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 |
|---|---|---|---|---|---|---|---|---|---|---|
continuous_proj_Icc : continuous (proj_Icc a b h) | (continuous_const.max $ continuous_const.min continuous_id).subtype_mk _ | lemma | continuous_proj_Icc | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_proj_Icc : quotient_map (proj_Icc a b h) | quotient_map_iff.2 ⟨proj_Icc_surjective h, λ s,
⟨λ hs, hs.preimage continuous_proj_Icc,
λ hs, ⟨_, hs, by { ext, simp }⟩⟩⟩ | lemma | quotient_map_proj_Icc | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous_proj_Icc",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Icc_extend_iff {f : Icc a b → β} :
continuous (Icc_extend h f) ↔ continuous f | quotient_map_proj_Icc.continuous_iff.symm | lemma | continuous_Icc_extend_iff | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.Icc_extend {f : γ → Icc a b → β} {g : γ → α}
(hf : continuous ↿f) (hg : continuous g) : continuous (λ a, Icc_extend h (f a) (g a)) | hf.comp $ continuous_id.prod_mk $ continuous_proj_Icc.comp hg | lemma | continuous.Icc_extend | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous"
] | See Note [continuity lemma statement]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.Icc_extend' {f : Icc a b → β} (hf : continuous f) : continuous (Icc_extend h f) | hf.comp continuous_proj_Icc | lemma | continuous.Icc_extend' | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous",
"continuous_proj_Icc"
] | A useful special case of `continuous.Icc_extend`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.Icc_extend {x : γ} (f : γ → Icc a b → β) {g : γ → α}
(hf : continuous_at ↿f (x, proj_Icc a b h (g x))) (hg : continuous_at g x) :
continuous_at (λ a, Icc_extend h (f a) (g a)) x | show continuous_at (↿f ∘ λ x, (x, proj_Icc a b h (g x))) x, from
continuous_at.comp hf $ continuous_at_id.prod $ continuous_proj_Icc.continuous_at.comp hg | lemma | continuous_at.Icc_extend | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"continuous_at",
"continuous_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_connected_component_mem_nhds :
ord_connected_component s a ∈ 𝓝 a ↔ s ∈ 𝓝 a | begin
refine ⟨λ h, mem_of_superset h ord_connected_component_subset, λ h, _⟩,
rcases exists_Icc_mem_subset_of_mem_nhds h with ⟨b, c, ha, ha', hs⟩,
exact mem_of_superset ha' (subset_ord_connected_component ha hs)
end | lemma | set.ord_connected_component_mem_nhds | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"exists_Icc_mem_subset_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_section_ord_separating_set_mem_nhds_within_Ici (hd : disjoint s (closure t))
(ha : a ∈ s) :
(ord_connected_section $ ord_separating_set s t)ᶜ ∈ 𝓝[≥] a | begin
have hmem : tᶜ ∈ 𝓝[≥] a,
{ refine mem_nhds_within_of_mem_nhds _,
rw [← mem_interior_iff_mem_nhds, interior_compl],
exact disjoint_left.1 hd ha },
rcases exists_Icc_mem_subset_of_mem_nhds_within_Ici hmem with ⟨b, hab, hmem', hsub⟩,
by_cases H : disjoint (Icc a b) (ord_connected_section $ ord_separ... | lemma | set.compl_section_ord_separating_set_mem_nhds_within_Ici | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"Ico_mem_nhds_within_Ici",
"closure",
"disjoint",
"exists_Icc_mem_subset_of_mem_nhds_within_Ici",
"interior_compl",
"lt_of_not_le",
"mem_interior_iff_mem_nhds",
"mem_nhds_within_of_mem_nhds",
"ne_of_mem_of_not_mem",
"not_forall",
"not_not",
"set.disjoint_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_section_ord_separating_set_mem_nhds_within_Iic (hd : disjoint s (closure t))
(ha : a ∈ s) : (ord_connected_section $ ord_separating_set s t)ᶜ ∈ 𝓝[≤] a | have hd' : disjoint (of_dual ⁻¹' s) (closure $ of_dual ⁻¹' t) := hd,
have ha' : to_dual a ∈ of_dual ⁻¹' s := ha,
by simpa only [dual_ord_separating_set, dual_ord_connected_section]
using compl_section_ord_separating_set_mem_nhds_within_Ici hd' ha' | lemma | set.compl_section_ord_separating_set_mem_nhds_within_Iic | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"closure",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_section_ord_separating_set_mem_nhds (hd : disjoint s (closure t)) (ha : a ∈ s) :
(ord_connected_section $ ord_separating_set s t)ᶜ ∈ 𝓝 a | begin
rw [← nhds_left_sup_nhds_right, mem_sup],
exact ⟨compl_section_ord_separating_set_mem_nhds_within_Iic hd ha,
compl_section_ord_separating_set_mem_nhds_within_Ici hd ha⟩
end | lemma | set.compl_section_ord_separating_set_mem_nhds | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"closure",
"disjoint",
"nhds_left_sup_nhds_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_t5_nhd_mem_nhds_set (hd : disjoint s (closure t)) : ord_t5_nhd s t ∈ 𝓝ˢ s | bUnion_mem_nhds_set $ λ x hx, ord_connected_component_mem_nhds.2 $
inter_mem (by { rw [← mem_interior_iff_mem_nhds, interior_compl], exact disjoint_left.1 hd hx })
(compl_section_ord_separating_set_mem_nhds hd hx) | lemma | set.ord_t5_nhd_mem_nhds_set | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"bUnion_mem_nhds_set",
"closure",
"disjoint",
"interior_compl",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology.t5_space : t5_space X | ⟨λ s t h₁ h₂, filter.disjoint_iff.2 ⟨ord_t5_nhd s t, ord_t5_nhd_mem_nhds_set h₂, ord_t5_nhd t s,
ord_t5_nhd_mem_nhds_set h₁.symm, disjoint_ord_t5_nhd⟩⟩ | instance | order_topology.t5_space | topology.algebra.order | src/topology/algebra/order/t5.lean | [
"topology.order.basic",
"data.set.intervals.ord_connected_component"
] | [
"t5_space"
] | A linear order with order topology is a completely normal Hausdorff topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_upper_lower_closure (α : Type*) [topological_space α] [preorder α] : Prop | (is_upper_set_closure : ∀ s : set α, is_upper_set s → is_upper_set (closure s))
(is_lower_set_closure : ∀ s : set α, is_lower_set s → is_lower_set (closure s))
(is_open_upper_closure : ∀ s : set α, is_open s → is_open (upper_closure s : set α))
(is_open_lower_closure : ∀ s : set α, is_open s → is_open (lower_closure s ... | class | has_upper_lower_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure",
"is_lower_set",
"is_open",
"is_upper_set",
"lower_closure",
"topological_space",
"upper_closure"
] | Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_group.to_has_upper_lower_closure [ordered_comm_group α]
[has_continuous_const_smul α α] : has_upper_lower_closure α | { is_upper_set_closure := λ s h x y hxy hx, closure_mono (h.smul_subset $ one_le_div'.2 hxy) $
by { rw closure_smul, exact ⟨x, hx, div_mul_cancel' _ _⟩ },
is_lower_set_closure := λ s h x y hxy hx, closure_mono (h.smul_subset $ div_le_one'.2 hxy) $
by { rw closure_smul, exact ⟨x, hx, div_mul_cancel' _ _⟩ },
... | instance | ordered_comm_group.to_has_upper_lower_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure_mono",
"closure_smul",
"div_mul_cancel'",
"has_continuous_const_smul",
"has_upper_lower_closure",
"ordered_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bounds_closure (s : set α) :
upper_bounds (closure s : set α) = upper_bounds s | ext $ λ a, by simp_rw [mem_upper_bounds_iff_subset_Iic, is_closed_Iic.closure_subset_iff] | lemma | upper_bounds_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure",
"mem_upper_bounds_iff_subset_Iic",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_bounds_closure (s : set α) :
lower_bounds (closure s : set α) = lower_bounds s | ext $ λ a, by simp_rw [mem_lower_bounds_iff_subset_Ici, is_closed_Ici.closure_subset_iff] | lemma | lower_bounds_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure",
"lower_bounds",
"mem_lower_bounds_iff_subset_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_closure : bdd_above (closure s) ↔ bdd_above s | by simp_rw [bdd_above, upper_bounds_closure] | lemma | bdd_above_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"bdd_above",
"closure",
"upper_bounds_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_closure : bdd_below (closure s) ↔ bdd_below s | by simp_rw [bdd_below, lower_bounds_closure] | lemma | bdd_below_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"bdd_below",
"closure",
"lower_bounds_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.closure : is_upper_set s → is_upper_set (closure s) | has_upper_lower_closure.is_upper_set_closure _ | lemma | is_upper_set.closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.closure : is_lower_set s → is_lower_set (closure s) | has_upper_lower_closure.is_lower_set_closure _ | lemma | is_lower_set.closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"closure",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.upper_closure : is_open s → is_open (upper_closure s : set α) | has_upper_lower_closure.is_open_upper_closure _ | lemma | is_open.upper_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"is_open",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.lower_closure : is_open s → is_open (lower_closure s : set α) | has_upper_lower_closure.is_open_lower_closure _ | lemma | is_open.lower_closure | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"is_open",
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.interior (h : is_upper_set s) : is_upper_set (interior s) | by { rw [←is_lower_set_compl, ←closure_compl], exact h.compl.closure } | lemma | is_upper_set.interior | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"interior",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.interior (h : is_lower_set s) : is_lower_set (interior s) | h.to_dual.interior | lemma | is_lower_set.interior | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"interior",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.interior (h : s.ord_connected) : (interior s).ord_connected | begin
rw [←h.upper_closure_inter_lower_closure, interior_inter],
exact (upper_closure s).upper.interior.ord_connected.inter
(lower_closure s).lower.interior.ord_connected,
end | lemma | set.ord_connected.interior | topology.algebra.order | src/topology/algebra/order/upper_lower.lean | [
"algebra.order.upper_lower",
"topology.algebra.group.basic",
"topology.order.basic"
] | [
"interior",
"interior_inter",
"lower_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_semiring [topological_space α] [non_unital_non_assoc_semiring α]
extends has_continuous_add α, has_continuous_mul α : Prop | class | topological_semiring | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"has_continuous_add",
"has_continuous_mul",
"non_unital_non_assoc_semiring",
"topological_space"
] | a topological semiring is a semiring `R` where addition and multiplication are continuous.
We allow for non-unital and non-associative semirings as well.
The `topological_semiring` class should *only* be instantiated in the presence of a
`non_unital_non_assoc_semiring` instance; if there is an instance of `non_unital_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_ring [topological_space α] [non_unital_non_assoc_ring α]
extends topological_semiring α, has_continuous_neg α : Prop | class | topological_ring | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"has_continuous_neg",
"non_unital_non_assoc_ring",
"topological_semiring",
"topological_space"
] | A topological ring is a ring `R` where addition, multiplication and negation are continuous.
If `R` is a (unital) ring, then continuity of negation can be derived from continuity of
multiplication as it is multiplication with `-1`. (See
`topological_semiring.has_continuous_neg_of_mul` and
`topological_semiring.to_topo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_semiring.has_continuous_neg_of_mul [topological_space α] [non_assoc_ring α]
[has_continuous_mul α] : has_continuous_neg α | { continuous_neg :=
by simpa using (continuous_const.mul continuous_id : continuous (λ x : α, (-1) * x)) } | lemma | topological_semiring.has_continuous_neg_of_mul | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"continuous",
"continuous_id",
"has_continuous_mul",
"has_continuous_neg",
"non_assoc_ring",
"topological_space"
] | If `R` is a ring with a continuous multiplication, then negation is continuous as well since it
is just multiplication with `-1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_semiring.to_topological_ring [topological_space α] [non_assoc_ring α]
(h : topological_semiring α) : topological_ring α | { ..h,
..(by { haveI := h.to_has_continuous_mul,
exact topological_semiring.has_continuous_neg_of_mul } : has_continuous_neg α) } | lemma | topological_semiring.to_topological_ring | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"has_continuous_neg",
"non_assoc_ring",
"topological_ring",
"topological_semiring",
"topological_semiring.has_continuous_neg_of_mul",
"topological_space"
] | If `R` is a ring which is a topological semiring, then it is automatically a topological
ring. This exists so that one can place a topological ring structure on `R` without explicitly
proving `continuous_neg`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_ring.to_topological_add_group [non_unital_non_assoc_ring α]
[topological_space α] [topological_ring α] : topological_add_group α | { ..topological_ring.to_topological_semiring.to_has_continuous_add,
..topological_ring.to_has_continuous_neg } | instance | topological_ring.to_topological_add_group | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"non_unital_non_assoc_ring",
"topological_add_group",
"topological_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.topological_semiring [topological_space α]
[non_unital_non_assoc_semiring α] [discrete_topology α] : topological_semiring α | ⟨⟩ | instance | discrete_topology.topological_semiring | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"discrete_topology",
"non_unital_non_assoc_semiring",
"topological_semiring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.topological_ring [topological_space α]
[non_unital_non_assoc_ring α] [discrete_topology α] : topological_ring α | ⟨⟩ | instance | discrete_topology.topological_ring | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"discrete_topology",
"non_unital_non_assoc_ring",
"topological_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.topological_closure (s : subsemiring α) : subsemiring α | { carrier := closure (s : set α),
..(s.to_submonoid.topological_closure),
..(s.to_add_submonoid.topological_closure ) } | def | subsemiring.topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"closure",
"subsemiring"
] | The (topological-space) closure of a subsemiring of a topological semiring is
itself a subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsemiring.topological_closure_coe (s : subsemiring α) :
(s.topological_closure : set α) = closure (s : set α) | rfl | lemma | subsemiring.topological_closure_coe | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"closure",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.le_topological_closure (s : subsemiring α) :
s ≤ s.topological_closure | subset_closure | lemma | subsemiring.le_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"subsemiring",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.is_closed_topological_closure (s : subsemiring α) :
is_closed (s.topological_closure : set α) | by convert is_closed_closure | lemma | subsemiring.is_closed_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"is_closed",
"is_closed_closure",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.topological_closure_minimal
(s : subsemiring α) {t : subsemiring α} (h : s ≤ t) (ht : is_closed (t : set α)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | subsemiring.topological_closure_minimal | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"closure_minimal",
"is_closed",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring.comm_semiring_topological_closure [t2_space α] (s : subsemiring α)
(hs : ∀ (x y : s), x * y = y * x) : comm_semiring s.topological_closure | { ..s.topological_closure.to_semiring,
..s.to_submonoid.comm_monoid_topological_closure hs } | def | subsemiring.comm_semiring_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"comm_semiring",
"subsemiring",
"t2_space"
] | If a subsemiring of a topological semiring is commutative, then so is its
topological closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_ring.of_add_group_of_nhds_zero [topological_add_group R]
(hmul : tendsto (uncurry ((*) : R → R → R)) ((𝓝 0) ×ᶠ (𝓝 0)) $ 𝓝 0)
(hmul_left : ∀ (x₀ : R), tendsto (λ x : R, x₀ * x) (𝓝 0) $ 𝓝 0)
(hmul_right : ∀ (x₀ : R), tendsto (λ x : R, x * x₀) (𝓝 0) $ 𝓝 0) : topological_ring R | begin
refine {..‹topological_add_group R›, ..},
have hleft : ∀ x₀ : R, 𝓝 x₀ = map (λ x, x₀ + x) (𝓝 0), by simp,
have hadd : tendsto (uncurry ((+) : R → R → R)) ((𝓝 0) ×ᶠ (𝓝 0)) (𝓝 0),
{ rw ← nhds_prod_eq,
convert continuous_add.tendsto ((0 : R), (0 : R)),
rw zero_add },
rw continuous_iff_continuo... | lemma | topological_ring.of_add_group_of_nhds_zero | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"continuous_at",
"continuous_iff_continuous_at",
"filter.prod_map_map_eq",
"nhds_prod_eq",
"topological_add_group",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_ring.of_nhds_zero
(hadd : tendsto (uncurry ((+) : R → R → R)) ((𝓝 0) ×ᶠ (𝓝 0)) $ 𝓝 0)
(hneg : tendsto (λ x, -x : R → R) (𝓝 0) (𝓝 0))
(hmul : tendsto (uncurry ((*) : R → R → R)) ((𝓝 0) ×ᶠ (𝓝 0)) $ 𝓝 0)
(hmul_left : ∀ (x₀ : R), tendsto (λ x : R, x₀ * x) (𝓝 0) $ 𝓝 0)
(hmul_right : ∀ (x₀ : R... | begin
haveI := topological_add_group.of_comm_of_nhds_zero hadd hneg hleft,
exact topological_ring.of_add_group_of_nhds_zero hmul hmul_left hmul_right
end | lemma | topological_ring.of_nhds_zero | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"topological_ring",
"topological_ring.of_add_group_of_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_continuous (x : α) : continuous (add_monoid_hom.mul_left x) | continuous_const.mul continuous_id | lemma | mul_left_continuous | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"add_monoid_hom.mul_left",
"continuous",
"continuous_id"
] | In a topological semiring, the left-multiplication `add_monoid_hom` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_continuous (x : α) : continuous (add_monoid_hom.mul_right x) | continuous_id.mul continuous_const | lemma | mul_right_continuous | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"add_monoid_hom.mul_right",
"continuous",
"continuous_const"
] | In a topological semiring, the right-multiplication `add_monoid_hom` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subring.topological_closure (S : subring α) : subring α | { carrier := closure (S : set α),
..S.to_submonoid.topological_closure,
..S.to_add_subgroup.topological_closure } | def | subring.topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"closure",
"subring"
] | The (topological-space) closure of a subring of a topological ring is
itself a subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subring.le_topological_closure (s : subring α) :
s ≤ s.topological_closure | subset_closure | lemma | subring.le_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"subring",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring.is_closed_topological_closure (s : subring α) :
is_closed (s.topological_closure : set α) | by convert is_closed_closure | lemma | subring.is_closed_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"is_closed",
"is_closed_closure",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring.topological_closure_minimal
(s : subring α) {t : subring α} (h : s ≤ t) (ht : is_closed (t : set α)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | subring.topological_closure_minimal | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"closure_minimal",
"is_closed",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring.comm_ring_topological_closure [t2_space α] (s : subring α)
(hs : ∀ (x y : s), x * y = y * x) : comm_ring s.topological_closure | { ..s.topological_closure.to_ring,
..s.to_submonoid.comm_monoid_topological_closure hs } | def | subring.comm_ring_topological_closure | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"comm_ring",
"subring",
"t2_space"
] | If a subring of a topological ring is commutative, then so is its topological closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_topology (α : Type u) [ring α]
extends topological_space α, topological_ring α : Type u | structure | ring_topology | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"ring",
"topological_ring",
"topological_space"
] | A ring topology on a ring `α` is a topology for which addition, negation and multiplication
are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited {α : Type u} [ring α] : inhabited (ring_topology α) | ⟨{to_topological_space := ⊤,
continuous_add := continuous_top,
continuous_mul := continuous_top,
continuous_neg := continuous_top}⟩ | instance | ring_topology.inhabited | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"continuous_mul",
"continuous_top",
"ring",
"ring_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext' {f g : ring_topology α} (h : f.is_open = g.is_open) : f = g | by { ext : 2, exact h } | lemma | ring_topology.ext' | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"ring_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
def_Inf (S : set (ring_topology α)) : ring_topology α | let Inf_S' := Inf (to_topological_space '' S) in
{ to_topological_space := Inf_S',
continuous_add :=
begin
apply continuous_Inf_rng.2,
rintros _ ⟨⟨t, tr⟩, haS, rfl⟩, resetI,
have h := continuous_Inf_dom (set.mem_image_of_mem to_topological_space haS) continuous_id,
have h_continuous_id := @con... | def | ring_topology.def_Inf | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"continuous.comp",
"continuous.prod_map",
"continuous_Inf_dom",
"continuous_id",
"continuous_mul",
"ring_topology",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced {α β : Type*} [t : topological_space α] [ring β] (f : α → β) :
ring_topology β | Inf {b : ring_topology β | (topological_space.coinduced f t) ≤ b.to_topological_space} | def | ring_topology.coinduced | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"ring",
"ring_topology",
"topological_space",
"topological_space.coinduced"
] | Given `f : α → β` and a topology on `α`, the coinduced ring topology on `β` is the finest
topology such that `f` is continuous and `β` is a topological ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coinduced_continuous {α β : Type*} [t : topological_space α] [ring β] (f : α → β) :
cont t (coinduced f).to_topological_space f | begin
rw continuous_iff_coinduced_le,
refine le_Inf _,
rintros _ ⟨t', ht', rfl⟩,
exact ht',
end | lemma | ring_topology.coinduced_continuous | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"cont",
"continuous_iff_coinduced_le",
"le_Inf",
"ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_group_topology (t : ring_topology α) : add_group_topology α | { to_topological_space := t.to_topological_space,
to_topological_add_group := @topological_ring.to_topological_add_group _ _ t.to_topological_space
t.to_topological_ring } | def | ring_topology.to_add_group_topology | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"add_group_topology",
"ring_topology",
"topological_ring.to_topological_add_group"
] | The forgetful functor from ring topologies on `a` to additive group topologies on `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_group_topology.order_embedding : order_embedding (ring_topology α)
(add_group_topology α) | order_embedding.of_map_le_iff to_add_group_topology $ λ _ _, iff.rfl | def | ring_topology.to_add_group_topology.order_embedding | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"add_group_topology",
"order_embedding",
"order_embedding.of_map_le_iff",
"ring_topology"
] | The order embedding from ring topologies on `a` to additive group topologies on `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absolute_value.comp {R S T : Type*} [semiring T] [semiring R] [ordered_semiring S]
(v : absolute_value R S) {f : T →+* R} (hf : function.injective f) :
absolute_value T S | { to_fun := v ∘ f,
map_mul' := by simp only [function.comp_app, map_mul, eq_self_iff_true, forall_const],
nonneg' := by simp only [v.nonneg, forall_const],
eq_zero' := by simp only [map_eq_zero_iff f hf, v.eq_zero, forall_const, iff_self],
add_le' := by simp only [function.comp_app, map_add, v.add_le, forall_co... | def | absolute_value.comp | topology.algebra.ring | src/topology/algebra/ring/basic.lean | [
"algebra.ring.prod",
"ring_theory.subring.basic",
"topology.algebra.group.basic"
] | [
"absolute_value",
"forall_const",
"map_mul",
"ordered_semiring",
"semiring"
] | Construct an absolute value on a semiring `T` from an absolute value on a semiring `R`
and an injective ring homomorphism `f : T →+* R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.closure (I : ideal R) : ideal R | { carrier := closure I,
smul_mem' := λ c x hx, map_mem_closure (mul_left_continuous _) hx $ λ a, I.mul_mem_left c,
..(add_submonoid.topological_closure I.to_add_submonoid) } | def | ideal.closure | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"closure",
"ideal",
"map_mem_closure",
"mul_left_continuous"
] | The closure of an ideal in a topological ring as an ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.coe_closure (I : ideal R) : (I.closure : set R) = closure I | rfl | lemma | ideal.coe_closure | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"closure",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.closure_eq_of_is_closed (I : ideal R) [hI : is_closed (I : set R)] :
I.closure = I | set_like.ext' hI.closure_eq | lemma | ideal.closure_eq_of_is_closed | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"ideal",
"is_closed",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_ring_quotient_topology : topological_space (R ⧸ N) | quotient.topological_space | instance | topological_ring_quotient_topology | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ring.is_open_map_coe : is_open_map (mk N) | begin
intros s s_op,
change is_open (mk N ⁻¹' (mk N '' s)),
rw quotient_ring_saturate,
exact is_open_Union (λ ⟨n, _⟩, is_open_map_add_left n s s_op)
end | lemma | quotient_ring.is_open_map_coe | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"is_open",
"is_open_Union",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ring.quotient_map_coe_coe : quotient_map (λ p : R × R, (mk N p.1, mk N p.2)) | is_open_map.to_quotient_map
((quotient_ring.is_open_map_coe N).prod (quotient_ring.is_open_map_coe N))
((continuous_quot_mk.comp continuous_fst).prod_mk (continuous_quot_mk.comp continuous_snd))
(by rintro ⟨⟨x⟩, ⟨y⟩⟩; exact ⟨(x, y), rfl⟩) | lemma | quotient_ring.quotient_map_coe_coe | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"continuous_fst",
"continuous_snd",
"is_open_map.to_quotient_map",
"quotient_map",
"quotient_ring.is_open_map_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_ring_quotient : topological_ring (R ⧸ N) | topological_semiring.to_topological_ring
{ continuous_add :=
have cont : continuous (mk N ∘ (λ (p : R × R), p.fst + p.snd)) :=
continuous_quot_mk.comp continuous_add,
(quotient_map.continuous_iff (quotient_ring.quotient_map_coe_coe N)).mpr cont,
continuous_mul :=
have cont : continuous (mk N ∘ (λ (p... | instance | topological_ring_quotient | topology.algebra.ring | src/topology/algebra/ring/ideal.lean | [
"topology.algebra.ring.basic",
"ring_theory.ideal.quotient"
] | [
"cont",
"continuous",
"continuous_mul",
"quotient_map.continuous_iff",
"quotient_ring.quotient_map_coe_coe",
"topological_ring",
"topological_semiring.to_topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology (α : Type*) | (cobounded [] : filter α)
(le_cofinite [] : cobounded ≤ cofinite) | class | bornology | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"filter"
] | A **bornology** on a type `α` is a filter of cobounded sets which contains the cofinite filter.
Such spaces are equivalently specified by their bounded sets, see `bornology.of_bounded`
and `bornology.ext_iff_is_bounded` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bornology.of_bounded {α : Type*} (B : set (set α))
(empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ : set α, s₂ ⊆ s₁ → s₂ ∈ B)
(union_mem : ∀ s₁ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (singleton_mem : ∀ x, {x} ∈ B) :
bornology α | { cobounded :=
{ sets := {s : set α | sᶜ ∈ B},
univ_sets := by rwa ←compl_univ at empty_mem,
sets_of_superset := λ x y hx hy, subset_mem xᶜ hx yᶜ (compl_subset_compl.mpr hy),
inter_sets := λ x y hx hy, by simpa [compl_inter] using union_mem xᶜ hx yᶜ hy, },
le_cofinite :=
begin
rw le_cofinite_iff_c... | def | bornology.of_bounded | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"compl_compl"
] | A constructor for bornologies by specifying the bounded sets,
and showing that they satisfy the appropriate conditions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bornology.of_bounded' {α : Type*} (B : set (set α))
(empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ : set α, s₂ ⊆ s₁ → s₂ ∈ B)
(union_mem : ∀ s₁ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (sUnion_univ : ⋃₀ B = univ) :
bornology α | bornology.of_bounded B empty_mem subset_mem union_mem $ λ x,
begin
rw sUnion_eq_univ_iff at sUnion_univ,
rcases sUnion_univ x with ⟨s, hs, hxs⟩,
exact subset_mem s hs {x} (singleton_subset_iff.mpr hxs)
end | def | bornology.of_bounded' | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"bornology.of_bounded"
] | A constructor for bornologies by specifying the bounded sets,
and showing that they satisfy the appropriate conditions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cobounded (s : set α) : Prop | s ∈ cobounded α | def | bornology.is_cobounded | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | `is_cobounded` is the predicate that `s` is in the filter of cobounded sets in the ambient
bornology on `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded (s : set α) : Prop | is_cobounded sᶜ | def | bornology.is_bounded | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | `is_bounded` is the predicate that `s` is bounded relative to the ambient bornology on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cobounded_def {s : set α} : is_cobounded s ↔ s ∈ cobounded α | iff.rfl | lemma | bornology.is_cobounded_def | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_def {s : set α} : is_bounded s ↔ sᶜ ∈ cobounded α | iff.rfl | lemma | bornology.is_bounded_def | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_compl_iff : is_bounded sᶜ ↔ is_cobounded s | by rw [is_bounded_def, is_cobounded_def, compl_compl] | lemma | bornology.is_bounded_compl_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_compl_iff : is_cobounded sᶜ ↔ is_bounded s | iff.rfl | lemma | bornology.is_cobounded_compl_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_empty : is_bounded (∅ : set α) | by { rw [is_bounded_def, compl_empty], exact univ_mem} | lemma | bornology.is_bounded_empty | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_singleton : is_bounded ({x} : set α) | by {rw [is_bounded_def], exact le_cofinite _ (finite_singleton x).compl_mem_cofinite} | lemma | bornology.is_bounded_singleton | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_univ : is_cobounded (univ : set α) | univ_mem | lemma | bornology.is_cobounded_univ | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_inter : is_cobounded (s ∩ t) ↔ is_cobounded s ∧ is_cobounded t | inter_mem_iff | lemma | bornology.is_cobounded_inter | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded.inter (hs : is_cobounded s) (ht : is_cobounded t) : is_cobounded (s ∩ t) | is_cobounded_inter.2 ⟨hs, ht⟩ | lemma | bornology.is_cobounded.inter | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_union : is_bounded (s ∪ t) ↔ is_bounded s ∧ is_bounded t | by simp only [← is_cobounded_compl_iff, compl_union, is_cobounded_inter] | lemma | bornology.is_bounded_union | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.union (hs : is_bounded s) (ht : is_bounded t) : is_bounded (s ∪ t) | is_bounded_union.2 ⟨hs, ht⟩ | lemma | bornology.is_bounded.union | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded.superset (hs : is_cobounded s) (ht : s ⊆ t) : is_cobounded t | mem_of_superset hs ht | lemma | bornology.is_cobounded.superset | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.subset (ht : is_bounded t) (hs : s ⊆ t) : is_bounded s | ht.superset (compl_subset_compl.mpr hs) | lemma | bornology.is_bounded.subset | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_bounded_univ : (⋃₀ {s : set α | is_bounded s}) = univ | sUnion_eq_univ_iff.2 $ λ a, ⟨{a}, is_bounded_singleton, mem_singleton a⟩ | lemma | bornology.sUnion_bounded_univ | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_cobounded_le_iff [bornology β] {f : α → β} :
(cobounded β).comap f ≤ cobounded α ↔ ∀ ⦃s⦄, is_bounded s → is_bounded (f '' s) | begin
refine ⟨λ h s hs, _, λ h t ht,
⟨(f '' tᶜ)ᶜ, h $ is_cobounded.compl ht, compl_subset_comm.1 $ subset_preimage_image _ _⟩⟩,
obtain ⟨t, ht, hts⟩ := h hs.compl,
rw [subset_compl_comm, ←preimage_compl] at hts,
exact (is_cobounded.compl ht).subset ((image_subset f hts).trans $ image_preimage_subset _ _),
en... | lemma | bornology.comap_cobounded_le_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff' {t t' : bornology α} :
t = t' ↔ ∀ s, (@cobounded α t).sets s ↔ (@cobounded α t').sets s | (ext_iff _ _).trans filter.ext_iff | lemma | bornology.ext_iff' | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"filter.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff_is_bounded {t t' : bornology α} :
t = t' ↔ ∀ s, @is_bounded α t s ↔ @is_bounded α t' s | ⟨λ h s, h ▸ iff.rfl, λ h, by { ext, simpa only [is_bounded_def, compl_compl] using h sᶜ, }⟩ | lemma | bornology.ext_iff_is_bounded | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_of_bounded_iff (B : set (set α)) {empty_mem subset_mem union_mem sUnion_univ} :
@is_cobounded _ (of_bounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ sᶜ ∈ B | iff.rfl | lemma | bornology.is_cobounded_of_bounded_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_of_bounded_iff (B : set (set α)) {empty_mem subset_mem union_mem sUnion_univ} :
@is_bounded _ (of_bounded B empty_mem subset_mem union_mem sUnion_univ) s ↔ s ∈ B | by rw [is_bounded_def, ←filter.mem_sets, of_bounded_cobounded_sets, set.mem_set_of_eq, compl_compl] | lemma | bornology.is_bounded_of_bounded_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_bInter {s : set ι} {f : ι → set α} (hs : s.finite) :
is_cobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, is_cobounded (f i) | bInter_mem hs | lemma | bornology.is_cobounded_bInter | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_bInter_finset (s : finset ι) {f : ι → set α} :
is_cobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, is_cobounded (f i) | bInter_finset_mem s | lemma | bornology.is_cobounded_bInter_finset | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_Inter [finite ι] {f : ι → set α} :
is_cobounded (⋂ i, f i) ↔ ∀ i, is_cobounded (f i) | Inter_mem | lemma | bornology.is_cobounded_Inter | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_sInter {S : set (set α)} (hs : S.finite) :
is_cobounded (⋂₀ S) ↔ ∀ s ∈ S, is_cobounded s | sInter_mem hs | lemma | bornology.is_cobounded_sInter | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bUnion {s : set ι} {f : ι → set α} (hs : s.finite) :
is_bounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, is_bounded (f i) | by simp only [← is_cobounded_compl_iff, compl_Union, is_cobounded_bInter hs] | lemma | bornology.is_bounded_bUnion | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bUnion_finset (s : finset ι) {f : ι → set α} :
is_bounded (⋃ i ∈ s, f i) ↔ ∀ i ∈ s, is_bounded (f i) | is_bounded_bUnion s.finite_to_set | lemma | bornology.is_bounded_bUnion_finset | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_sUnion {S : set (set α)} (hs : S.finite) :
is_bounded (⋃₀ S) ↔ (∀ s ∈ S, is_bounded s) | by rw [sUnion_eq_bUnion, is_bounded_bUnion hs] | lemma | bornology.is_bounded_sUnion | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_Union [finite ι] {s : ι → set α} :
is_bounded (⋃ i, s i) ↔ ∀ i, is_bounded (s i) | by rw [← sUnion_range, is_bounded_sUnion (finite_range s), forall_range_iff] | lemma | bornology.is_bounded_Union | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_bounded [bornology α] {s : set α} (hs : s.finite) : is_bounded s | bornology.le_cofinite α hs.compl_mem_cofinite | lemma | set.finite.is_bounded | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology.cofinite : bornology α | { cobounded := cofinite,
le_cofinite := le_rfl } | def | bornology.cofinite | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"le_rfl"
] | The cofinite filter as a bornology | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_space (α : Type*) [bornology α] : Prop | (bounded_univ : bornology.is_bounded (univ : set α)) | class | bounded_space | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bornology",
"bornology.is_bounded"
] | A space with a `bornology` is a **bounded space** if `set.univ : set α` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_univ : is_bounded (univ : set α) ↔ bounded_space α | ⟨λ h, ⟨h⟩, λ h, h.1⟩ | lemma | bornology.is_bounded_univ | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bounded_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cobounded_eq_bot_iff : cobounded α = ⊥ ↔ bounded_space α | by rw [← is_bounded_univ, is_bounded_def, compl_univ, empty_mem_iff_bot] | lemma | bornology.cobounded_eq_bot_iff | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"bounded_space"
] | 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.