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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|---|---|---|---|---|---|---|---|---|---|---|
dist_coe_int (x y : ℕ) : dist (x : ℤ) (y : ℤ) = dist x y | rfl | lemma | nat.dist_coe_int | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_cast_real (x y : ℕ) : dist (x : ℝ) y = dist x y | rfl | theorem | nat.dist_cast_real | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise_one_le_dist : pairwise (λ m n : ℕ, 1 ≤ dist m n) | begin
intros m n hne,
rw ← dist_coe_int,
apply int.pairwise_one_le_dist,
exact_mod_cast hne
end | lemma | nat.pairwise_one_le_dist | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [
"int.pairwise_one_le_dist",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_coe_real : uniform_embedding (coe : ℕ → ℝ) | uniform_embedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | lemma | nat.uniform_embedding_coe_real | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [
"uniform_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_coe_real : closed_embedding (coe : ℕ → ℝ) | closed_embedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | lemma | nat.closed_embedding_coe_real | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [
"closed_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ball (x : ℕ) (r : ℝ) : coe ⁻¹' (ball (x : ℝ) r) = ball x r | rfl | theorem | nat.preimage_ball | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (x : ℕ) (r : ℝ) :
coe ⁻¹' (closed_ball (x : ℝ) r) = closed_ball x r | rfl | theorem | nat.preimage_closed_ball | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_Icc (x : ℕ) (r : ℝ) :
closed_ball x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊ | begin
rcases le_or_lt 0 r with hr|hr,
{ rw [← preimage_closed_ball, real.closed_ball_eq_Icc, preimage_Icc],
exact add_nonneg (cast_nonneg x) hr },
{ rw closed_ball_eq_empty.2 hr,
apply (Icc_eq_empty _).symm,
rw not_le,
calc ⌊(x : ℝ) + r⌋₊ ≤ ⌊(x : ℝ)⌋₊ : by { apply floor_mono, linarith }
... < ... | theorem | nat.closed_ball_eq_Icc | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [
"nat.lt_ceil",
"real.closed_ball_eq_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_real_to_nnreal : continuous real.to_nnreal | (continuous_id.max continuous_const).subtype_mk _ | lemma | continuous_real_to_nnreal | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"continuous",
"continuous_const",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : ℝ≥0 → ℝ) | continuous_subtype_val | lemma | nnreal.continuous_coe | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"continuous",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_map.coe_nnreal_real : C(ℝ≥0, ℝ) | ⟨coe, continuous_coe⟩ | def | continuous_map.coe_nnreal_real | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [] | Embedding of `ℝ≥0` to `ℝ` as a bundled continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.can_lift {X : Type*} [topological_space X] :
can_lift C(X, ℝ) C(X, ℝ≥0) continuous_map.coe_nnreal_real.comp (λ f, ∀ x, 0 ≤ f x) | { prf := λ f hf, ⟨⟨λ x, ⟨f x, hf x⟩, f.2.subtype_mk _⟩, fun_like.ext' rfl⟩ } | instance | nnreal.continuous_map.can_lift | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"can_lift",
"continuous_map.can_lift",
"fun_like.ext'",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe {f : filter α} {m : α → ℝ≥0} {x : ℝ≥0} :
tendsto (λa, (m a : ℝ)) f (𝓝 (x : ℝ)) ↔ tendsto m f (𝓝 x) | tendsto_subtype_rng.symm | lemma | nnreal.tendsto_coe | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe' {f : filter α} [ne_bot f] {m : α → ℝ≥0} {x : ℝ} :
tendsto (λ a, m a : α → ℝ) f (𝓝 x) ↔ ∃ hx : 0 ≤ x, tendsto m f (𝓝 ⟨x, hx⟩) | ⟨λ h, ⟨ge_of_tendsto' h (λ c, (m c).2), tendsto_coe.1 h⟩, λ ⟨hx, hm⟩, tendsto_coe.2 hm⟩ | lemma | nnreal.tendsto_coe' | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_at_top : map (coe : ℝ≥0 → ℝ) at_top = at_top | map_coe_Ici_at_top 0 | lemma | nnreal.map_coe_at_top | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_coe_at_top : comap (coe : ℝ≥0 → ℝ) at_top = at_top | (at_top_Ici_eq 0).symm | lemma | nnreal.comap_coe_at_top | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe_at_top {f : filter α} {m : α → ℝ≥0} :
tendsto (λ a, (m a : ℝ)) f at_top ↔ tendsto m f at_top | tendsto_Ici_at_top.symm | lemma | nnreal.tendsto_coe_at_top | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.tendsto_real_to_nnreal {f : filter α} {m : α → ℝ} {x : ℝ} (h : tendsto m f (𝓝 x)) :
tendsto (λa, real.to_nnreal (m a)) f (𝓝 (real.to_nnreal x)) | (continuous_real_to_nnreal.tendsto _).comp h | lemma | tendsto_real_to_nnreal | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"filter",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.tendsto_real_to_nnreal_at_top : tendsto real.to_nnreal at_top at_top | begin
rw ← tendsto_coe_at_top,
apply tendsto_id.congr' _,
filter_upwards [Ici_mem_at_top (0 : ℝ)] with x hx,
simp only [max_eq_left (set.mem_Ici.1 hx), id.def, real.coe_to_nnreal'],
end | lemma | tendsto_real_to_nnreal_at_top | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"real.coe_to_nnreal'",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero : 𝓝 (0 : ℝ≥0) = ⨅a ≠ 0, 𝓟 (Iio a) | nhds_bot_order.trans $ by simp [bot_lt_iff_ne_bot] | lemma | nnreal.nhds_zero | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"bot_lt_iff_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero_basis : (𝓝 (0 : ℝ≥0)).has_basis (λ a : ℝ≥0, 0 < a) (λ a, Iio a) | nhds_bot_basis | lemma | nnreal.nhds_zero_basis | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"nhds_bot_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_coe {f : α → ℝ≥0} {r : ℝ≥0} :
has_sum (λa, (f a : ℝ)) (r : ℝ) ↔ has_sum f r | by simp only [has_sum, coe_sum.symm, tendsto_coe] | lemma | nnreal.has_sum_coe | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_real_to_nnreal_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : summable f) :
has_sum (λ n, real.to_nnreal (f n)) (real.to_nnreal (∑' n, f n)) | begin
have h_sum : (λ s, ∑ b in s, real.to_nnreal (f b)) = λ s, real.to_nnreal (∑ b in s, f b),
from funext (λ _, (real.to_nnreal_sum_of_nonneg (λ n _, hf_nonneg n)).symm),
simp_rw [has_sum, h_sum],
exact tendsto_real_to_nnreal hf.has_sum,
end | lemma | nnreal.has_sum_real_to_nnreal_of_nonneg | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"has_sum",
"real.to_nnreal",
"real.to_nnreal_sum_of_nonneg",
"summable",
"tendsto_real_to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_coe {f : α → ℝ≥0} : summable (λa, (f a : ℝ)) ↔ summable f | begin
split,
exact assume ⟨a, ha⟩, ⟨⟨a, has_sum_le (λa, (f a).2) has_sum_zero ha⟩, has_sum_coe.1 ha⟩,
exact assume ⟨a, ha⟩, ⟨a.1, has_sum_coe.2 ha⟩
end | lemma | nnreal.summable_coe | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"has_sum_le",
"has_sum_zero",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_coe_of_nonneg {f : α → ℝ} (hf₁ : ∀ n, 0 ≤ f n) :
@summable (ℝ≥0) _ _ _ (λ n, ⟨f n, hf₁ n⟩) ↔ summable f | begin
lift f to α → ℝ≥0 using hf₁ with f rfl hf₁,
simp only [summable_coe, subtype.coe_eta]
end | lemma | nnreal.summable_coe_of_nonneg | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"lift",
"subtype.coe_eta",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_tsum {f : α → ℝ≥0} : ↑∑'a, f a = ∑'a, (f a : ℝ) | if hf : summable f
then (eq.symm $ (has_sum_coe.2 $ hf.has_sum).tsum_eq)
else by simp [tsum, hf, mt summable_coe.1 hf] | lemma | nnreal.coe_tsum | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_tsum_of_nonneg {f : α → ℝ} (hf₁ : ∀ n, 0 ≤ f n) :
(⟨∑' n, f n, tsum_nonneg hf₁⟩ : ℝ≥0) = (∑' n, ⟨f n, hf₁ n⟩ : ℝ≥0) | begin
lift f to α → ℝ≥0 using hf₁ with f rfl hf₁,
simp_rw [← nnreal.coe_tsum, subtype.coe_eta]
end | lemma | nnreal.coe_tsum_of_nonneg | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"lift",
"nnreal.coe_tsum",
"subtype.coe_eta",
"tsum_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_left (a : ℝ≥0) (f : α → ℝ≥0) : ∑' x, a * f x = a * ∑' x, f x | nnreal.eq $ by simp only [coe_tsum, nnreal.coe_mul, tsum_mul_left] | lemma | nnreal.tsum_mul_left | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"nnreal.coe_mul",
"nnreal.eq",
"tsum_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_right (f : α → ℝ≥0) (a : ℝ≥0) : (∑' x, f x * a) = (∑' x, f x) * a | nnreal.eq $ by simp only [coe_tsum, nnreal.coe_mul, tsum_mul_right] | lemma | nnreal.tsum_mul_right | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"nnreal.coe_mul",
"nnreal.eq",
"tsum_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_comp_injective {β : Type*} {f : α → ℝ≥0} (hf : summable f)
{i : β → α} (hi : function.injective i) :
summable (f ∘ i) | nnreal.summable_coe.1 $
show summable ((coe ∘ f) ∘ i), from (nnreal.summable_coe.2 hf).comp_injective hi | lemma | nnreal.summable_comp_injective | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_nat_add (f : ℕ → ℝ≥0) (hf : summable f) (k : ℕ) : summable (λ i, f (i + k)) | summable_comp_injective hf $ add_left_injective k | lemma | nnreal.summable_nat_add | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) : summable (λ i, f (i + k)) ↔ summable f | begin
rw [← summable_coe, ← summable_coe],
exact @summable_nat_add_iff ℝ _ _ _ (λ i, (f i : ℝ)) k,
end | lemma | nnreal.summable_nat_add_iff | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"summable",
"summable_nat_add_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) {a : ℝ≥0} :
has_sum (λ n, f (n + k)) a ↔ has_sum f (a + ∑ i in range k, f i) | by simp [← has_sum_coe, coe_sum, nnreal.coe_add, ← has_sum_nat_add_iff k] | lemma | nnreal.has_sum_nat_add_iff | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"has_sum",
"has_sum_nat_add_iff",
"nnreal.coe_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_tsum_nat_add {f : ℕ → ℝ≥0} (k : ℕ) (hf : summable f) :
∑' i, f i = (∑ i in range k, f i) + ∑' i, f (i + k) | by rw [←nnreal.coe_eq, coe_tsum, nnreal.coe_add, coe_sum, coe_tsum,
sum_add_tsum_nat_add k (nnreal.summable_coe.2 hf)] | lemma | nnreal.sum_add_tsum_nat_add | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"nnreal.coe_add",
"sum_add_tsum_nat_add",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_real_pos_eq_infi_nnreal_pos [complete_lattice α] {f : ℝ → α} :
(⨅ (n : ℝ) (h : 0 < n), f n) = (⨅ (n : ℝ≥0) (h : 0 < n), f n) | le_antisymm (infi_mono' $ λ r, ⟨r, le_rfl⟩) (infi₂_mono' $ λ r hr, ⟨⟨r, hr.le⟩, hr, le_rfl⟩) | lemma | nnreal.infi_real_pos_eq_infi_nnreal_pos | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"complete_lattice",
"infi_mono'",
"infi₂_mono'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cofinite_zero_of_summable {α} {f : α → ℝ≥0} (hf : summable f) :
tendsto f cofinite (𝓝 0) | begin
have h_f_coe : f = λ n, real.to_nnreal (f n : ℝ), from funext (λ n, real.to_nnreal_coe.symm),
rw [h_f_coe, ← @real.to_nnreal_coe 0],
exact tendsto_real_to_nnreal ((summable_coe.mpr hf).tendsto_cofinite_zero),
end | lemma | nnreal.tendsto_cofinite_zero_of_summable | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"real.to_nnreal",
"real.to_nnreal_coe",
"summable",
"tendsto_real_to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_zero_of_summable {f : ℕ → ℝ≥0} (hf : summable f) :
tendsto f at_top (𝓝 0) | by { rw ←nat.cofinite_eq_at_top, exact tendsto_cofinite_zero_of_summable hf } | lemma | nnreal.tendsto_at_top_zero_of_summable | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_tsum_compl_at_top_zero {α : Type*} (f : α → ℝ≥0) :
tendsto (λ (s : finset α), ∑' b : {x // x ∉ s}, f b) at_top (𝓝 0) | begin
simp_rw [← tendsto_coe, coe_tsum, nnreal.coe_zero],
exact tendsto_tsum_compl_at_top_zero (λ (a : α), (f a : ℝ))
end | lemma | nnreal.tendsto_tsum_compl_at_top_zero | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"finset",
"nnreal.coe_zero",
"tendsto_tsum_compl_at_top_zero"
] | The sum over the complement of a finset tends to `0` when the finset grows to cover the whole
space. This does not need a summability assumption, as otherwise all sums are zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_order_iso (n : ℕ) (hn : n ≠ 0) : ℝ≥0 ≃o ℝ≥0 | strict_mono.order_iso_of_surjective (λ x, x ^ n)
(λ x y h, strict_mono_on_pow hn.bot_lt (zero_le x) (zero_le y) h) $
(continuous_id.pow _).surjective (tendsto_pow_at_top hn) $
by simpa [order_bot.at_bot_eq, pos_iff_ne_zero] | def | nnreal.pow_order_iso | topology.instances | src/topology/instances/nnreal.lean | [
"topology.algebra.infinite_sum.order",
"topology.algebra.infinite_sum.ring",
"topology.instances.real"
] | [
"strict_mono.order_iso_of_surjective",
"strict_mono_on_pow"
] | `x ↦ x ^ n` as an order isomorphism of `ℝ≥0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq (x y : ℚ) : dist x y = |x - y| | rfl | theorem | rat.dist_eq | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_cast (x y : ℚ) : dist (x : ℝ) y = dist x y | rfl | lemma | rat.dist_cast | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_coe_real : uniform_continuous (coe : ℚ → ℝ) | uniform_continuous_comap | theorem | rat.uniform_continuous_coe_real | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"uniform_continuous",
"uniform_continuous_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_coe_real : uniform_embedding (coe : ℚ → ℝ) | uniform_embedding_comap rat.cast_injective | theorem | rat.uniform_embedding_coe_real | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"rat.cast_injective",
"uniform_embedding",
"uniform_embedding_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_embedding_coe_real : dense_embedding (coe : ℚ → ℝ) | uniform_embedding_coe_real.dense_embedding rat.dense_range_cast | theorem | rat.dense_embedding_coe_real | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"dense_embedding",
"rat.dense_range_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_coe_real : embedding (coe : ℚ → ℝ) | dense_embedding_coe_real.to_embedding | theorem | rat.embedding_coe_real | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe_real : continuous (coe : ℚ → ℝ) | uniform_continuous_coe_real.continuous | theorem | rat.continuous_coe_real | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.dist_cast_rat (x y : ℕ) : dist (x : ℚ) y = dist x y | by rw [← nat.dist_cast_real, ← rat.dist_cast]; congr' 1; norm_cast | theorem | nat.dist_cast_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"nat.dist_cast_real",
"rat.dist_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.uniform_embedding_coe_rat : uniform_embedding (coe : ℕ → ℚ) | uniform_embedding_bot_of_pairwise_le_dist zero_lt_one $ by simpa using nat.pairwise_one_le_dist | lemma | nat.uniform_embedding_coe_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"nat.pairwise_one_le_dist",
"uniform_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.closed_embedding_coe_rat : closed_embedding (coe : ℕ → ℚ) | closed_embedding_of_pairwise_le_dist zero_lt_one $ by simpa using nat.pairwise_one_le_dist | lemma | nat.closed_embedding_coe_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"closed_embedding",
"nat.pairwise_one_le_dist",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.dist_cast_rat (x y : ℤ) : dist (x : ℚ) y = dist x y | by rw [← int.dist_cast_real, ← rat.dist_cast]; congr' 1; norm_cast | theorem | int.dist_cast_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"int.dist_cast_real",
"rat.dist_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.uniform_embedding_coe_rat : uniform_embedding (coe : ℤ → ℚ) | uniform_embedding_bot_of_pairwise_le_dist zero_lt_one $ by simpa using int.pairwise_one_le_dist | lemma | int.uniform_embedding_coe_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"int.pairwise_one_le_dist",
"uniform_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.closed_embedding_coe_rat : closed_embedding (coe : ℤ → ℚ) | closed_embedding_of_pairwise_le_dist zero_lt_one $ by simpa using int.pairwise_one_le_dist | lemma | int.closed_embedding_coe_rat | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"closed_embedding",
"int.pairwise_one_le_dist",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_add : uniform_continuous (λp : ℚ × ℚ, p.1 + p.2) | rat.uniform_embedding_coe_real.to_uniform_inducing.uniform_continuous_iff.2 $
by simp only [(∘), rat.cast_add]; exact real.uniform_continuous_add.comp
(rat.uniform_continuous_coe_real.prod_map rat.uniform_continuous_coe_real) | theorem | rat.uniform_continuous_add | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"rat.cast_add",
"rat.uniform_continuous_coe_real",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_neg : uniform_continuous (@has_neg.neg ℚ _) | metric.uniform_continuous_iff.2 $ λ ε ε0, ⟨_, ε0, λ a b h,
by rw dist_comm at h; simpa [rat.dist_eq] using h⟩ | theorem | rat.uniform_continuous_neg | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"dist_comm",
"rat.dist_eq",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_abs : uniform_continuous (abs : ℚ → ℚ) | metric.uniform_continuous_iff.2 $ λ ε ε0,
⟨ε, ε0, λ a b h, lt_of_le_of_lt
(by simpa [rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _) h⟩ | lemma | rat.uniform_continuous_abs | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"abs_abs_sub_abs_le_abs_sub",
"rat.dist_eq",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul : continuous (λp : ℚ × ℚ, p.1 * p.2) | rat.embedding_coe_real.continuous_iff.2 $ by simp [(∘)]; exact
real.continuous_mul.comp ((rat.continuous_coe_real.prod_map rat.continuous_coe_real)) | lemma | rat.continuous_mul | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"continuous",
"continuous_mul",
"rat.continuous_coe_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_Icc (a b : ℚ) : totally_bounded (Icc a b) | by simpa only [preimage_cast_Icc]
using totally_bounded_preimage rat.uniform_embedding_coe_real (totally_bounded_Icc a b) | lemma | rat.totally_bounded_Icc | topology.instances | src/topology/instances/rat.lean | [
"topology.metric_space.basic",
"topology.algebra.order.archimedean",
"topology.instances.int",
"topology.instances.nat",
"topology.instances.real"
] | [
"rat.uniform_embedding_coe_real",
"totally_bounded",
"totally_bounded_Icc",
"totally_bounded_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_compact_eq_empty (hs : is_compact s) :
interior s = ∅ | dense_embedding_coe_real.to_dense_inducing.interior_compact_eq_empty dense_irrational hs | lemma | rat.interior_compact_eq_empty | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [
"dense_irrational",
"interior",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_compl_compact (hs : is_compact s) : dense sᶜ | interior_eq_empty_iff_dense_compl.1 (interior_compact_eq_empty hs) | lemma | rat.dense_compl_compact | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [
"dense",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocompact_inf_nhds_ne_bot : ne_bot (cocompact ℚ ⊓ 𝓝 p) | begin
refine (has_basis_cocompact.inf (nhds_basis_opens _)).ne_bot_iff.2 _,
rintro ⟨s, o⟩ ⟨hs, hpo, ho⟩, rw inter_comm,
exact (dense_compl_compact hs).inter_open_nonempty _ ho ⟨p, hpo⟩
end | instance | rat.cocompact_inf_nhds_ne_bot | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_countably_generated_cocompact : ¬is_countably_generated (cocompact ℚ) | begin
introI H,
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩,
rw tendsto_inf at hx, rcases hx with ⟨hxc, hx0⟩,
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x),
from (hxc.eventually hx0.is_compact_insert_range.compl_mem_cocompact).exists,
exact hn (or.inr ⟨n, rfl⟩)
end | lemma | rat.not_countably_generated_cocompact | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_countably_generated_nhds_infty_alexandroff :
¬is_countably_generated (𝓝 (∞ : ℚ∞)) | begin
introI,
have : is_countably_generated (comap (coe : ℚ → ℚ∞) (𝓝 ∞)), by apply_instance,
rw [alexandroff.comap_coe_nhds_infty, coclosed_compact_eq_cocompact] at this,
exact not_countably_generated_cocompact this
end | lemma | rat.not_countably_generated_nhds_infty_alexandroff | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [
"alexandroff.comap_coe_nhds_infty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_first_countable_topology_alexandroff :
¬first_countable_topology ℚ∞ | by { introI, exact not_countably_generated_nhds_infty_alexandroff infer_instance } | lemma | rat.not_first_countable_topology_alexandroff | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_second_countable_topology_alexandroff :
¬second_countable_topology ℚ∞ | by { introI, exact not_first_countable_topology_alexandroff infer_instance } | lemma | rat.not_second_countable_topology_alexandroff | topology.instances | src/topology/instances/rat_lemmas.lean | [
"topology.instances.irrational",
"topology.instances.rat",
"topology.alexandroff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_add : uniform_continuous (λp : ℝ × ℝ, p.1 + p.2) | metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_add_continuous_lemma abs ε0 in
⟨δ, δ0, λ a b h, let ⟨h₁, h₂⟩ := max_lt_iff.1 h in Hδ h₁ h₂⟩ | theorem | real.uniform_continuous_add | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"rat_add_continuous_lemma",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_neg : uniform_continuous (@has_neg.neg ℝ _) | metric.uniform_continuous_iff.2 $ λ ε ε0, ⟨_, ε0, λ a b h,
by rw dist_comm at h; simpa [real.dist_eq] using h⟩ | theorem | real.uniform_continuous_neg | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"dist_comm",
"real.dist_eq",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.is_topological_basis_Ioo_rat :
@is_topological_basis ℝ _ (⋃(a b : ℚ) (h : a < b), {Ioo a b}) | is_topological_basis_of_open_of_nhds
(by simp [is_open_Ioo] {contextual:=tt})
(assume a v hav hv,
let ⟨l, u, ⟨hl, hu⟩, h⟩ := mem_nhds_iff_exists_Ioo_subset.mp (is_open.mem_nhds hv hav),
⟨q, hlq, hqa⟩ := exists_rat_btwn hl,
⟨p, hap, hpu⟩ := exists_rat_btwn hu in
⟨Ioo q p,
by { simp only... | lemma | real.is_topological_basis_Ioo_rat | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"exists_rat_btwn",
"is_open.mem_nhds",
"is_open_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.cocompact_eq : cocompact ℝ = at_bot ⊔ at_top | by simp only [← comap_dist_right_at_top_eq_cocompact (0 : ℝ), real.dist_eq, sub_zero,
comap_abs_at_top] | lemma | real.cocompact_eq | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"comap_dist_right_at_top_eq_cocompact",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.mem_closure_iff {s : set ℝ} {x : ℝ} :
x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε | by simp [mem_closure_iff_nhds_basis nhds_basis_ball, real.dist_eq] | lemma | real.mem_closure_iff | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"closure",
"mem_closure_iff_nhds_basis",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_inv (s : set ℝ) {r : ℝ} (r0 : 0 < r) (H : ∀ x ∈ s, r ≤ |x|) :
uniform_continuous (λp:s, p.1⁻¹) | metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_inv_continuous_lemma abs ε0 r0 in
⟨δ, δ0, λ a b h, Hδ (H _ a.2) (H _ b.2) h⟩ | lemma | real.uniform_continuous_inv | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"rat_inv_continuous_lemma",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_abs : uniform_continuous (abs : ℝ → ℝ) | metric.uniform_continuous_iff.2 $ λ ε ε0,
⟨ε, ε0, λ a b, lt_of_le_of_lt (abs_abs_sub_abs_le_abs_sub _ _)⟩ | lemma | real.uniform_continuous_abs | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"abs_abs_sub_abs_le_abs_sub",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.tendsto_inv {r : ℝ} (r0 : r ≠ 0) : tendsto (λq, q⁻¹) (𝓝 r) (𝓝 r⁻¹) | by rw ← abs_pos at r0; exact
tendsto_of_uniform_continuous_subtype
(real.uniform_continuous_inv {x | |r| / 2 < |x|} (half_pos r0) (λ x h, le_of_lt h))
(is_open.mem_nhds ((is_open_lt' (|r| / 2)).preimage continuous_abs) (half_lt_self r0)) | lemma | real.tendsto_inv | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"abs_pos",
"continuous_abs",
"half_pos",
"is_open.mem_nhds",
"is_open_lt'",
"real.uniform_continuous_inv",
"tendsto_of_uniform_continuous_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.continuous_inv : continuous (λa:{r:ℝ // r ≠ 0}, a.val⁻¹) | continuous_iff_continuous_at.mpr $ assume ⟨r, hr⟩,
tendsto.comp (real.tendsto_inv hr) (continuous_iff_continuous_at.mp continuous_subtype_val _) | lemma | real.continuous_inv | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"continuous_subtype_val",
"real.tendsto_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.continuous.inv [topological_space α] {f : α → ℝ} (h : ∀a, f a ≠ 0) (hf : continuous f) :
continuous (λa, (f a)⁻¹) | show continuous ((has_inv.inv ∘ @subtype.val ℝ (λr, r ≠ 0)) ∘ λa, ⟨f a, h a⟩),
from real.continuous_inv.comp (hf.subtype_mk _) | lemma | real.continuous.inv | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_const_mul {x : ℝ} : uniform_continuous ((*) x) | uniform_continuous_const_smul x | lemma | real.uniform_continuous_const_mul | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.uniform_continuous_mul (s : set (ℝ × ℝ))
{r₁ r₂ : ℝ} (H : ∀ x ∈ s, |(x : ℝ × ℝ).1| < r₁ ∧ |x.2| < r₂) :
uniform_continuous (λp:s, p.1.1 * p.1.2) | metric.uniform_continuous_iff.2 $ λ ε ε0,
let ⟨δ, δ0, Hδ⟩ := rat_mul_continuous_lemma abs ε0 in
⟨δ, δ0, λ a b h,
let ⟨h₁, h₂⟩ := max_lt_iff.1 h in Hδ (H _ a.2).1 (H _ b.2).2 h₁ h₂⟩ | lemma | real.uniform_continuous_mul | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"rat_mul_continuous_lemma",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.continuous_mul : continuous (λp : ℝ × ℝ, p.1 * p.2) | continuous_iff_continuous_at.2 $ λ ⟨a₁, a₂⟩,
tendsto_of_uniform_continuous_subtype
(real.uniform_continuous_mul
({x | |x| < |a₁| + 1} ×ˢ {x | |x| < |a₂| + 1})
(λ x, id))
(is_open.mem_nhds
(((is_open_gt' (|a₁| + 1)).preimage continuous_abs).prod
((is_open_gt' (|a₂| + 1)).preimage continuous_abs ))
... | lemma | real.continuous_mul | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"continuous_abs",
"is_open.mem_nhds",
"is_open_gt'",
"lt_add_one",
"real.uniform_continuous_mul",
"tendsto_of_uniform_continuous_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.totally_bounded_ball (x ε : ℝ) : totally_bounded (ball x ε) | by rw real.ball_eq_Ioo; apply totally_bounded_Ioo | lemma | real.totally_bounded_ball | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"real.ball_eq_Ioo",
"totally_bounded",
"totally_bounded_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_of_rat_image_lt {q : ℚ} : closure ((coe:ℚ → ℝ) '' {x | q < x}) = {r | ↑q ≤ r} | subset.antisymm
((is_closed_ge' _).closure_subset_iff.2
(image_subset_iff.2 $ λ p h, le_of_lt $ (@rat.cast_lt ℝ _ _ _).2 h)) $
λ x hx, mem_closure_iff_nhds.2 $ λ t ht,
let ⟨ε, ε0, hε⟩ := metric.mem_nhds_iff.1 ht in
let ⟨p, h₁, h₂⟩ := exists_rat_btwn ((lt_add_iff_pos_right x).2 ε0) in
⟨_, hε (show abs _ < _,
b... | lemma | closure_of_rat_image_lt | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"abs_of_nonneg",
"closure",
"exists_rat_btwn",
"is_closed_ge'",
"rat.cast_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.bounded_iff_bdd_below_bdd_above {s : set ℝ} : bounded s ↔ bdd_below s ∧ bdd_above s | ⟨begin
assume bdd,
rcases (bounded_iff_subset_ball 0).1 bdd with ⟨r, hr⟩, -- hr : s ⊆ closed_ball 0 r
rw real.closed_ball_eq_Icc at hr, -- hr : s ⊆ Icc (0 - r) (0 + r)
exact ⟨bdd_below_Icc.mono hr, bdd_above_Icc.mono hr⟩
end,
λ h, bounded_of_bdd_above_of_bdd_below h.2 h.1⟩ | lemma | real.bounded_iff_bdd_below_bdd_above | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"bdd_above",
"bdd_below",
"real.closed_ball_eq_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.subset_Icc_Inf_Sup_of_bounded {s : set ℝ} (h : bounded s) :
s ⊆ Icc (Inf s) (Sup s) | subset_Icc_cInf_cSup (real.bounded_iff_bdd_below_bdd_above.1 h).1
(real.bounded_iff_bdd_below_bdd_above.1 h).2 | lemma | real.subset_Icc_Inf_Sup_of_bounded | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"subset_Icc_cInf_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.compact_of_continuous' [topological_space α] {f : ℝ → α} {c : ℝ}
(hp : periodic f c) (hc : 0 < c) (hf : continuous f) :
is_compact (range f) | begin
convert is_compact_Icc.image hf,
ext x,
refine ⟨_, mem_range_of_mem_image f (Icc 0 c)⟩,
rintros ⟨y, h1⟩,
obtain ⟨z, hz, h2⟩ := hp.exists_mem_Ico₀ hc y,
exact ⟨z, mem_Icc_of_Ico hz, h2.symm.trans h1⟩,
end | lemma | function.periodic.compact_of_continuous' | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"is_compact",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.compact_of_continuous [topological_space α] {f : ℝ → α} {c : ℝ}
(hp : periodic f c) (hc : c ≠ 0) (hf : continuous f) :
is_compact (range f) | begin
cases lt_or_gt_of_ne hc with hneg hpos,
exacts [hp.neg.compact_of_continuous' (neg_pos.mpr hneg) hf, hp.compact_of_continuous' hpos hf],
end | lemma | function.periodic.compact_of_continuous | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"is_compact",
"topological_space"
] | A continuous, periodic function has compact range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.bounded_of_continuous [pseudo_metric_space α] {f : ℝ → α} {c : ℝ}
(hp : periodic f c) (hc : c ≠ 0) (hf : continuous f) :
bounded (range f) | (hp.compact_of_continuous hc hf).bounded | lemma | function.periodic.bounded_of_continuous | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"continuous",
"pseudo_metric_space"
] | A continuous, periodic function is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_coe_cofinite : tendsto (coe : ℤ → ℝ) cofinite (cocompact ℝ) | begin
refine tendsto_cocompact_of_tendsto_dist_comp_at_top (0 : ℝ) _,
simp only [filter.tendsto_at_top, eventually_cofinite, not_le, ← mem_ball],
change ∀ r : ℝ, (coe ⁻¹' (ball (0 : ℝ) r)).finite,
simp [real.ball_eq_Ioo, set.finite_Ioo],
end | lemma | int.tendsto_coe_cofinite | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"filter.tendsto_at_top",
"finite",
"real.ball_eq_Ioo",
"set.finite_Ioo",
"tendsto_cocompact_of_tendsto_dist_comp_at_top"
] | Under the coercion from `ℤ` to `ℝ`, inverse images of compact sets are finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_zmultiples_hom_cofinite {a : ℝ} (ha : a ≠ 0) :
tendsto (zmultiples_hom ℝ a) cofinite (cocompact ℝ) | begin
convert (tendsto_cocompact_mul_right₀ ha).comp int.tendsto_coe_cofinite,
ext n,
simp,
end | lemma | int.tendsto_zmultiples_hom_cofinite | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"int.tendsto_coe_cofinite",
"zmultiples_hom"
] | For nonzero `a`, the "multiples of `a`" map `zmultiples_hom` from `ℤ` to `ℝ` is discrete, i.e.
inverse images of compact sets are finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_zmultiples_subtype_cofinite (a : ℝ) :
tendsto (zmultiples a).subtype cofinite (cocompact ℝ) | begin
rcases eq_or_ne a 0 with rfl | ha,
{ rw add_subgroup.zmultiples_zero_eq_bot,
intros K hK,
rw [filter.mem_map, mem_cofinite],
apply set.to_finite },
intros K hK,
have H := int.tendsto_zmultiples_hom_cofinite ha hK,
simp only [filter.mem_map, mem_cofinite, ← preimage_compl] at ⊢ H,
rw [← (zm... | lemma | add_subgroup.tendsto_zmultiples_subtype_cofinite | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"eq_or_ne",
"filter.mem_map",
"int.tendsto_zmultiples_hom_cofinite",
"set.to_finite",
"zmultiples_hom"
] | The subgroup "multiples of `a`" (`zmultiples a`) is a discrete subgroup of `ℝ`, i.e. its
intersection with compact sets is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.subgroup_dense_of_no_min {G : add_subgroup ℝ} {g₀ : ℝ} (g₀_in : g₀ ∈ G) (g₀_ne : g₀ ≠ 0)
(H' : ¬ ∃ a : ℝ, is_least {g : ℝ | g ∈ G ∧ 0 < g} a) :
dense (G : set ℝ) | begin
let G_pos := {g : ℝ | g ∈ G ∧ 0 < g},
push_neg at H',
intros x,
suffices : ∀ ε > (0 : ℝ), ∃ g ∈ G, |x - g| < ε,
by simpa only [real.mem_closure_iff, abs_sub_comm],
intros ε ε_pos,
obtain ⟨g₁, g₁_in, g₁_pos⟩ : ∃ g₁ : ℝ, g₁ ∈ G ∧ 0 < g₁,
{ cases lt_or_gt_of_ne g₀_ne with Hg₀ Hg₀,
{ exact ⟨-g₀,... | lemma | real.subgroup_dense_of_no_min | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"abs_of_nonneg",
"abs_sub_comm",
"add_subgroup",
"add_subgroup.int_mul_mem",
"dense",
"is_glb",
"is_glb_cInf",
"is_least",
"real.mem_closure_iff"
] | Given a nontrivial subgroup `G ⊆ ℝ`, if `G ∩ ℝ_{>0}` has no minimum then `G` is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.subgroup_dense_or_cyclic (G : add_subgroup ℝ) :
dense (G : set ℝ) ∨ ∃ a : ℝ, G = add_subgroup.closure {a} | begin
cases add_subgroup.bot_or_exists_ne_zero G with H H,
{ right,
use 0,
rw [H, add_subgroup.closure_singleton_zero] },
{ let G_pos := {g : ℝ | g ∈ G ∧ 0 < g},
by_cases H' : ∃ a, is_least G_pos a,
{ right,
rcases H' with ⟨a, ha⟩,
exact ⟨a, add_subgroup.cyclic_of_min ha⟩ },
{ left... | lemma | real.subgroup_dense_or_cyclic | topology.instances | src/topology/instances/real.lean | [
"topology.metric_space.basic",
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.algebra.ring.basic",
"topology.algebra.star",
"topology.algebra.order.field",
"ring_theory.subring.basic",
"group_theory.archimedean",
"algebra.order.group.bounds",
"algebra.periodic",... | [
"add_subgroup",
"add_subgroup.cyclic_of_min",
"dense",
"is_least",
"real.subgroup_dense_of_no_min"
] | Subgroups of `ℝ` are either dense or cyclic. See `real.subgroup_dense_of_no_min` and
`subgroup_cyclic_of_min` for more precise statements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_real_smul {G} [add_monoid_hom_class G E F] (f : G) (hf : continuous f) (c : ℝ) (x : E) :
f (c • x) = c • f x | suffices (λ c : ℝ, f (c • x)) = λ c : ℝ, c • f x, from _root_.congr_fun this c,
rat.dense_embedding_coe_real.dense.equalizer
(hf.comp $ continuous_id.smul continuous_const)
(continuous_id.smul continuous_const)
(funext $ λ r, map_rat_cast_smul f ℝ ℝ r x) | lemma | map_real_smul | topology.instances | src/topology/instances/real_vector_space.lean | [
"topology.algebra.module.basic",
"topology.instances.rat"
] | [
"add_monoid_hom_class",
"continuous",
"continuous_const",
"map_rat_cast_smul"
] | A continuous additive map between two vector spaces over `ℝ` is `ℝ`-linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_real_linear_map (f : E →+ F) (hf : continuous f) : E →L[ℝ] F | ⟨{ to_fun := f, map_add' := f.map_add, map_smul' := map_real_smul f hf }, hf⟩ | def | add_monoid_hom.to_real_linear_map | topology.instances | src/topology/instances/real_vector_space.lean | [
"topology.algebra.module.basic",
"topology.instances.rat"
] | [
"continuous",
"map_real_smul"
] | Reinterpret a continuous additive homomorphism between two real vector spaces
as a continuous real-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_real_linear_map (f : E →+ F) (hf : continuous f) :
⇑(f.to_real_linear_map hf) = f | rfl | lemma | add_monoid_hom.coe_to_real_linear_map | topology.instances | src/topology/instances/real_vector_space.lean | [
"topology.algebra.module.basic",
"topology.instances.rat"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv.to_real_linear_equiv (e : E ≃+ F) (h₁ : continuous e)
(h₂ : continuous e.symm) : E ≃L[ℝ] F | { .. e,
.. e.to_add_monoid_hom.to_real_linear_map h₁ } | def | add_equiv.to_real_linear_equiv | topology.instances | src/topology/instances/real_vector_space.lean | [
"topology.algebra.module.basic",
"topology.instances.rat"
] | [
"continuous"
] | Reinterpret a continuous additive equivalence between two real vector spaces
as a continuous real-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.is_scalar_tower [t2_space E] {A : Type*} [topological_space A]
[ring A] [algebra ℝ A] [module A E] [has_continuous_smul ℝ A]
[has_continuous_smul A E] : is_scalar_tower ℝ A E | ⟨λ r x y, map_real_smul ((smul_add_hom A E).flip y) (continuous_id.smul continuous_const) r x⟩ | instance | real.is_scalar_tower | topology.instances | src/topology/instances/real_vector_space.lean | [
"topology.algebra.module.basic",
"topology.instances.rat"
] | [
"algebra",
"continuous_const",
"has_continuous_smul",
"is_scalar_tower",
"map_real_smul",
"module",
"ring",
"smul_add_hom",
"t2_space",
"topological_space"
] | A topological group carries at most one structure of a topological `ℝ`-module, so for any
topological `ℝ`-algebra `A` (e.g. `A = ℂ`) and any topological group that is both a topological
`ℝ`-module and a topological `A`-module, these structures agree. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_sign_of_pos {a : α} (h : 0 < a) : continuous_at sign a | begin
refine (continuous_at_const : continuous_at (λ x, (1 : sign_type)) a).congr _,
rw [filter.eventually_eq, eventually_nhds_iff],
exact ⟨{x | 0 < x}, λ x hx, (sign_pos hx).symm, is_open_lt' 0, h⟩
end | lemma | continuous_at_sign_of_pos | topology.instances | src/topology/instances/sign.lean | [
"data.sign",
"topology.order.basic"
] | [
"continuous_at",
"continuous_at_const",
"eventually_nhds_iff",
"filter.eventually_eq",
"is_open_lt'",
"sign",
"sign_pos",
"sign_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_sign_of_neg {a : α} (h : a < 0) : continuous_at sign a | begin
refine (continuous_at_const : continuous_at (λ x, (-1 : sign_type)) a).congr _,
rw [filter.eventually_eq, eventually_nhds_iff],
exact ⟨{x | x < 0}, λ x hx, (sign_neg hx).symm, is_open_gt' 0, h⟩
end | lemma | continuous_at_sign_of_neg | topology.instances | src/topology/instances/sign.lean | [
"data.sign",
"topology.order.basic"
] | [
"continuous_at",
"continuous_at_const",
"eventually_nhds_iff",
"filter.eventually_eq",
"is_open_gt'",
"sign",
"sign_neg",
"sign_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_sign_of_ne_zero {a : α} (h : a ≠ 0) : continuous_at sign a | begin
rcases h.lt_or_lt with h_neg|h_pos,
{ exact continuous_at_sign_of_neg h_neg },
{ exact continuous_at_sign_of_pos h_pos }
end | lemma | continuous_at_sign_of_ne_zero | topology.instances | src/topology/instances/sign.lean | [
"data.sign",
"topology.order.basic"
] | [
"continuous_at",
"continuous_at_sign_of_neg",
"continuous_at_sign_of_pos",
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_def (x : tsze R M) : nhds x = (nhds x.fst).prod (nhds x.snd) | by cases x; exact nhds_prod_eq | lemma | triv_sq_zero_ext.nhds_def | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"nhds",
"nhds_def",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inl [has_zero M] (x : R) : nhds (inl x : tsze R M) = (nhds x).prod (nhds 0) | nhds_def _ | lemma | triv_sq_zero_ext.nhds_inl | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"nhds",
"nhds_def",
"nhds_inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inr [has_zero R] (m : M) : nhds (inr m : tsze R M) = (nhds 0).prod (nhds m) | nhds_def _ | lemma | triv_sq_zero_ext.nhds_inr | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"nhds",
"nhds_def",
"nhds_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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