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
value |
|---|---|---|---|---|---|---|---|---|---|---|
eval_continuous (y : E) : continuous (λ x : weak_dual 𝕜 E, x y) | continuous_pi_iff.mp coe_fn_continuous y | lemma | weak_dual.eval_continuous | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_eval [topological_space α] {g : α → weak_dual 𝕜 E}
(h : ∀ y, continuous (λ a, (g a) y)) : continuous g | continuous_induced_rng.2 (continuous_pi_iff.mpr h) | lemma | weak_dual.continuous_of_continuous_eval | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"topological_space",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weak_space (𝕜 E) [comm_semiring 𝕜] [topological_space 𝕜] [has_continuous_add 𝕜]
[has_continuous_const_smul 𝕜 𝕜] [add_comm_monoid E] [module 𝕜 E] [topological_space E] | weak_bilin (top_dual_pairing 𝕜 E).flip | def | weak_space | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"add_comm_monoid",
"comm_semiring",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"top_dual_pairing",
"topological_space",
"weak_bilin"
] | The weak topology is the topology coarsest topology on `E` such that all
functionals `λ x, top_dual_pairing 𝕜 E v x` are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : E →L[𝕜] F) :
weak_space 𝕜 E →L[𝕜] weak_space 𝕜 F | { cont := weak_bilin.continuous_of_continuous_eval _ (λ l, weak_bilin.eval_continuous _ (l ∘L f)),
..f } | def | weak_space.map | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"cont",
"weak_bilin.continuous_of_continuous_eval",
"weak_bilin.eval_continuous",
"weak_space"
] | A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_apply (f : E →L[𝕜] F) (x : E) : weak_space.map f x = f x | rfl | lemma | weak_space.map_apply | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"weak_space.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map (f : E →L[𝕜] F) : (weak_space.map f : E → F) = f | rfl | lemma | weak_space.coe_map | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"weak_space.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_forall_eval_tendsto_top_dual_pairing
{l : filter α} {f : α → weak_dual 𝕜 E} {x : weak_dual 𝕜 E} :
tendsto f l (𝓝 x) ↔
∀ y, tendsto (λ i, top_dual_pairing 𝕜 E (f i) y) l (𝓝 (top_dual_pairing 𝕜 E x y)) | weak_bilin.tendsto_iff_forall_eval_tendsto _ continuous_linear_map.coe_injective | theorem | tendsto_iff_forall_eval_tendsto_top_dual_pairing | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous_linear_map.coe_injective",
"filter",
"top_dual_pairing",
"weak_bilin.tendsto_iff_forall_eval_tendsto",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_basis (I : ideal R) : submodules_ring_basis (λ n : ℕ, (I^n • ⊤ : ideal R)) | { inter := begin
suffices : ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j, by simpa,
intros i j,
exact ⟨max i j, pow_le_pow (le_max_left i j), pow_le_pow (le_max_right i j)⟩
end,
left_mul := begin
suffices : ∀ (a : R) (i : ℕ), ∃ j : ℕ, a • I ^ j ≤ I ^ i, by simpa,
intros r n,
use n,
rin... | lemma | ideal.adic_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"left_mul",
"pow_le_pow",
"submodules_ring_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_filter_basis (I : ideal R) | I.adic_basis.to_ring_subgroups_basis.to_ring_filter_basis | def | ideal.ring_filter_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"ring_filter_basis"
] | The adic ring filter basis associated to an ideal `I` is made of powers of `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adic_topology (I : ideal R) : topological_space R | (adic_basis I).topology | def | ideal.adic_topology | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"topological_space"
] | The adic topology associated to an ideal `I`. This topology admits powers of `I` as a basis of
neighborhoods of zero. It is compatible with the ring structure and is non-archimedean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean (I : ideal R) : @nonarchimedean_ring R _ I.adic_topology | I.adic_basis.to_ring_subgroups_basis.nonarchimedean | lemma | ideal.nonarchimedean | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"nonarchimedean_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_zero_adic (I : ideal R) :
has_basis (@nhds R I.adic_topology (0 : R)) (λ n : ℕ, true) (λ n, ((I^n : ideal R) : set R)) | ⟨begin
intros U,
rw I.ring_filter_basis.to_add_group_filter_basis.nhds_zero_has_basis.mem_iff,
split,
{ rintros ⟨-, ⟨i, rfl⟩, h⟩,
replace h : ↑(I ^ i) ⊆ U := by simpa using h,
use [i, trivial, h] },
{ rintros ⟨i, -, h⟩,
exact ⟨(I^i : ideal R), ⟨i, by simp⟩, h⟩ }
end⟩ | lemma | ideal.has_basis_nhds_zero_adic | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"nhds"
] | For the `I`-adic topology, the neighborhoods of zero has basis given by the powers of `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_adic (I : ideal R) (x : R) :
has_basis (@nhds R I.adic_topology x) (λ n : ℕ, true) (λ n, (λ y, x + y) '' (I^n : ideal R)) | begin
letI := I.adic_topology,
have := I.has_basis_nhds_zero_adic.map (λ y, x + y),
rwa map_add_left_nhds_zero x at this
end | lemma | ideal.has_basis_nhds_adic | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_module_basis :
I.ring_filter_basis.submodules_basis (λ n : ℕ, (I^n) • (⊤ : submodule R M)) | { inter := λ i j, ⟨max i j, le_inf_iff.mpr ⟨smul_mono_left $ pow_le_pow (le_max_left i j),
smul_mono_left $ pow_le_pow (le_max_right i j)⟩⟩,
smul := λ m i, ⟨(I^i • ⊤ : ideal R), ⟨i, rfl⟩,
λ a a_in, by { replace a_in : a ∈ I^i := by simpa [(I^i).mul_top] us... | lemma | ideal.adic_module_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"pow_le_pow",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_module_topology : topological_space M | @module_filter_basis.topology R M _ I.adic_basis.topology _ _
(I.ring_filter_basis.module_filter_basis (I.adic_module_basis M)) | def | ideal.adic_module_topology | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"module_filter_basis.topology",
"topological_space"
] | The topology on a `R`-module `M` associated to an ideal `M`. Submodules $I^n M$,
written `I^n • ⊤` form a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_add_subgroup (n : ℕ) : @open_add_subgroup R _ I.adic_topology | { is_open' := begin
letI := I.adic_topology,
convert (I.adic_basis.to_ring_subgroups_basis.open_add_subgroup n).is_open,
simp
end,
..(I^n).to_add_subgroup} | def | ideal.open_add_subgroup | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"is_open",
"open_add_subgroup"
] | The elements of the basis of neighborhoods of zero for the `I`-adic topology
on a `R`-module `M`, seen as open additive subgroups of `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adic [H : topological_space R] (J : ideal R) : Prop | H = J.adic_topology | def | is_adic | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"topological_space"
] | Given a topology on a ring `R` and an ideal `J`, `is_adic J` means the topology is the
`J`-adic one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adic_iff [top : topological_space R] [topological_ring R] {J : ideal R} :
is_adic J ↔ (∀ n : ℕ, is_open ((J^n : ideal R) : set R)) ∧
(∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J^n : ideal R) : set R) ⊆ s) | begin
split,
{ intro H,
change _ = _ at H,
rw H,
letI := J.adic_topology,
split,
{ intro n,
exact (J.open_add_subgroup n).is_open' },
{ intros s hs,
simpa using J.has_basis_nhds_zero_adic.mem_iff.mp hs } },
{ rintro ⟨H₁, H₂⟩,
apply topological_add_group.ext,
{ apply @to... | lemma | is_adic_iff | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"ideal.adic_basis",
"is_adic",
"is_open",
"mem_nhds_iff",
"ring_subgroups_basis.to_ring_filter_basis",
"topological_ring",
"topological_ring.to_topological_add_group",
"topological_space"
] | A topological ring is `J`-adic if and only if it admits the powers of `J` as a basis of
open neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_ideal_adic_pow {J : ideal R} (h : is_adic J) {n : ℕ} (hn : 0 < n) :
is_adic (J^n) | begin
rw is_adic_iff at h ⊢,
split,
{ intro m, rw ← pow_mul, apply h.left },
{ intros V hV,
cases h.right V hV with m hm,
use m,
refine set.subset.trans _ hm,
cases n, { exfalso, exact nat.not_succ_le_zero 0 hn },
rw [← pow_mul, nat.succ_mul],
apply ideal.pow_le_pow,
apply nat.le_add... | lemma | is_ideal_adic_pow | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"ideal",
"ideal.pow_le_pow",
"is_adic",
"is_adic_iff",
"pow_mul",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bot_adic_iff {A : Type*} [comm_ring A] [topological_space A] [topological_ring A] :
is_adic (⊥ : ideal A) ↔ discrete_topology A | begin
rw is_adic_iff,
split,
{ rintro ⟨h, h'⟩,
rw discrete_topology_iff_open_singleton_zero,
simpa using h 1 },
{ introsI,
split,
{ simp, },
{ intros U U_nhds,
use 1,
simp [mem_of_mem_nhds U_nhds] } },
end | lemma | is_bot_adic_iff | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"comm_ring",
"discrete_topology",
"ideal",
"is_adic",
"is_adic_iff",
"mem_of_mem_nhds",
"topological_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_ideal (R : Type*) [comm_ring R] | (I : ideal R) | class | with_ideal | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"comm_ring",
"ideal"
] | The ring `R` is equipped with a preferred ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space_module (M : Type*) [add_comm_group M] [module R M] :
topological_space M | (I : ideal R).adic_module_topology M | def | with_ideal.topological_space_module | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/adic_topology.lean | [
"ring_theory.ideal.operations",
"topology.algebra.nonarchimedean.bases",
"topology.uniform_space.completion",
"topology.algebra.uniform_ring"
] | [
"add_comm_group",
"ideal",
"module",
"topological_space"
] | The adic topology on a `R` module coming from the ideal `with_ideal.I`.
This cannot be an instance because `R` cannot be inferred from `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_subgroups_basis {A ι : Type*} [ring A] (B : ι → add_subgroup A) : Prop | (inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j)
(mul : ∀ i, ∃ j, (B j : set A) * B j ⊆ B i)
(left_mul : ∀ x : A, ∀ i, ∃ j, (B j : set A) ⊆ (λ y : A, x*y) ⁻¹' (B i))
(right_mul : ∀ x : A, ∀ i, ∃ j, (B j : set A) ⊆ (λ y : A, y*x) ⁻¹' (B i)) | structure | ring_subgroups_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"add_subgroup",
"left_mul",
"right_mul",
"ring"
] | A family of additive subgroups on a ring `A` is a subgroups basis if it satisfies some
axioms ensuring there is a topology on `A` which is compatible with the ring structure and
admits this family as a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_comm {A ι : Type*} [comm_ring A] (B : ι → add_subgroup A)
(inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j)
(mul : ∀ i, ∃ j, (B j : set A) * B j ⊆ B i)
(left_mul : ∀ x : A, ∀ i, ∃ j, (B j : set A) ⊆ (λ y : A, x*y) ⁻¹' (B i)) :
ring_subgroups_basis B | { inter := inter,
mul := mul,
left_mul := left_mul,
right_mul := begin
intros x i,
cases left_mul x i with j hj,
use j,
simpa [mul_comm] using hj
end } | lemma | ring_subgroups_basis.of_comm | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"add_subgroup",
"comm_ring",
"left_mul",
"mul_comm",
"right_mul",
"ring_subgroups_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_filter_basis [nonempty ι] {B : ι → add_subgroup A}
(hB : ring_subgroups_basis B) : ring_filter_basis A | { sets := {U | ∃ i, U = B i},
nonempty := by { inhabit ι, exact ⟨B default, default, rfl⟩ },
inter_sets := begin
rintros _ _ ⟨i, rfl⟩ ⟨j, rfl⟩,
cases hB.inter i j with k hk,
use [B k, k, rfl, hk]
end,
zero' := by { rintros _ ⟨i, rfl⟩, exact (B i).zero_mem },
add' := begin
rintros _ ⟨i, rfl⟩,
... | def | ring_subgroups_basis.to_ring_filter_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"add_subgroup",
"ring_filter_basis",
"ring_subgroups_basis"
] | Every subgroups basis on a ring leads to a ring filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_add_group_filter_basis_iff {V : set A} :
V ∈ hB.to_ring_filter_basis.to_add_group_filter_basis ↔ ∃ i, V = B i | iff.rfl | lemma | ring_subgroups_basis.mem_add_group_filter_basis_iff | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_add_group_filter_basis (i) :
(B i : set A) ∈ hB.to_ring_filter_basis.to_add_group_filter_basis | ⟨i, rfl⟩ | lemma | ring_subgroups_basis.mem_add_group_filter_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology : topological_space A | hB.to_ring_filter_basis.to_add_group_filter_basis.topology | def | ring_subgroups_basis.topology | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"topological_space"
] | The topology defined from a subgroups basis, admitting the given subgroups as a basis
of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_zero : has_basis (@nhds A hB.topology 0) (λ _, true) (λ i, B i) | ⟨begin
intros s,
rw hB.to_ring_filter_basis.to_add_group_filter_basis.nhds_zero_has_basis.mem_iff,
split,
{ rintro ⟨-, ⟨i, rfl⟩, hi⟩,
exact ⟨i, trivial, hi⟩ },
{ rintro ⟨i, -, hi⟩,
exact ⟨B i, ⟨i, rfl⟩, hi⟩ }
end⟩ | lemma | ring_subgroups_basis.has_basis_nhds_zero | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds (a : A) :
has_basis (@nhds A hB.topology a) (λ _, true) (λ i, {b | b - a ∈ B i}) | ⟨begin
intros s,
rw (hB.to_ring_filter_basis.to_add_group_filter_basis.nhds_has_basis a).mem_iff,
simp only [exists_prop, exists_true_left],
split,
{ rintro ⟨-, ⟨i, rfl⟩, hi⟩,
use i,
convert hi,
ext b,
split,
{ intros h,
use [b - a, h],
abel },
{ rintros ⟨c, hc, rfl⟩,
... | lemma | ring_subgroups_basis.has_basis_nhds | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"exists_prop",
"exists_true_left",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_add_subgroup (i : ι) : @open_add_subgroup A _ hB.topology | { is_open' := begin
letI := hB.topology,
rw is_open_iff_mem_nhds,
intros a a_in,
rw (hB.has_basis_nhds a).mem_iff,
use [i, trivial],
rintros b b_in,
simpa using (B i).add_mem a_in b_in
end,
..B i } | def | ring_subgroups_basis.open_add_subgroup | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"is_open_iff_mem_nhds",
"open_add_subgroup"
] | Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean : @nonarchimedean_ring A _ hB.topology | begin
letI := hB.topology,
constructor,
intros U hU,
obtain ⟨i, -, hi : (B i : set A) ⊆ U⟩ := hB.has_basis_nhds_zero.mem_iff.mp hU,
exact ⟨hB.open_add_subgroup i, hi⟩
end | lemma | ring_subgroups_basis.nonarchimedean | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"nonarchimedean_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodules_ring_basis (B : ι → submodule R A) : Prop | (inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j)
(left_mul : ∀ (a : A) i, ∃ j, a • B j ≤ B i)
(mul : ∀ i, ∃ j, (B j : set A) * B j ⊆ B i) | structure | submodules_ring_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"left_mul",
"submodule"
] | A family of submodules in a commutative `R`-algebra `A` is a submodules basis if it satisfies
some axioms ensuring there is a topology on `A` which is compatible with the ring structure and
admits this family as a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ring_subgroups_basis (hB : submodules_ring_basis B) :
ring_subgroups_basis (λ i, (B i).to_add_subgroup) | begin
apply ring_subgroups_basis.of_comm (λ i, (B i).to_add_subgroup) hB.inter hB.mul,
intros a i,
rcases hB.left_mul a i with ⟨j, hj⟩,
use j,
rintros b (b_in : b ∈ B j),
exact hj ⟨b, b_in, rfl⟩
end | lemma | submodules_ring_basis.to_ring_subgroups_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"ring_subgroups_basis",
"ring_subgroups_basis.of_comm",
"submodules_ring_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology [nonempty ι] (hB : submodules_ring_basis B) : topological_space A | hB.to_ring_subgroups_basis.topology | def | submodules_ring_basis.topology | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"submodules_ring_basis",
"topological_space"
] | The topology associated to a basis of submodules in an algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodules_basis [topological_space R]
(B : ι → submodule R M) : Prop | (inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j)
(smul : ∀ (m : M) (i : ι), ∀ᶠ a in 𝓝 (0 : R), a • m ∈ B i) | structure | submodules_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"submodule",
"topological_space"
] | A family of submodules in an `R`-module `M` is a submodules basis if it satisfies
some axioms ensuring there is a topology on `M` which is compatible with the module structure and
admits this family as a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_filter_basis : module_filter_basis R M | { sets := {U | ∃ i, U = B i},
nonempty := by { inhabit ι, exact ⟨B default, default, rfl⟩ },
inter_sets := begin
rintros _ _ ⟨i, rfl⟩ ⟨j, rfl⟩,
cases hB.inter i j with k hk,
use [B k, k, rfl, hk]
end,
zero' := by { rintros _ ⟨i, rfl⟩, exact (B i).zero_mem },
add' := begin
rintros _ ⟨i, rfl⟩,
... | def | submodules_basis.to_module_filter_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"module_filter_basis"
] | The image of a submodules basis is a module filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topology : topological_space M | hB.to_module_filter_basis.to_add_group_filter_basis.topology | def | submodules_basis.topology | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"topological_space"
] | The topology associated to a basis of submodules in a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_add_subgroup (i : ι) : @open_add_subgroup M _ hB.topology | { is_open' := begin
letI := hB.topology,
rw is_open_iff_mem_nhds,
intros a a_in,
rw (hB.to_module_filter_basis.to_add_group_filter_basis.nhds_has_basis a).mem_iff,
use [B i, i, rfl],
rintros - ⟨b, b_in, rfl⟩,
exact (B i).add_mem a_in b_in
end,
..(B i).to_add_subgroup } | def | submodules_basis.open_add_subgroup | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"is_open_iff_mem_nhds",
"open_add_subgroup"
] | Given a submodules basis, the basis elements as open additive subgroups in the associated
topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean (hB : submodules_basis B) : @nonarchimedean_add_group M _ hB.topology | begin
letI := hB.topology,
constructor,
intros U hU,
obtain ⟨-, ⟨i, rfl⟩, hi : (B i : set M) ⊆ U⟩ :=
hB.to_module_filter_basis.to_add_group_filter_basis.nhds_zero_has_basis.mem_iff.mp hU,
exact ⟨hB.open_add_subgroup i, hi⟩
end | lemma | submodules_basis.nonarchimedean | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"nonarchimedean_add_group",
"submodules_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodules_ring_basis.to_submodules_basis : submodules_basis B | { inter := hB.inter,
smul := hsmul } | lemma | submodules_ring_basis.to_submodules_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"submodules_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_filter_basis.submodules_basis (BR : ring_filter_basis R)
(B : ι → submodule R M) : Prop | (inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j)
(smul : ∀ (m : M) (i : ι), ∃ U ∈ BR, U ⊆ (λ a, a • m) ⁻¹' B i) | structure | ring_filter_basis.submodules_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"ring_filter_basis",
"submodule"
] | Given a ring filter basis on a commutative ring `R`, define a compatibility condition
on a family of submodules of a `R`-module `M`. This compatibility condition allows to get
a topological module structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_filter_basis.submodules_basis_is_basis (BR : ring_filter_basis R) {B : ι → submodule R M}
(hB : BR.submodules_basis B) : @submodules_basis ι R _ M _ _ BR.topology B | { inter := hB.inter,
smul := begin
letI := BR.topology,
intros m i,
rcases hB.smul m i with ⟨V, V_in, hV⟩,
exact mem_of_superset (BR.to_add_group_filter_basis.mem_nhds_zero V_in) hV
end } | lemma | ring_filter_basis.submodules_basis_is_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"ring_filter_basis",
"submodule",
"submodules_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_filter_basis.module_filter_basis [nonempty ι] (BR : ring_filter_basis R)
{B : ι → submodule R M} (hB : BR.submodules_basis B) :
@module_filter_basis R M _ BR.topology _ _ | @submodules_basis.to_module_filter_basis ι R _ M _ _ BR.topology _ _
(BR.submodules_basis_is_basis hB) | def | ring_filter_basis.module_filter_basis | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/bases.lean | [
"topology.algebra.nonarchimedean.basic",
"topology.algebra.filter_basis",
"algebra.module.submodule.pointwise"
] | [
"module_filter_basis",
"ring_filter_basis",
"submodule",
"submodules_basis.to_module_filter_basis"
] | The module filter basis associated to a ring filter basis and a compatible submodule basis.
This allows to build a topological module structure compatible with the given module structure
and the topology associated to the given ring filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_add_group (G : Type*)
[add_group G] [topological_space G] extends topological_add_group G : Prop | (is_nonarchimedean : ∀ U ∈ nhds (0 : G), ∃ V : open_add_subgroup G, (V : set G) ⊆ U) | class | nonarchimedean_add_group | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"add_group",
"nhds",
"open_add_subgroup",
"topological_add_group",
"topological_space"
] | An topological additive group is nonarchimedean if every neighborhood of 0
contains an open subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_group (G : Type*)
[group G] [topological_space G] extends topological_group G : Prop | (is_nonarchimedean : ∀ U ∈ nhds (1 : G), ∃ V : open_subgroup G, (V : set G) ⊆ U) | class | nonarchimedean_group | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"group",
"nhds",
"open_subgroup",
"topological_group",
"topological_space"
] | A topological group is nonarchimedean if every neighborhood of 1 contains an open subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_ring (R : Type*)
[ring R] [topological_space R] extends topological_ring R : Prop | (is_nonarchimedean : ∀ U ∈ nhds (0 : R), ∃ V : open_add_subgroup R, (V : set R) ⊆ U) | class | nonarchimedean_ring | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"nhds",
"open_add_subgroup",
"ring",
"topological_ring",
"topological_space"
] | An topological ring is nonarchimedean if its underlying topological additive
group is nonarchimedean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_ring.to_nonarchimedean_add_group
(R : Type*) [ring R] [topological_space R] [t: nonarchimedean_ring R] :
nonarchimedean_add_group R | {..t} | instance | nonarchimedean_ring.to_nonarchimedean_add_group | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"nonarchimedean_add_group",
"nonarchimedean_ring",
"ring",
"topological_space"
] | Every nonarchimedean ring is naturally a nonarchimedean additive group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_of_emb (f : G →* H) (emb : open_embedding f) : nonarchimedean_group H | { is_nonarchimedean := λ U hU, have h₁ : (f ⁻¹' U) ∈ nhds (1 : G), from
by {apply emb.continuous.tendsto, rwa f.map_one},
let ⟨V, hV⟩ := is_nonarchimedean (f ⁻¹' U) h₁ in
⟨{is_open' := emb.is_open_map _ V.is_open, ..subgroup.map f V},
set.image_subset_iff.2 hV⟩ } | lemma | nonarchimedean_group.nonarchimedean_of_emb | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"nhds",
"nonarchimedean_group",
"open_embedding",
"subgroup.map"
] | If a topological group embeds into a nonarchimedean group, then it is nonarchimedean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_subset {U} (hU : U ∈ nhds (1 : G × K)) :
∃ (V : open_subgroup G) (W : open_subgroup K), (V : set G) ×ˢ (W : set K) ⊆ U | begin
erw [nhds_prod_eq, filter.mem_prod_iff] at hU,
rcases hU with ⟨U₁, hU₁, U₂, hU₂, h⟩,
cases is_nonarchimedean _ hU₁ with V hV,
cases is_nonarchimedean _ hU₂ with W hW,
use V, use W,
rw set.prod_subset_iff,
intros x hX y hY,
exact set.subset.trans (set.prod_mono hV hW) h (set.mem_sep hX hY),
end | lemma | nonarchimedean_group.prod_subset | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"filter.mem_prod_iff",
"nhds",
"nhds_prod_eq",
"open_subgroup",
"set.mem_sep",
"set.prod_mono",
"set.prod_subset_iff",
"set.subset.trans"
] | An open neighborhood of the identity in the cartesian product of two nonarchimedean groups
contains the cartesian product of an open neighborhood in each group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_self_subset {U} (hU : U ∈ nhds (1 : G × G)) :
∃ (V : open_subgroup G), (V : set G) ×ˢ (V : set G) ⊆ U | let ⟨V, W, h⟩ := prod_subset hU in
⟨V ⊓ W, by {refine set.subset.trans (set.prod_mono _ _) ‹_›; simp}⟩ | lemma | nonarchimedean_group.prod_self_subset | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"nhds",
"open_subgroup",
"set.prod_mono",
"set.subset.trans"
] | An open neighborhood of the identity in the cartesian square of a nonarchimedean group
contains the cartesian square of an open neighborhood in the group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_mul_subset (U : open_add_subgroup R) (r : R) :
∃ V : open_add_subgroup R, r • (V : set R) ⊆ U | ⟨U.comap (add_monoid_hom.mul_left r) (continuous_mul_left r),
(U : set R).image_preimage_subset _⟩ | lemma | nonarchimedean_ring.left_mul_subset | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"add_monoid_hom.mul_left",
"continuous_mul_left",
"open_add_subgroup"
] | Given an open subgroup `U` and an element `r` of a nonarchimedean ring, there is an open
subgroup `V` such that `r • V` is contained in `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_subset (U : open_add_subgroup R) :
∃ V : open_add_subgroup R, (V : set R) * V ⊆ U | let ⟨V, H⟩ := prod_self_subset (is_open.mem_nhds (is_open.preimage continuous_mul U.is_open)
begin
simpa only [set.mem_preimage, set_like.mem_coe, prod.snd_zero, mul_zero] using U.zero_mem,
end) in
begin
use V,
rintros v ⟨a, b, ha, hb, hv⟩,
have hy := H (set.mk_mem_prod ha hb),
simp only [set.mem_preima... | lemma | nonarchimedean_ring.mul_subset | topology.algebra.nonarchimedean | src/topology/algebra/nonarchimedean/basic.lean | [
"group_theory.subgroup.basic",
"topology.algebra.open_subgroup",
"topology.algebra.ring.basic"
] | [
"continuous_mul",
"is_open.mem_nhds",
"is_open.preimage",
"mul_zero",
"open_add_subgroup",
"set.mem_preimage",
"set.mk_mem_prod",
"set_like.mem_coe"
] | An open subgroup of a nonarchimedean ring contains the square of another one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rat.dense_range_cast : dense_range (coe : ℚ → 𝕜) | dense_of_exists_between $ λ a b h, set.exists_range_iff.2 $ exists_rat_btwn h | lemma | rat.dense_range_cast | topology.algebra.order | src/topology/algebra/order/archimedean.lean | [
"topology.order.basic",
"algebra.order.archimedean"
] | [
"dense_of_exists_between",
"dense_range",
"exists_rat_btwn"
] | Rational numbers are dense in a linear ordered archimedean field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_Icc_space (α : Type*) [topological_space α] [preorder α] : Prop | (is_compact_Icc : ∀ {a b : α}, is_compact (Icc a b)) | class | compact_Icc_space | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"topological_space"
] | This typeclass says that all closed intervals in `α` are compact. This is true for all
conditionally complete linear orders with order topology and products (finite or infinite)
of such spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conditionally_complete_linear_order.to_compact_Icc_space
(α : Type*) [conditionally_complete_linear_order α] [topological_space α] [order_topology α] :
compact_Icc_space α | begin
refine ⟨λ a b, _⟩,
cases le_or_lt a b with hab hab, swap, { simp [hab] },
refine is_compact_iff_ultrafilter_le_nhds.2 (λ f hf, _),
contrapose! hf,
rw [le_principal_iff],
have hpt : ∀ x ∈ Icc a b, {x} ∉ f,
from λ x hx hxf, hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x)),
set s := {x ∈ Icc a ... | instance | conditionally_complete_linear_order.to_compact_Icc_space | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"Ioc_mem_nhds_within_Iic",
"bdd_above",
"compact_Icc_space",
"conditionally_complete_linear_order",
"is_lub",
"is_lub_cSup",
"mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset",
"mem_nhds_within_Iic_iff_exists_Ioc_subset'",
"mem_nhds_within_of_mem_nhds",
"order_topology",
"pure_le_nhds",
"topo... | A closed interval in a conditionally complete linear order is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.compact_Icc_space' {α β : Type*} [preorder β] [topological_space β]
[compact_Icc_space β] : compact_Icc_space (α → β) | pi.compact_Icc_space | instance | pi.compact_Icc_space' | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"compact_Icc_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_uIcc {α : Type*} [linear_order α] [topological_space α] [compact_Icc_space α]
{a b : α} : is_compact (uIcc a b) | is_compact_Icc | lemma | is_compact_uIcc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"compact_Icc_space",
"is_compact",
"topological_space"
] | An unordered closed interval is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space_of_complete_linear_order {α : Type*} [complete_linear_order α]
[topological_space α] [order_topology α] :
compact_space α | ⟨by simp only [← Icc_bot_top, is_compact_Icc]⟩ | instance | compact_space_of_complete_linear_order | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"compact_space",
"complete_linear_order",
"order_topology",
"topological_space"
] | A complete linear order is a compact space.
We do not register an instance for a `[compact_Icc_space α]` because this would only add instances
for products (indexed or not) of complete linear orders, and we have instances with higher priority
that cover these cases. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space_Icc (a b : α) : compact_space (Icc a b) | is_compact_iff_compact_space.mp is_compact_Icc | instance | compact_space_Icc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_least s x | begin
haveI : nonempty s := ne_s.to_subtype,
suffices : (s ∩ ⋂ x ∈ s, Iic x).nonempty,
from ⟨this.some, this.some_spec.1, mem_Inter₂.mp this.some_spec.2⟩,
rw bInter_eq_Inter,
by_contra H,
rw not_nonempty_iff_eq_empty at H,
rcases hs.elim_directed_family_closed (λ x : s, Iic ↑x) (λ x, is_closed_Iic) H
... | lemma | is_compact.exists_is_least | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"by_contra",
"is_closed_Iic",
"is_compact",
"is_least",
"is_total.directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_greatest s x | @is_compact.exists_is_least αᵒᵈ _ _ _ _ hs ne_s | lemma | is_compact.exists_is_greatest | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_compact.exists_is_least",
"is_greatest"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_glb s x | exists_imp_exists (λ x (hx : is_least s x), ⟨hx.1, hx.is_glb⟩) (hs.exists_is_least ne_s) | lemma | is_compact.exists_is_glb | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_glb",
"is_least"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_lub s x | @is_compact.exists_is_glb αᵒᵈ _ _ _ _ hs ne_s | lemma | is_compact.exists_is_lub | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_compact.exists_is_glb",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_forall_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃x∈s, ∀y∈s, f x ≤ f y | begin
rcases (hs.image_of_continuous_on hf).exists_is_least (ne_s.image f)
with ⟨_, ⟨x, hxs, rfl⟩, hx⟩,
exact ⟨x, hxs, ball_image_iff.1 hx⟩
end | lemma | is_compact.exists_forall_le | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_compact"
] | The **extreme value theorem**: a continuous function realizes its minimum on a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.exists_forall_ge :
∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
∃x∈s, ∀y∈s, f y ≤ f x | @is_compact.exists_forall_le αᵒᵈ _ _ _ _ _ | lemma | is_compact.exists_forall_ge | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_compact",
"is_compact.exists_forall_le"
] | The **extreme value theorem**: a continuous function realizes its maximum on a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.exists_forall_le' {s : set β} {f : β → α} (hf : continuous_on f s)
(hsc : is_closed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) :
∃ x ∈ s, ∀ y ∈ s, f x ≤ f y | begin
rcases (has_basis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩,
have hsub : insert x₀ (K ∩ s) ⊆ s, from insert_subset.2 ⟨h₀, inter_subset_right _ _⟩,
obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
((hK.inter_right hsc).insert x₀).exists_forall_le... | lemma | continuous_on.exists_forall_le' | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_closed"
] | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
larger than a value in its image away from compact sets, then it has a minimum on this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.exists_forall_ge' {s : set β} {f : β → α} (hf : continuous_on f s)
(hsc : is_closed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) :
∃ x ∈ s, ∀ y ∈ s, f y ≤ f x | @continuous_on.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc | lemma | continuous_on.exists_forall_ge' | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"continuous_on.exists_forall_le'",
"is_closed"
] | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
smaller than a value in its image away from compact sets, then it has a maximum on this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous.exists_forall_le' {f : β → α} (hf : continuous f) (x₀ : β)
(h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ (x : β), ∀ (y : β), f x ≤ f y | let ⟨x, _, hx⟩ := hf.continuous_on.exists_forall_le' is_closed_univ (mem_univ x₀)
(by rwa [principal_univ, inf_top_eq])
in ⟨x, λ y, hx y (mem_univ y)⟩ | lemma | continuous.exists_forall_le' | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"inf_top_eq",
"is_closed_univ"
] | The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
away from compact sets, then it has a global minimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous.exists_forall_ge' {f : β → α} (hf : continuous f) (x₀ : β)
(h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ (x : β), ∀ (y : β), f y ≤ f x | @continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h | lemma | continuous.exists_forall_ge' | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"continuous.exists_forall_le'"
] | The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
away from compact sets, then it has a global maximum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous.exists_forall_le [nonempty β] {f : β → α}
(hf : continuous f) (hlim : tendsto f (cocompact β) at_top) :
∃ x, ∀ y, f x ≤ f y | by { inhabit β, exact hf.exists_forall_le' default (hlim.eventually $ eventually_ge_at_top _) } | lemma | continuous.exists_forall_le | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous"
] | The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.exists_forall_ge [nonempty β] {f : β → α}
(hf : continuous f) (hlim : tendsto f (cocompact β) at_bot) :
∃ x, ∀ y, f y ≤ f x | @continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim | lemma | continuous.exists_forall_ge | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"continuous.exists_forall_le"
] | The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
compact sets, then it has a global maximum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
{f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
∃ (x : β), ∀ (y : β), f x ≤ f y | begin
obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
rw [mem_lower_bounds, forall_range_iff] at hx,
exact ⟨x, hx⟩,
end | lemma | continuous.exists_forall_le_of_has_compact_mul_support | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"has_compact_mul_support",
"mem_lower_bounds"
] | A continuous function with compact support has a global minimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
{f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
∃ (x : β), ∀ (y : β), f y ≤ f x | @continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h | lemma | continuous.exists_forall_ge_of_has_compact_mul_support | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"continuous.exists_forall_le_of_has_compact_mul_support",
"has_compact_mul_support"
] | A continuous function with compact support has a global maximum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.bdd_below [nonempty α] {s : set α} (hs : is_compact s) : bdd_below s | begin
cases s.eq_empty_or_nonempty,
{ rw h,
exact bdd_below_empty },
{ obtain ⟨a, ha, has⟩ := hs.exists_is_least h,
exact ⟨a, has⟩ },
end | lemma | is_compact.bdd_below | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_below",
"bdd_below_empty",
"is_compact"
] | A compact set is bounded below | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.bdd_above [nonempty α] {s : set α} (hs : is_compact s) : bdd_above s | @is_compact.bdd_below αᵒᵈ _ _ _ _ _ hs | lemma | is_compact.bdd_above | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_above",
"is_compact",
"is_compact.bdd_below"
] | A compact set is bounded above | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.bdd_below_image [nonempty α] {f : β → α} {K : set β}
(hK : is_compact K) (hf : continuous_on f K) : bdd_below (f '' K) | (hK.image_of_continuous_on hf).bdd_below | lemma | is_compact.bdd_below_image | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_below",
"continuous_on",
"is_compact"
] | A continuous function is bounded below on a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.bdd_above_image [nonempty α] {f : β → α} {K : set β}
(hK : is_compact K) (hf : continuous_on f K) : bdd_above (f '' K) | @is_compact.bdd_below_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf | lemma | is_compact.bdd_above_image | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_above",
"continuous_on",
"is_compact",
"is_compact.bdd_below_image"
] | A continuous function is bounded above on a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.bdd_below_range_of_has_compact_mul_support [has_one α] {f : β → α}
(hf : continuous f) (h : has_compact_mul_support f) : bdd_below (range f) | (h.is_compact_range hf).bdd_below | lemma | continuous.bdd_below_range_of_has_compact_mul_support | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_below",
"continuous",
"has_compact_mul_support"
] | A continuous function with compact support is bounded below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.bdd_above_range_of_has_compact_mul_support [has_one α]
{f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
bdd_above (range f) | @continuous.bdd_below_range_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ hf h | lemma | continuous.bdd_above_range_of_has_compact_mul_support | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_above",
"continuous",
"continuous.bdd_below_range_of_has_compact_mul_support",
"has_compact_mul_support"
] | A continuous function with compact support is bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.Sup_lt_iff_of_continuous {f : β → α}
{K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
Sup (f '' K) < y ↔ ∀ x ∈ K, f x < y | begin
refine ⟨λ h x hx, (le_cSup (hK.bdd_above_image hf) $ mem_image_of_mem f hx).trans_lt h, λ h, _⟩,
obtain ⟨x, hx, h2x⟩ := hK.exists_forall_ge h0K hf,
refine (cSup_le (h0K.image f) _).trans_lt (h x hx),
rintro _ ⟨x', hx', rfl⟩, exact h2x x' hx'
end | lemma | is_compact.Sup_lt_iff_of_continuous | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"cSup_le",
"continuous_on",
"is_compact",
"le_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.lt_Inf_iff_of_continuous {α β : Type*}
[conditionally_complete_linear_order α] [topological_space α]
[order_topology α] [topological_space β] {f : β → α}
{K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
y < Inf (f '' K) ↔ ∀ x ∈ K, y < f x | @is_compact.Sup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y | lemma | is_compact.lt_Inf_iff_of_continuous | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"conditionally_complete_linear_order",
"continuous_on",
"is_compact",
"is_compact.Sup_lt_iff_of_continuous",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Inf s ∈ s | let ⟨a, ha⟩ := hs.exists_is_least ne_s in
ha.Inf_mem | lemma | is_compact.Inf_mem | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s | @is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s | lemma | is_compact.Sup_mem | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_compact.Inf_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_glb s (Inf s) | is_glb_cInf ne_s hs.bdd_below | lemma | is_compact.is_glb_Inf | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_glb",
"is_glb_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_lub s (Sup s) | @is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s | lemma | is_compact.is_lub_Sup | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_compact.is_glb_Inf",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_least s (Inf s) | ⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩ | lemma | is_compact.is_least_Inf | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_least"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_greatest s (Sup s) | @is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s | lemma | is_compact.is_greatest_Sup | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"is_compact",
"is_compact.is_least_Inf",
"is_greatest"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y | let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
in ⟨x, hxs, hx.symm, λ y hy,
hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩ | lemma | is_compact.exists_Inf_image_eq_and_le | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"Inf_mem",
"bdd_below",
"cInf_le",
"continuous_on",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x | @is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf | lemma | is_compact.exists_Sup_image_eq_and_ge | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_compact",
"is_compact.exists_Inf_image_eq_and_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
{f : β → α} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x | let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩ | lemma | is_compact.exists_Inf_image_eq | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_Sup_image_eq :
∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
∃ x ∈ s, Sup (f '' s) = f x | @is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _ | lemma | is_compact.exists_Sup_image_eq | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"is_compact",
"is_compact.exists_Inf_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
s = Icc (Inf s) (Sup s) | eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed | lemma | eq_Icc_of_connected_compact | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"eq_Icc_cInf_cSup_of_connected_bdd_closed",
"is_compact",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.continuous_Sup {f : γ → β → α}
{K : set β} (hK : is_compact K) (hf : continuous ↿f) :
continuous (λ x, Sup (f x '' K)) | begin
rcases eq_empty_or_nonempty K with rfl|h0K,
{ simp_rw [image_empty], exact continuous_const },
rw [continuous_iff_continuous_at],
intro x,
obtain ⟨y, hyK, h2y, hy⟩ :=
hK.exists_Sup_image_eq_and_ge h0K
(show continuous (λ y, f x y), from hf.comp $ continuous.prod.mk x).continuous_on,
rw [cont... | lemma | is_compact.continuous_Sup | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"continuous.prod.mk",
"continuous_at",
"continuous_const",
"continuous_iff_continuous_at",
"continuous_on",
"generalized_tube_lemma",
"is_compact",
"is_compact_singleton",
"le_cSup",
"tendsto_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.continuous_Inf {f : γ → β → α}
{K : set β} (hK : is_compact K) (hf : continuous ↿f) :
continuous (λ x, Inf (f x '' K)) | @is_compact.continuous_Sup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf | lemma | is_compact.continuous_Inf | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous",
"is_compact",
"is_compact.continuous_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_Icc (hab : a ≤ b) (h : continuous_on f $ Icc a b) :
f '' Icc a b = Icc (Inf $ f '' Icc a b) (Sup $ f '' Icc a b) | eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, is_preconnected_Icc.image f h⟩
(is_compact_Icc.image_of_continuous_on h) | lemma | continuous_on.image_Icc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"eq_Icc_of_connected_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_uIcc_eq_Icc (h : continuous_on f $ [a, b]) :
f '' [a, b] = Icc (Inf (f '' [a, b])) (Sup (f '' [a, b])) | begin
cases le_total a b with h2 h2,
{ simp_rw [uIcc_of_le h2] at h ⊢, exact h.image_Icc h2 },
{ simp_rw [uIcc_of_ge h2] at h ⊢, exact h.image_Icc h2 },
end | lemma | continuous_on.image_uIcc_eq_Icc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_uIcc (h : continuous_on f $ [a, b]) :
f '' [a, b] = [Inf (f '' [a, b]), Sup (f '' [a, b])] | begin
refine h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm,
refine cInf_le_cSup _ _ (nonempty_uIcc.image _); rw h.image_uIcc_eq_Icc,
exacts [bdd_below_Icc, bdd_above_Icc]
end | lemma | continuous_on.image_uIcc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_above_Icc",
"bdd_below_Icc",
"cInf_le_cSup",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_image_Icc_le (h : continuous_on f $ Icc a b) (hc : c ∈ Icc a b) :
Inf (f '' (Icc a b)) ≤ f c | begin
rw h.image_Icc (nonempty_Icc.mp (set.nonempty_of_mem hc)),
exact cInf_le bdd_below_Icc (mem_Icc.mpr ⟨cInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
le_cSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩),
end | lemma | continuous_on.Inf_image_Icc_le | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_below_Icc",
"cInf_le",
"continuous_on",
"le_cSup",
"set.nonempty_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_Sup_image_Icc (h : continuous_on f $ Icc a b) (hc : c ∈ Icc a b) :
f c ≤ Sup (f '' (Icc a b)) | begin
rw h.image_Icc (nonempty_Icc.mp (set.nonempty_of_mem hc)),
exact le_cSup bdd_above_Icc (mem_Icc.mpr ⟨cInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
le_cSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩),
end | lemma | continuous_on.le_Sup_image_Icc | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"bdd_above_Icc",
"continuous_on",
"le_cSup",
"set.nonempty_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.