dist_tendsto_zero, simp [dist_eq_norm_div] } | lemma | tendsto_iff_norm_tendsto_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_eq_norm_div",
"filter",
"tendsto_iff_dist_tendsto_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_one_iff_norm_tendsto_one {f : α → E} {a : filter α} :
tendsto f a (𝓝 1) ↔ tendsto (λ e, ‖f e‖) a (𝓝 0) | by { rw tendsto_iff_norm_tendsto_one, simp only [div_one] } | lemma | tendsto_one_iff_norm_tendsto_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"
] | [
"div_one",
"filter",
"tendsto_iff_norm_tendsto_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_norm_nhds_one : comap norm (𝓝 0) = 𝓝 (1 : E) | by simpa only [dist_one_right] using nhds_comap_dist (1 : E) | lemma | comap_norm_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",
"nhds_comap_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squeeze_one_norm' {f : α → E} {a : α → ℝ} {t₀ : filter α} (h : ∀ᶠ n in t₀, ‖f n‖ ≤ a n)
(h' : tendsto a t₀ (𝓝 0)) : tendsto f t₀ (𝓝 1) | tendsto_one_iff_norm_tendsto_one.2 $ squeeze_zero' (eventually_of_forall $ λ n, norm_nonneg' _) h h' | lemma | squeeze_one_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"
] | [
"filter",
"norm_nonneg'",
"squeeze_zero'"
] | Special case of the sandwich theorem: if the norm of `f` is eventually bounded by a real
function `a` which tends to `0`, then `f` tends to `1`. In this pair of lemmas (`squeeze_one_norm'`
and `squeeze_one_norm`), following a convention of similar lemmas in `topology.metric_space.basic`
and `topology.algebra.order`, th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squeeze_one_norm {f : α → E} {a : α → ℝ} {t₀ : filter α} (h : ∀ n, ‖f n‖ ≤ a n) :
tendsto a t₀ (𝓝 0) → tendsto f t₀ (𝓝 1) | squeeze_one_norm' $ eventually_of_forall h | lemma | squeeze_one_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"
] | [
"filter",
"squeeze_one_norm'"
] | Special case of the sandwich theorem: if the norm of `f` is bounded by a real function `a` which
tends to `0`, then `f` tends to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_norm_div_self (x : E) : tendsto (λ a, ‖a / x‖) (𝓝 x) (𝓝 0) | by simpa [dist_eq_norm_div] using
tendsto_id.dist (tendsto_const_nhds : tendsto (λ a, (x:E)) (𝓝 x) _) | lemma | tendsto_norm_div_self | 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",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm' {x : E} : tendsto (λ a, ‖a‖) (𝓝 x) (𝓝 ‖x‖) | by simpa using tendsto_id.dist (tendsto_const_nhds : tendsto (λ a, (1:E)) _ _) | lemma | tendsto_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"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_one : tendsto (λ a : E, ‖a‖) (𝓝 1) (𝓝 0) | by simpa using tendsto_norm_div_self (1:E) | lemma | tendsto_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"
] | [
"tendsto_norm_div_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_norm' : continuous (λ a : E, ‖a‖) | by simpa using continuous_id.dist (continuous_const : continuous (λ a, (1:E))) | lemma | continuous_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"
] | [
"continuous",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_nnnorm' : continuous (λ a : E, ‖a‖₊) | continuous_norm'.subtype_mk _ | lemma | continuous_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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_one_norm' :
lipschitz_with 1 (norm : E → ℝ) | by simpa only [dist_one_left] using lipschitz_with.dist_right (1 : E) | lemma | lipschitz_with_one_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_one_left",
"lipschitz_with",
"lipschitz_with.dist_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_one_nnnorm' :
lipschitz_with 1 (has_nnnorm.nnnorm : E → ℝ≥0) | lipschitz_with_one_norm' | lemma | lipschitz_with_one_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"
] | [
"lipschitz_with",
"lipschitz_with_one_norm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_norm' : uniform_continuous (norm : E → ℝ) | lipschitz_with_one_norm'.uniform_continuous | lemma | uniform_continuous_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"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_nnnorm' : uniform_continuous (λ (a : E), ‖a‖₊) | uniform_continuous_norm'.subtype_mk _ | lemma | uniform_continuous_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"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_one_iff_norm {x : E} : x ∈ closure ({1} : set E) ↔ ‖x‖ = 0 | by rw [←closed_ball_zero', mem_closed_ball_one_iff, (norm_nonneg' x).le_iff_eq] | lemma | mem_closure_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"
] | [
"closure",
"mem_closed_ball_one_iff",
"norm_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_one_eq : closure ({1} : set E) = {x | ‖x‖ = 0} | set.ext (λ x, mem_closure_one_iff_norm) | lemma | closure_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"
] | [
"closure",
"mem_closure_one_iff_norm",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.op_one_is_bounded_under_le' {f : α → E} {g : α → F} {l : filter α}
(hf : tendsto f l (𝓝 1)) (hg : is_bounded_under (≤) l (norm ∘ g)) (op : E → F → G)
(h_op : ∃ A, ∀ x y, ‖op x y‖ ≤ A * ‖x‖ * ‖y‖) :
tendsto (λ x, op (f x) (g x)) l (𝓝 1) | begin
cases h_op with A h_op,
rcases hg with ⟨C, hC⟩, rw eventually_map at hC,
rw normed_comm_group.tendsto_nhds_one at hf ⊢,
intros ε ε₀,
rcases exists_pos_mul_lt ε₀ (A * C) with ⟨δ, δ₀, hδ⟩,
filter_upwards [hf δ δ₀, hC] with i hf hg,
refine (h_op _ _).trans_lt _,
cases le_total A 0 with hA hA,
{ exa... | lemma | filter.tendsto.op_one_is_bounded_under_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"
] | [
"exists_pos_mul_lt",
"filter",
"mul_le_mul",
"mul_le_mul_of_nonneg_left",
"mul_nonpos_of_nonpos_of_nonneg",
"mul_right_comm",
"norm_nonneg'",
"normed_comm_group.tendsto_nhds_one"
] | A helper lemma used to prove that the (scalar or usual) product of a function that tends to one
and a bounded function tends to one. This lemma is formulated for any binary operation
`op : E → F → G` with an estimate `‖op x y‖ ≤ A * ‖x‖ * ‖y‖` for some constant A instead of
multiplication so that it can be applied to `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.op_one_is_bounded_under_le {f : α → E} {g : α → F} {l : filter α}
(hf : tendsto f l (𝓝 1)) (hg : is_bounded_under (≤) l (norm ∘ g)) (op : E → F → G)
(h_op : ∀ x y, ‖op x y‖ ≤ ‖x‖ * ‖y‖) :
tendsto (λ x, op (f x) (g x)) l (𝓝 1) | hf.op_one_is_bounded_under_le' hg op ⟨1, λ x y, (one_mul (‖x‖)).symm ▸ h_op x y⟩ | lemma | filter.tendsto.op_one_is_bounded_under_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"
] | [
"filter",
"one_mul"
] | A helper lemma used to prove that the (scalar or usual) product of a function that tends to one
and a bounded function tends to one. This lemma is formulated for any binary operation
`op : E → F → G` with an estimate `‖op x y‖ ≤ ‖x‖ * ‖y‖` instead of multiplication so that it
can be applied to `(*)`, `flip (*)`, `(•)`,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.norm' (h : tendsto f l (𝓝 a)) :
tendsto (λ x, ‖f x‖) l (𝓝 ‖a‖) | tendsto_norm'.comp h | lemma | filter.tendsto.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.nnnorm' (h : tendsto f l (𝓝 a)) :
tendsto (λ x, ‖f x‖₊) l (𝓝 (‖a‖₊)) | tendsto.comp continuous_nnnorm'.continuous_at h | lemma | filter.tendsto.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 | |
continuous.norm' : continuous f → continuous (λ x, ‖f x‖) | continuous_norm'.comp | lemma | continuous.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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.nnnorm' : continuous f → continuous (λ x, ‖f x‖₊) | continuous_nnnorm'.comp | lemma | continuous.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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.norm' {a : α} (h : continuous_at f a) : continuous_at (λ x, ‖f x‖) a | h.norm' | lemma | continuous_at.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"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.nnnorm' {a : α} (h : continuous_at f a) : continuous_at (λ x, ‖f x‖₊) a | h.nnnorm' | lemma | continuous_at.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"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.norm' {s : set α} {a : α} (h : continuous_within_at f s a) :
continuous_within_at (λ x, ‖f x‖) s a | h.norm' | lemma | continuous_within_at.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"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.nnnorm' {s : set α} {a : α} (h : continuous_within_at f s a) :
continuous_within_at (λ x, ‖f x‖₊) s a | h.nnnorm' | lemma | continuous_within_at.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"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.norm' {s : set α} (h : continuous_on f s) : continuous_on (λ x, ‖f x‖) s | λ x hx, (h x hx).norm' | lemma | continuous_on.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"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.nnnorm' {s : set α} (h : continuous_on f s) : continuous_on (λ x, ‖f x‖₊) s | λ x hx, (h x hx).nnnorm' | lemma | continuous_on.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"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_ne_of_tendsto_norm_at_top' {l : filter α} {f : α → E}
(h : tendsto (λ y, ‖f y‖) l at_top) (x : E) :
∀ᶠ y in l, f y ≠ x | (h.eventually_ne_at_top _).mono $ λ x, ne_of_apply_ne norm | lemma | eventually_ne_of_tendsto_norm_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"
] | [
"filter",
"ne_of_apply_ne"
] | If `‖y‖ → ∞`, then we can assume `y ≠ x` for any fixed `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group.mem_closure_iff :
a ∈ closure s ↔ ∀ ε, 0 < ε → ∃ b ∈ s, ‖a / b‖ < ε | by simp [metric.mem_closure_iff, dist_eq_norm_div] | lemma | seminormed_comm_group.mem_closure_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"
] | [
"closure",
"dist_eq_norm_div",
"metric.mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_zero_iff''' [t0_space E] {a : E} : ‖a‖ ≤ 0 ↔ a = 1 | begin
letI : normed_group E :=
{ to_metric_space := metric_space.of_t0_pseudo_metric_space E, ..‹seminormed_group E› },
rw [←dist_one_right, dist_le_zero],
end | lemma | norm_le_zero_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_le_zero",
"metric_space.of_t0_pseudo_metric_space",
"normed_group",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_zero''' [t0_space E] {a : E} : ‖a‖ = 0 ↔ a = 1 | (norm_nonneg' a).le_iff_eq.symm.trans norm_le_zero_iff''' | lemma | 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"
] | [
"norm_le_zero_iff'''",
"norm_nonneg'",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pos_iff''' [t0_space E] {a : E} : 0 < ‖a‖ ↔ a ≠ 1 | by rw [← not_le, norm_le_zero_iff'''] | lemma | norm_pos_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"
] | [
"norm_le_zero_iff'''",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group.tendsto_uniformly_on_one {f : ι → κ → G} {s : set κ} {l : filter ι} :
tendsto_uniformly_on f 1 l s ↔ ∀ ε > 0, ∀ᶠ i in l, ∀ x ∈ s, ‖f i x‖ < ε | by simp_rw [tendsto_uniformly_on_iff, pi.one_apply, dist_one_left] | lemma | seminormed_group.tendsto_uniformly_on_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_left",
"filter",
"pi.one_apply",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_one
{f : ι → κ → G} {l : filter ι} {l' : filter κ} : uniform_cauchy_seq_on_filter f l l' ↔
tendsto_uniformly_on_filter (λ n : ι × ι, λ z, f n.fst z / f n.snd z) 1 (l ×ᶠ l) l' | begin
refine ⟨λ hf u hu, _, λ hf u hu, _⟩,
{ obtain ⟨ε, hε, H⟩ := uniformity_basis_dist.mem_uniformity_iff.mp hu,
refine (hf {p : G × G | dist p.fst p.snd < ε} $ dist_mem_uniformity hε).mono (λ x hx,
H 1 (f x.fst.fst x.snd / f x.fst.snd x.snd) _),
simpa [dist_eq_norm_div, norm_div_rev] using hx },
{... | lemma | seminormed_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_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_eq_norm_div",
"filter",
"norm_div_rev",
"tendsto_uniformly_on_filter",
"uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group.uniform_cauchy_seq_on_iff_tendsto_uniformly_on_one
{f : ι → κ → G} {s : set κ} {l : filter ι} :
uniform_cauchy_seq_on f l s ↔
tendsto_uniformly_on (λ n : ι × ι, λ z, f n.fst z / f n.snd z) 1 (l ×ᶠ l) s | by rw [tendsto_uniformly_on_iff_tendsto_uniformly_on_filter,
uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter,
seminormed_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_one] | lemma | seminormed_group.uniform_cauchy_seq_on_iff_tendsto_uniformly_on_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"
] | [
"filter",
"seminormed_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_one",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter",
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group.induced [group E] [seminormed_group F] [monoid_hom_class 𝓕 E F] (f : 𝓕) :
seminormed_group E | { norm := λ x, ‖f x‖,
dist_eq := λ x y, by simpa only [map_div, ←dist_eq_norm_div],
..pseudo_metric_space.induced f _ } | def | seminormed_group.induced | 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",
"map_div",
"monoid_hom_class",
"pseudo_metric_space.induced",
"seminormed_group"
] | A group homomorphism from a `group` to a `seminormed_group` induces a `seminormed_group`
structure on the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group.induced [comm_group E] [seminormed_group F] [monoid_hom_class 𝓕 E F]
(f : 𝓕) : seminormed_comm_group E | { ..seminormed_group.induced E F f } | def | seminormed_comm_group.induced | 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",
"monoid_hom_class",
"seminormed_comm_group",
"seminormed_group",
"seminormed_group.induced"
] | A group homomorphism from a `comm_group` to a `seminormed_group` induces a
`seminormed_comm_group` structure on the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group.induced [group E] [normed_group F] [monoid_hom_class 𝓕 E F] (f : 𝓕)
(h : injective f) : normed_group E | { ..seminormed_group.induced E F f, ..metric_space.induced f h _ } | def | normed_group.induced | 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",
"metric_space.induced",
"monoid_hom_class",
"normed_group",
"seminormed_group.induced"
] | An injective group homomorphism from a `group` to a `normed_group` induces a `normed_group`
structure on the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_group.induced [comm_group E] [normed_group F] [monoid_hom_class 𝓕 E F] (f : 𝓕)
(h : injective f) : normed_comm_group E | { ..seminormed_group.induced E F f, ..metric_space.induced f h _ } | def | normed_comm_group.induced | 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",
"metric_space.induced",
"monoid_hom_class",
"normed_comm_group",
"normed_group",
"seminormed_group.induced"
] | An injective group homomorphism from an `comm_group` to a `normed_group` induces a
`normed_comm_group` structure on the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_group.to_has_isometric_smul_left : has_isometric_smul E E | ⟨λ a, isometry.of_dist_eq $ λ b c, by simp [dist_eq_norm_div]⟩ | instance | normed_group.to_has_isometric_smul_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_eq_norm_div",
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_inv (x y : E) : dist x⁻¹ y = dist x y⁻¹ | by simp_rw [dist_eq_norm_div, ←norm_inv' (x⁻¹ / y), inv_div, div_inv_eq_mul, mul_comm] | lemma | dist_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"
] | [
"dist_eq_norm_div",
"div_inv_eq_mul",
"inv_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_mul_right (a b : E) : dist a (a * b) = ‖b‖ | by rw [←dist_one_left, ←dist_mul_left a 1 b, mul_one] | lemma | dist_self_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"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_mul_left (a b : E) : dist (a * b) a = ‖b‖ | by rw [dist_comm, dist_self_mul_right] | lemma | dist_self_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"
] | [
"dist_comm",
"dist_self_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_div_right (a b : E) : dist a (a / b) = ‖b‖ | by rw [div_eq_mul_inv, dist_self_mul_right, norm_inv'] | lemma | dist_self_div_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_self_mul_right",
"div_eq_mul_inv",
"norm_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_div_left (a b : E) : dist (a / b) a = ‖b‖ | by rw [dist_comm, dist_self_div_right] | lemma | dist_self_div_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_self_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂ | by simpa only [dist_mul_left, dist_mul_right] using dist_triangle (a₁ * a₂) (b₁ * a₂) (b₁ * b₂) | lemma | dist_mul_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_mul_left",
"dist_mul_right",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_mul_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
dist (a₁ * a₂) (b₁ * b₂) ≤ r₁ + r₂ | (dist_mul_mul_le a₁ a₂ b₁ b₂).trans $ add_le_add h₁ h₂ | lemma | dist_mul_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"
] | [
"dist_mul_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_div_le (a₁ a₂ b₁ b₂ : E) :
dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂ | by simpa only [div_eq_mul_inv, dist_inv_inv] using dist_mul_mul_le a₁ a₂⁻¹ b₁ b₂⁻¹ | lemma | dist_div_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_inv_inv",
"dist_mul_mul_le",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_div_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
dist (a₁ / a₂) (b₁ / b₂) ≤ r₁ + r₂ | (dist_div_div_le a₁ a₂ b₁ b₂).trans $ add_le_add h₁ h₂ | lemma | dist_div_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"
] | [
"dist_div_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_dist_sub_le_dist_mul_mul (a₁ a₂ b₁ b₂ : E) : | |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂) :=
by simpa only [dist_mul_left, dist_mul_right, dist_comm b₂]
using abs_dist_sub_le (a₁ * a₂) (b₁ * b₂) (b₁ * a₂) | lemma | abs_dist_sub_le_dist_mul_mul | 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_comm",
"dist_mul_left",
"dist_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_multiset_sum_le {E} [seminormed_add_comm_group E] (m : multiset E) :
‖m.sum‖ ≤ (m.map (λ x, ‖x‖)).sum | m.le_sum_of_subadditive norm norm_zero norm_add_le | lemma | norm_multiset_sum_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"
] | [
"multiset",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_multiset_prod_le (m : multiset E) : ‖m.prod‖ ≤ (m.map $ λ x, ‖x‖).sum | begin
rw [←multiplicative.of_add_le, of_add_multiset_prod, multiset.map_map],
refine multiset.le_prod_of_submultiplicative (multiplicative.of_add ∘ norm) _ (λ x y, _) _,
{ simp only [comp_app, norm_one', of_add_zero] },
{ exact norm_mul_le' _ _ }
end | lemma | norm_multiset_prod_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"
] | [
"multiplicative.of_add",
"multiset",
"multiset.le_prod_of_submultiplicative",
"multiset.map_map",
"norm_mul_le'",
"norm_one'",
"of_add_multiset_prod",
"of_add_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sum_le {E} [seminormed_add_comm_group E] (s : finset ι) (f : ι → E) :
‖∑ i in s, f i‖ ≤ ∑ i in s, ‖f i‖ | s.le_sum_of_subadditive norm norm_zero norm_add_le f | lemma | norm_sum_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"
] | [
"finset",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_prod_le (s : finset ι) (f : ι → E) : ‖∏ i in s, f i‖ ≤ ∑ i in s, ‖f i‖ | begin
rw [←multiplicative.of_add_le, of_add_sum],
refine finset.le_prod_of_submultiplicative (multiplicative.of_add ∘ norm) _ (λ x y, _) _ _,
{ simp only [comp_app, norm_one', of_add_zero] },
{ exact norm_mul_le' _ _ }
end | lemma | norm_prod_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"
] | [
"finset",
"finset.le_prod_of_submultiplicative",
"multiplicative.of_add",
"norm_mul_le'",
"norm_one'",
"of_add_sum",
"of_add_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_prod_le_of_le (s : finset ι) {f : ι → E} {n : ι → ℝ} (h : ∀ b ∈ s, ‖f b‖ ≤ n b) :
‖∏ b in s, f b‖ ≤ ∑ b in s, n b | (norm_prod_le s f).trans $ finset.sum_le_sum h | lemma | norm_prod_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"
] | [
"finset",
"norm_prod_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_prod_prod_le_of_le (s : finset ι) {f a : ι → E} {d : ι → ℝ}
(h : ∀ b ∈ s, dist (f b) (a b) ≤ d b) :
dist (∏ b in s, f b) (∏ b in s, a b) ≤ ∑ b in s, d b | by { simp only [dist_eq_norm_div, ← finset.prod_div_distrib] at *, exact norm_prod_le_of_le s h } | lemma | dist_prod_prod_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"
] | [
"dist_eq_norm_div",
"finset",
"finset.prod_div_distrib",
"norm_prod_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_prod_prod_le (s : finset ι) (f a : ι → E) :
dist (∏ b in s, f b) (∏ b in s, a b) ≤ ∑ b in s, dist (f b) (a b) | dist_prod_prod_le_of_le s $ λ _ _, le_rfl | lemma | dist_prod_prod_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_prod_prod_le_of_le",
"finset",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_ball_iff_norm : a * b ∈ ball a r ↔ ‖b‖ < r | by rw [mem_ball_iff_norm'', mul_div_cancel'''] | lemma | mul_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"
] | [
"mem_ball_iff_norm''",
"mul_div_cancel'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_closed_ball_iff_norm : a * b ∈ closed_ball a r ↔ ‖b‖ ≤ r | by rw [mem_closed_ball_iff_norm'', mul_div_cancel'''] | lemma | mul_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"
] | [
"mem_closed_ball_iff_norm''",
"mul_div_cancel'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_ball (a b : E) (r : ℝ) :
((*) b) ⁻¹' ball a r = ball (a / b) r | by { ext c, simp only [dist_eq_norm_div, set.mem_preimage, mem_ball, div_div_eq_mul_div, mul_comm] } | lemma | preimage_mul_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"
] | [
"dist_eq_norm_div",
"div_div_eq_mul_div",
"mul_comm",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_closed_ball (a b : E) (r : ℝ) :
((*) b) ⁻¹' (closed_ball a r) = closed_ball (a / b) r | by { ext c,
simp only [dist_eq_norm_div, set.mem_preimage, mem_closed_ball, div_div_eq_mul_div, mul_comm] } | lemma | preimage_mul_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"
] | [
"dist_eq_norm_div",
"div_div_eq_mul_div",
"mul_comm",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_sphere (a b : E) (r : ℝ) :
((*) b) ⁻¹' sphere a r = sphere (a / b) r | by { ext c, simp only [set.mem_preimage, mem_sphere_iff_norm', div_div_eq_mul_div, mul_comm] } | lemma | preimage_mul_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"
] | [
"div_div_eq_mul_div",
"mem_sphere_iff_norm'",
"mul_comm",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a^n‖ ≤ n * ‖a‖ | begin
induction n with n ih, { simp, },
simpa only [pow_succ', nat.cast_succ, add_mul, one_mul] using norm_mul_le_of_le ih le_rfl,
end | lemma | norm_pow_le_mul_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"
] | [
"ih",
"le_rfl",
"nat.cast_succ",
"norm_mul_le_of_le",
"one_mul",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a^n‖₊ ≤ n * ‖a‖₊ | by simpa only [← nnreal.coe_le_coe, nnreal.coe_mul, nnreal.coe_nat_cast]
using norm_pow_le_mul_norm n a | lemma | nnnorm_pow_le_mul_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"
] | [
"nnreal.coe_le_coe",
"nnreal.coe_mul",
"nnreal.coe_nat_cast",
"norm_pow_le_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_closed_ball {n : ℕ} (h : a ∈ closed_ball b r) :
a^n ∈ closed_ball (b^n) (n • r) | begin
simp only [mem_closed_ball, dist_eq_norm_div, ← div_pow] at h ⊢,
refine (norm_pow_le_mul_norm n (a / b)).trans _,
simpa only [nsmul_eq_mul] using mul_le_mul_of_nonneg_left h n.cast_nonneg,
end | lemma | pow_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"
] | [
"dist_eq_norm_div",
"div_pow",
"mul_le_mul_of_nonneg_left",
"norm_pow_le_mul_norm",
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_ball {n : ℕ} (hn : 0 < n) (h : a ∈ ball b r) :
a^n ∈ ball (b^n) (n • r) | begin
simp only [mem_ball, dist_eq_norm_div, ← div_pow] at h ⊢,
refine lt_of_le_of_lt (norm_pow_le_mul_norm n (a / b)) _,
replace hn : 0 < (n : ℝ), { norm_cast, assumption, },
rw nsmul_eq_mul,
nlinarith,
end | lemma | pow_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"
] | [
"dist_eq_norm_div",
"div_pow",
"norm_pow_le_mul_norm",
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_closed_ball_mul_iff {c : E} :
a * c ∈ closed_ball (b * c) r ↔ a ∈ closed_ball b r | by simp only [mem_closed_ball, dist_eq_norm_div, mul_div_mul_right_eq_div] | lemma | mul_mem_closed_ball_mul_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_eq_norm_div",
"mul_div_mul_right_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_ball_mul_iff {c : E} :
a * c ∈ ball (b * c) r ↔ a ∈ ball b r | by simp only [mem_ball, dist_eq_norm_div, mul_div_mul_right_eq_div] | lemma | mul_mem_ball_mul_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_eq_norm_div",
"mul_div_mul_right_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball'' :
a • closed_ball b r = closed_ball (a • b) r | by { ext, simp [mem_closed_ball, set.mem_smul_set, dist_eq_norm_div, div_eq_inv_mul,
← eq_inv_mul_iff_mul_eq, mul_assoc], } | lemma | smul_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"
] | [
"dist_eq_norm_div",
"div_eq_inv_mul",
"eq_inv_mul_iff_mul_eq",
"mul_assoc",
"set.mem_smul_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball'' :
a • ball b r = ball (a • b) r | by { ext, simp [mem_ball, set.mem_smul_set, dist_eq_norm_div, div_eq_inv_mul,
← eq_inv_mul_iff_mul_eq, mul_assoc], } | lemma | smul_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"
] | [
"dist_eq_norm_div",
"div_eq_inv_mul",
"eq_inv_mul_iff_mul_eq",
"mul_assoc",
"set.mem_smul_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
controlled_prod_of_mem_closure {s : subgroup E} (hg : a ∈ closure (s : set E))
{b : ℕ → ℝ} (b_pos : ∀ n, 0 < b n) :
∃ v : ℕ → E,
tendsto (λ n, ∏ i in range (n+1), v i) at_top (𝓝 a) ∧
(∀ n, v n ∈ s) ∧
‖v 0 / a‖ < b 0 ∧
∀ n, 0 < n → ‖v n‖ < b n | begin
obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : tendsto u at_top (𝓝 a)⟩ :=
mem_closure_iff_seq_limit.mp hg,
obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0,
{ have : {x | ‖x / a‖ < b 0} ∈ 𝓝 a,
{ simp_rw ← dist_eq_norm_div,
exact metric.ball_mem_nhds _ (b_pos _) },
exact filter.tendsto... | lemma | controlled_prod_of_mem_closure | 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"
] | [
"closure",
"dist_eq_norm_div",
"finset.eq_prod_range_div'",
"metric.ball_mem_nhds",
"metric.dist_mem_uniformity",
"strict_mono",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
(hb : b ∈ closure (j.range : set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
∃ a : ℕ → E,
tendsto (λ n, ∏ i in range (n + 1), j (a i)) at_top (𝓝 b) ∧
‖j (a 0) / b‖ < f 0 ∧
∀ n, 0 < n → ‖j (a n)‖ < f n | begin
obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos,
choose g hg using v_in,
refine ⟨g, by simpa [← hg] using sum_v, by simpa [hg 0] using hv₀, λ n hn,
by simpa [hg] using hv_pos n hn⟩,
end | lemma | controlled_prod_of_mem_closure_range | 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"
] | [
"closure",
"controlled_prod_of_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂ | nnreal.coe_le_coe.1 $ dist_mul_mul_le a₁ a₂ b₁ b₂ | lemma | nndist_mul_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_mul_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_mul_mul_le (a₁ a₂ b₁ b₂ : E) : edist (a₁ * a₂) (b₁ * b₂) ≤ edist a₁ b₁ + edist a₂ b₂ | by { simp only [edist_nndist], norm_cast, apply nndist_mul_mul_le } | lemma | edist_mul_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"
] | [
"edist_nndist",
"nndist_mul_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_multiset_prod_le (m : multiset E) : ‖m.prod‖₊ ≤ (m.map (λ x, ‖x‖₊)).sum | nnreal.coe_le_coe.1 $ by { push_cast, rw multiset.map_map, exact norm_multiset_prod_le _ } | lemma | nnnorm_multiset_prod_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"
] | [
"multiset",
"multiset.map_map",
"norm_multiset_prod_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_prod_le (s : finset ι) (f : ι → E) :
‖∏ a in s, f a‖₊ ≤ ∑ a in s, ‖f a‖₊ | nnreal.coe_le_coe.1 $ by { push_cast, exact norm_prod_le _ _ } | lemma | nnnorm_prod_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"
] | [
"finset",
"norm_prod_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_prod_le_of_le (s : finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b) :
‖∏ b in s, f b‖₊ ≤ ∑ b in s, n b | (norm_prod_le_of_le s h).trans_eq nnreal.coe_sum.symm | lemma | nnnorm_prod_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"
] | [
"finset",
"norm_prod_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_abs (r : ℝ) : ‖r‖ = |r| | rfl | lemma | real.norm_eq_abs | 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_of_nonneg (hr : 0 ≤ r) : ‖r‖ = r | abs_of_nonneg hr | lemma | real.norm_of_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"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_nonpos (hr : r ≤ 0) : ‖r‖ = -r | abs_of_nonpos hr | lemma | real.norm_of_nonpos | 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_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_norm_self (r : ℝ) : r ≤ ‖r‖ | le_abs_self r | lemma | real.le_norm_self | 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_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_nat (n : ℕ) : ‖(n : ℝ)‖ = n | abs_of_nonneg n.cast_nonneg | lemma | real.norm_coe_nat | 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_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_coe_nat (n : ℕ) : ‖(n : ℝ)‖₊ = n | nnreal.eq $ norm_coe_nat _ | lemma | real.nnnorm_coe_nat | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_two : ‖(2 : ℝ)‖ = 2 | abs_of_pos zero_lt_two | lemma | real.norm_two | 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_of_pos",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_two : ‖(2 : ℝ)‖₊ = 2 | nnreal.eq $ by simp | lemma | real.nnnorm_two | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_nonneg (hr : 0 ≤ r) : ‖r‖₊ = ⟨r, hr⟩ | nnreal.eq $ norm_of_nonneg hr | lemma | real.nnnorm_of_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"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_abs (r : ℝ) : ‖(|r|)‖₊ = ‖r‖₊ | by simp [nnnorm] | lemma | real.nnnorm_abs | 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 | |
ennnorm_eq_of_real (hr : 0 ≤ r) : (‖r‖₊ : ℝ≥0∞) = ennreal.of_real r | by { rw [← of_real_norm_eq_coe_nnnorm, norm_of_nonneg hr] } | lemma | real.ennnorm_eq_of_real | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennnorm_eq_of_real_abs (r : ℝ) : (‖r‖₊ : ℝ≥0∞) = ennreal.of_real (|r|) | by rw [← real.nnnorm_abs r, real.ennnorm_eq_of_real (abs_nonneg _)] | lemma | real.ennnorm_eq_of_real_abs | 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_nonneg",
"ennreal.of_real",
"real.ennnorm_eq_of_real",
"real.nnnorm_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nnreal_eq_nnnorm_of_nonneg (hr : 0 ≤ r) : r.to_nnreal = ‖r‖₊ | begin
rw real.to_nnreal_of_nonneg hr,
congr,
rw [real.norm_eq_abs, abs_of_nonneg hr],
end | lemma | real.to_nnreal_eq_nnnorm_of_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"
] | [
"abs_of_nonneg",
"real.norm_eq_abs",
"real.to_nnreal_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_le_ennnorm (r : ℝ) : ennreal.of_real r ≤ ‖r‖₊ | begin
obtain hr | hr := le_total 0 r,
{ exact (real.ennnorm_eq_of_real hr).ge },
{ rw [ennreal.of_real_eq_zero.2 hr],
exact bot_le }
end | lemma | real.of_real_le_ennnorm | 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"
] | [
"bot_le",
"ennreal.of_real",
"real.ennnorm_eq_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cast_real (m : ℤ) : ‖(m : ℝ)‖ = ‖m‖ | rfl | lemma | int.norm_cast_real | 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 |
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