statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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coe_eq_coe {x y : α} :
(x : connected_components α) = y ↔ connected_component x = connected_component y | quotient.eq' | lemma | connected_components.coe_eq_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_components",
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_coe {x y : α} :
(x : connected_components α) ≠ y ↔ connected_component x ≠ connected_component y | not_congr coe_eq_coe | lemma | connected_components.coe_ne_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_coe' {x y : α} :
(x : connected_components α) = y ↔ x ∈ connected_component y | coe_eq_coe.trans connected_component_eq_iff_mem | lemma | connected_components.coe_eq_coe' | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_eq_iff_mem",
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_coe : surjective (coe : α → connected_components α) | surjective_quot_mk _ | lemma | connected_components.surjective_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"surjective_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_coe : quotient_map (coe : α → connected_components α) | quotient_map_quot_mk | lemma | connected_components.quotient_map_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"quotient_map",
"quotient_map_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : α → connected_components α) | quotient_map_coe.continuous | lemma | connected_components.continuous_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coe : range (coe : α → connected_components α)= univ | surjective_coe.range_eq | lemma | connected_components.range_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.image_eq_of_connected_component_eq (h : continuous f) (a b : α)
(hab : connected_component a = connected_component b) : f a = f b | singleton_eq_singleton_iff.1 $
h.image_connected_component_eq_singleton a ▸
h.image_connected_component_eq_singleton b ▸ hab ▸ rfl | lemma | continuous.image_eq_of_connected_component_eq | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.connected_components_lift (h : continuous f) :
connected_components α → β | λ x, quotient.lift_on' x f h.image_eq_of_connected_component_eq | def | continuous.connected_components_lift | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"continuous",
"quotient.lift_on'"
] | The lift to `connected_components α` of a continuous map from `α` to a totally disconnected space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.connected_components_lift_continuous (h : continuous f) :
continuous h.connected_components_lift | h.quotient_lift_on' h.image_eq_of_connected_component_eq | lemma | continuous.connected_components_lift_continuous | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.connected_components_lift_apply_coe (h : continuous f) (x : α) :
h.connected_components_lift x = f x | rfl | lemma | continuous.connected_components_lift_apply_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.connected_components_lift_comp_coe (h : continuous f) :
h.connected_components_lift ∘ coe = f | rfl | lemma | continuous.connected_components_lift_comp_coe | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_components_lift_unique' {β : Sort*} {g₁ g₂ : connected_components α → β}
(hg : g₁ ∘ (coe : α → connected_components α) = g₂ ∘ coe) :
g₁ = g₂ | connected_components.surjective_coe.injective_comp_right hg | lemma | connected_components_lift_unique' | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.connected_components_lift_unique (h : continuous f)
(g : connected_components α → β) (hg : g ∘ coe = f) : g = h.connected_components_lift | connected_components_lift_unique' $ hg.trans h.connected_components_lift_comp_coe.symm | lemma | continuous.connected_components_lift_unique | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"connected_components_lift_unique'",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_components_preimage_singleton {x : α} :
coe ⁻¹' ({x} : set (connected_components α)) = connected_component x | by { ext y, simp [connected_components.coe_eq_coe'] } | lemma | connected_components_preimage_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_components",
"connected_components.coe_eq_coe'"
] | The preimage of a singleton in `connected_components` is the connected component
of an element in the equivalence class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_components_preimage_image (U : set α) :
coe ⁻¹' (coe '' U : set (connected_components α)) = ⋃ x ∈ U, connected_component x | by simp only [connected_components_preimage_singleton, preimage_Union₂, image_eq_Union] | lemma | connected_components_preimage_image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_components",
"connected_components_preimage_singleton"
] | The preimage of the image of a set under the quotient map to `connected_components α`
is the union of the connected components of the elements in it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_components.totally_disconnected_space :
totally_disconnected_space (connected_components α) | begin
rw totally_disconnected_space_iff_connected_component_singleton,
refine connected_components.surjective_coe.forall.2 (λ x, _),
rw [← connected_components.quotient_map_coe.image_connected_component,
← connected_components_preimage_singleton,
image_preimage_eq _ connected_components.surjective_coe],
... | instance | connected_components.totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"connected_components.surjective_coe",
"connected_components_preimage_singleton",
"is_connected_connected_component",
"totally_disconnected_space",
"totally_disconnected_space_iff_connected_component_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.connected_components_map {β : Type*} [topological_space β] {f : α → β}
(h : continuous f) : connected_components α → connected_components β | continuous.connected_components_lift (continuous_quotient_mk.comp h) | def | continuous.connected_components_map | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_components",
"continuous",
"continuous.connected_components_lift",
"topological_space"
] | Functoriality of `connected_components` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.connected_components_map_continuous {β : Type*} [topological_space β] {f : α → β}
(h : continuous f) : continuous h.connected_components_map | continuous.connected_components_lift_continuous (continuous_quotient_mk.comp h) | lemma | continuous.connected_components_map_continuous | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous",
"continuous.connected_components_lift_continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.constant {Y : Type*} [topological_space Y] [discrete_topology Y]
{s : set α} (hs : is_preconnected s) {f : α → Y} (hf : continuous_on f s)
{x y : α} (hx : x ∈ s) (hy : y ∈ s) : f x = f y | (hs.image f hf).subsingleton (mem_image_of_mem f hx) (mem_image_of_mem f hy) | lemma | is_preconnected.constant | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"discrete_topology",
"is_preconnected",
"topological_space"
] | A preconnected set `s` has the property that every map to a
discrete space that is continuous on `s` is constant on `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_of_forall_constant {s : set α}
(hs : ∀ f : α → bool, continuous_on f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y) : is_preconnected s | begin
unfold is_preconnected,
by_contra',
rcases this with ⟨u, v, u_op, v_op, hsuv, ⟨x, x_in_s, x_in_u⟩, ⟨y, y_in_s, y_in_v⟩, H⟩,
rw [not_nonempty_iff_eq_empty] at H,
have hy : y ∉ u,
from λ y_in_u, eq_empty_iff_forall_not_mem.mp H y ⟨y_in_s, ⟨y_in_u, y_in_v⟩⟩,
have : continuous_on u.bool_indicator s,
... | lemma | is_preconnected_of_forall_constant | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"continuous_on_indicator_iff_clopen",
"is_preconnected"
] | If every map to `bool` (a discrete two-element space), that is
continuous on a set `s`, is constant on s, then s is preconnected | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preconnected_space.constant {Y : Type*} [topological_space Y] [discrete_topology Y]
(hp : preconnected_space α) {f : α → Y} (hf : continuous f) {x y : α} : f x = f y | is_preconnected.constant hp.is_preconnected_univ (continuous.continuous_on hf) trivial trivial | lemma | preconnected_space.constant | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous",
"continuous.continuous_on",
"discrete_topology",
"is_preconnected.constant",
"preconnected_space",
"topological_space"
] | A `preconnected_space` version of `is_preconnected.constant` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preconnected_space_of_forall_constant (hs : ∀ f : α → bool, continuous f → ∀ x y, f x = f y) :
preconnected_space α | ⟨is_preconnected_of_forall_constant
(λ f hf x hx y hy, hs f (continuous_iff_continuous_on_univ.mpr hf) x y)⟩ | lemma | preconnected_space_of_forall_constant | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous",
"preconnected_space"
] | A `preconnected_space` version of `is_preconnected_of_forall_constant` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.constant_of_maps_to [topological_space β]
{S : set α} (hS : is_preconnected S) {T : set β} [discrete_topology T] {f : α → β}
(hc : continuous_on f S) (hTm : maps_to f S T)
{x y : α} (hx : x ∈ S) (hy : y ∈ S) : f x = f y | begin
let F : S → T := (λ x:S, ⟨f x.val, hTm x.property⟩),
suffices : F ⟨x, hx⟩ = F ⟨y, hy⟩,
{ rw ←subtype.coe_inj at this, exact this },
exact (is_preconnected_iff_preconnected_space.mp hS).constant
(continuous_induced_rng.mpr $ continuous_on_iff_continuous_restrict.mp hc)
end | lemma | is_preconnected.constant_of_maps_to | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"discrete_topology",
"is_preconnected",
"topological_space"
] | Refinement of `is_preconnected.constant` only assuming the map factors through a
discrete subset of the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.topological_space {β : α → Type v} [t₂ : Πa, topological_space (β a)] :
topological_space (Πa, β a) | ⨅a, induced (λf, f a) (t₂ a) | instance | Pi.topological_space | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.topological_space [t : topological_space α] : topological_space (ulift.{v u} α) | t.induced ulift.down | instance | ulift.topological_space | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_mul : continuous (of_mul : α → additive α) | continuous_id | lemma | continuous_of_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_mul : continuous (to_mul : additive α → α) | continuous_id | lemma | continuous_to_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_add : continuous (of_add : α → multiplicative α) | continuous_id | lemma | continuous_of_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_add : continuous (to_add : multiplicative α → α) | continuous_id | lemma | continuous_to_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_of_mul : is_open_map (of_mul : α → additive α) | is_open_map.id | lemma | is_open_map_of_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"is_open_map",
"is_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_to_mul : is_open_map (to_mul : additive α → α) | is_open_map.id | lemma | is_open_map_to_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"is_open_map",
"is_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_of_add : is_open_map (of_add : α → multiplicative α) | is_open_map.id | lemma | is_open_map_of_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map.id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_to_add : is_open_map (to_add : multiplicative α → α) | is_open_map.id | lemma | is_open_map_to_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map.id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_of_mul : is_closed_map (of_mul : α → additive α) | is_closed_map.id | lemma | is_closed_map_of_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"is_closed_map",
"is_closed_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_to_mul : is_closed_map (to_mul : additive α → α) | is_closed_map.id | lemma | is_closed_map_to_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"is_closed_map",
"is_closed_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_of_add : is_closed_map (of_add : α → multiplicative α) | is_closed_map.id | lemma | is_closed_map_of_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed_map",
"is_closed_map.id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_to_add : is_closed_map (to_add : multiplicative α → α) | is_closed_map.id | lemma | is_closed_map_to_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed_map",
"is_closed_map.id",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_mul (a : α) : 𝓝 (of_mul a) = map of_mul (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_of_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_add (a : α) : 𝓝 (of_add a) = map of_add (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_of_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_to_mul (a : additive α) : 𝓝 (to_mul a) = map to_mul (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_to_mul | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"additive",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_to_add (a : multiplicative α) : 𝓝 (to_add a) = map to_add (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_to_add | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"multiplicative",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_dual : continuous (to_dual : α → αᵒᵈ) | continuous_id | lemma | continuous_to_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_dual : continuous (of_dual : αᵒᵈ → α) | continuous_id | lemma | continuous_of_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_to_dual : is_open_map (to_dual : α → αᵒᵈ) | is_open_map.id | lemma | is_open_map_to_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_of_dual : is_open_map (of_dual : αᵒᵈ → α) | is_open_map.id | lemma | is_open_map_of_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_to_dual : is_closed_map (to_dual : α → αᵒᵈ) | is_closed_map.id | lemma | is_closed_map_to_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed_map",
"is_closed_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_of_dual : is_closed_map (of_dual : αᵒᵈ → α) | is_closed_map.id | lemma | is_closed_map_of_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed_map",
"is_closed_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_to_dual (a : α) : 𝓝 (to_dual a) = map to_dual (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_to_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_dual (a : α) : 𝓝 (of_dual a) = map of_dual (𝓝 a) | by { unfold nhds, refl, } | lemma | nhds_of_dual | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.preimage_mem_nhds [topological_space α] [s : setoid α]
{V : set $ quotient s} {a : α} (hs : V ∈ 𝓝 (quotient.mk a)) : quotient.mk ⁻¹' V ∈ 𝓝 a | preimage_nhds_coinduced hs | lemma | quotient.preimage_mem_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"preimage_nhds_coinduced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.quotient [setoid α] [topological_space α] {s : set α} (H : dense s) :
dense (quotient.mk '' s) | (surjective_quotient_mk α).dense_range.dense_image continuous_coinduced_rng H | lemma | dense.quotient | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_coinduced_rng",
"dense",
"dense_range.dense_image",
"surjective_quotient_mk",
"topological_space"
] | The image of a dense set under `quotient.mk` is a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.quotient [setoid α] [topological_space α] {f : β → α} (hf : dense_range f) :
dense_range (quotient.mk ∘ f) | (surjective_quotient_mk α).dense_range.comp hf continuous_coinduced_rng | lemma | dense_range.quotient | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_coinduced_rng",
"dense_range",
"dense_range.comp",
"surjective_quotient_mk",
"topological_space"
] | The composition of `quotient.mk` and a function with dense range has dense range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum.discrete_topology [topological_space α] [topological_space β]
[hα : discrete_topology α] [hβ : discrete_topology β] : discrete_topology (α ⊕ β) | ⟨by unfold sum.topological_space; simp [hα.eq_bot, hβ.eq_bot]⟩ | instance | sum.discrete_topology | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"discrete_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.discrete_topology {β : α → Type v} [Πa, topological_space (β a)]
[h : Πa, discrete_topology (β a)] : discrete_topology (sigma β) | ⟨by { unfold sigma.topological_space, simp [λ a, (h a).eq_bot] }⟩ | instance | sigma.discrete_topology | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"discrete_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_subtype (s : set α) (a : {x // x ∈ s}) (t : set {x // x ∈ s}) :
t ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (a : α), coe ⁻¹' u ⊆ t | mem_nhds_induced coe a t | theorem | mem_nhds_subtype | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"mem_nhds_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_subtype (s : set α) (a : {x // x ∈ s}) :
𝓝 a = comap coe (𝓝 (a : α)) | nhds_induced coe a | theorem | nhds_subtype | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_subtype_eq_bot_iff {s t : set α} {x : s} :
𝓝[(coe : s → α) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : α) ⊓ 𝓟 s = ⊥ | by rw [inf_principal_eq_bot_iff_comap, nhds_within, nhds_within, comap_inf, comap_principal,
nhds_induced] | lemma | nhds_within_subtype_eq_bot_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_induced",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_ne_subtype_eq_bot_iff {S : set α} {x : S} : 𝓝[{x}ᶜ] x = ⊥ ↔ 𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S = ⊥ | by rw [← nhds_within_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
subtype.coe_injective.preimage_image ] | lemma | nhds_ne_subtype_eq_bot_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_within_subtype_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_ne_subtype_ne_bot_iff {S : set α} {x : S} :
(𝓝[{x}ᶜ] x).ne_bot ↔ (𝓝[{x}ᶜ] (x : α) ⊓ 𝓟 S).ne_bot | by rw [ne_bot_iff, ne_bot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff] | lemma | nhds_ne_subtype_ne_bot_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_ne_subtype_eq_bot_iff",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_subtype_iff {S : set α} :
discrete_topology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ | by simp_rw [discrete_topology_iff_nhds_ne, set_coe.forall', nhds_ne_subtype_eq_bot_iff] | lemma | discrete_topology_subtype_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"discrete_topology",
"discrete_topology_iff_nhds_ne",
"nhds_ne_subtype_eq_bot_iff",
"set_coe.forall'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinite_topology (α : Type*) | α | def | cofinite_topology | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | A type synonym equiped with the topology whose open sets are the empty set and the sets with
finite complements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : α ≃ cofinite_topology α | equiv.refl α | def | cofinite_topology.of | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"equiv.refl"
] | The identity equivalence between `α` and `cofinite_topology α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff {s : set (cofinite_topology α)} :
is_open s ↔ (s.nonempty → (sᶜ).finite) | iff.rfl | lemma | cofinite_topology.is_open_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"finite",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff' {s : set (cofinite_topology α)} :
is_open s ↔ (s = ∅ ∨ (sᶜ).finite) | by simp only [is_open_iff, nonempty_iff_ne_empty, or_iff_not_imp_left] | lemma | cofinite_topology.is_open_iff' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"finite",
"is_open",
"or_iff_not_imp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff {s : set (cofinite_topology α)} :
is_closed s ↔ s = univ ∨ s.finite | by simp [← is_open_compl_iff, is_open_iff'] | lemma | cofinite_topology.is_closed_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"is_closed",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq (a : cofinite_topology α) : 𝓝 a = pure a ⊔ cofinite | begin
ext U,
rw mem_nhds_iff,
split,
{ rintro ⟨V, hVU, V_op, haV⟩,
exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩ },
{ rintros ⟨hU : a ∈ U, hU' : (Uᶜ).finite⟩,
exact ⟨U, subset.rfl, λ h, hU', hU⟩ }
end | lemma | cofinite_topology.nhds_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_iff {a : cofinite_topology α} {s : set (cofinite_topology α)} :
s ∈ 𝓝 a ↔ a ∈ s ∧ sᶜ.finite | by simp [nhds_eq] | lemma | cofinite_topology.mem_nhds_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"cofinite_topology",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_fst : continuous (@prod.fst α β) | continuous_inf_dom_left continuous_induced_dom | lemma | continuous_fst | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_dom",
"continuous_inf_dom_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.fst {f : α → β × γ} (hf : continuous f) : continuous (λ a : α, (f a).1) | continuous_fst.comp hf | lemma | continuous.fst | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | Postcomposing `f` with `prod.fst` is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.fst' {f : α → γ} (hf : continuous f) : continuous (λ x : α × β, f x.fst) | hf.comp continuous_fst | lemma | continuous.fst' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_fst"
] | Precomposing `f` with `prod.fst` is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_fst {p : α × β} : continuous_at prod.fst p | continuous_fst.continuous_at | lemma | continuous_at_fst | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.fst {f : α → β × γ} {x : α} (hf : continuous_at f x) :
continuous_at (λ a : α, (f a).1) x | continuous_at_fst.comp hf | lemma | continuous_at.fst | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | Postcomposing `f` with `prod.fst` is continuous at `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.fst' {f : α → γ} {x : α} {y : β} (hf : continuous_at f x) :
continuous_at (λ x : α × β, f x.fst) (x, y) | continuous_at.comp hf continuous_at_fst | lemma | continuous_at.fst' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_at_fst"
] | Precomposing `f` with `prod.fst` is continuous at `(x, y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.fst'' {f : α → γ} {x : α × β} (hf : continuous_at f x.fst) :
continuous_at (λ x : α × β, f x.fst) x | hf.comp continuous_at_fst | lemma | continuous_at.fst'' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_at_fst"
] | Precomposing `f` with `prod.fst` is continuous at `x : α × β` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_snd : continuous (@prod.snd α β) | continuous_inf_dom_right continuous_induced_dom | lemma | continuous_snd | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_dom",
"continuous_inf_dom_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.snd {f : α → β × γ} (hf : continuous f) : continuous (λ a : α, (f a).2) | continuous_snd.comp hf | lemma | continuous.snd | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | Postcomposing `f` with `prod.snd` is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.snd' {f : β → γ} (hf : continuous f) : continuous (λ x : α × β, f x.snd) | hf.comp continuous_snd | lemma | continuous.snd' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_snd"
] | Precomposing `f` with `prod.snd` is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_snd {p : α × β} : continuous_at prod.snd p | continuous_snd.continuous_at | lemma | continuous_at_snd | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.snd {f : α → β × γ} {x : α} (hf : continuous_at f x) :
continuous_at (λ a : α, (f a).2) x | continuous_at_snd.comp hf | lemma | continuous_at.snd | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | Postcomposing `f` with `prod.snd` is continuous at `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.snd' {f : β → γ} {x : α} {y : β} (hf : continuous_at f y) :
continuous_at (λ x : α × β, f x.snd) (x, y) | continuous_at.comp hf continuous_at_snd | lemma | continuous_at.snd' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_at_snd"
] | Precomposing `f` with `prod.snd` is continuous at `(x, y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.snd'' {f : β → γ} {x : α × β} (hf : continuous_at f x.snd) :
continuous_at (λ x : α × β, f x.snd) x | hf.comp continuous_at_snd | lemma | continuous_at.snd'' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_at_snd"
] | Precomposing `f` with `prod.snd` is continuous at `x : α × β` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.prod_mk {f : γ → α} {g : γ → β}
(hf : continuous f) (hg : continuous g) : continuous (λx, (f x, g x)) | continuous_inf_rng.2 ⟨continuous_induced_rng.2 hf, continuous_induced_rng.2 hg⟩ | lemma | continuous.prod_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_prod_mk {f : α → β} {g : α → γ} :
continuous (λ x, (f x, g x)) ↔ continuous f ∧ continuous g | ⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.prod_mk h.2⟩ | lemma | continuous_prod_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.prod.mk (a : α) : continuous (λ b : β, (a, b)) | continuous_const.prod_mk continuous_id' | lemma | continuous.prod.mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_id'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.prod.mk_left (b : β) : continuous (λ a : α, (a, b)) | continuous_id'.prod_mk continuous_const | lemma | continuous.prod.mk_left | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp₂ {g : α × β → γ} (hg : continuous g) {e : δ → α} (he : continuous e)
{f : δ → β} (hf : continuous f) : continuous (λ x, g (e x, f x)) | hg.comp $ he.prod_mk hf | lemma | continuous.comp₂ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp₃ {g : α × β × γ → ε} (hg : continuous g)
{e : δ → α} (he : continuous e) {f : δ → β} (hf : continuous f)
{k : δ → γ} (hk : continuous k) : continuous (λ x, g (e x, f x, k x)) | hg.comp₂ he $ hf.prod_mk hk | lemma | continuous.comp₃ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp₄ {g : α × β × γ × ζ → ε} (hg : continuous g)
{e : δ → α} (he : continuous e) {f : δ → β} (hf : continuous f)
{k : δ → γ} (hk : continuous k) {l : δ → ζ} (hl : continuous l) :
continuous (λ x, g (e x, f x, k x, l x)) | hg.comp₃ he hf $ hk.prod_mk hl | lemma | continuous.comp₄ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.prod_map {f : γ → α} {g : δ → β} (hf : continuous f) (hg : continuous g) :
continuous (λ x : γ × δ, (f x.1, g x.2)) | hf.fst'.prod_mk hg.snd' | lemma | continuous.prod_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf_dom_left₂ {α β γ} {f : α → β → γ}
{ta1 ta2 : topological_space α} {tb1 tb2 : topological_space β} {tc1 : topological_space γ}
(h : by haveI := ta1; haveI := tb1; exact continuous (λ p : α × β, f p.1 p.2)) :
by haveI | ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact continuous (λ p : α × β, f p.1 p.2) :=
begin
have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _)),
have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _)),
have h_continuous_id := @continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ... | lemma | continuous_inf_dom_left₂ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous.comp",
"continuous.prod_map",
"continuous_id",
"continuous_inf_dom_left",
"topological_space"
] | A version of `continuous_inf_dom_left` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inf_dom_right₂ {α β γ} {f : α → β → γ}
{ta1 ta2 : topological_space α} {tb1 tb2 : topological_space β} {tc1 : topological_space γ}
(h : by haveI := ta2; haveI := tb2; exact continuous (λ p : α × β, f p.1 p.2)) :
by haveI | ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact continuous (λ p : α × β, f p.1 p.2) :=
begin
have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _)),
have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _)),
have h_continuous_id := @continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ... | lemma | continuous_inf_dom_right₂ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous.comp",
"continuous.prod_map",
"continuous_id",
"continuous_inf_dom_right",
"topological_space"
] | A version of `continuous_inf_dom_right` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_Inf_dom₂ {α β γ} {f : α → β → γ}
{tas : set (topological_space α)} {tbs : set (topological_space β)}
{ta : topological_space α} {tb : topological_space β} {tc : topological_space γ}
(ha : ta ∈ tas) (hb : tb ∈ tbs)
(hf : continuous (λ p : α × β, f p.1 p.2)):
by haveI | Inf tas; haveI := Inf tbs; exact @continuous _ _ _ tc (λ p : α × β, f p.1 p.2) :=
begin
let t : topological_space (α × β) := prod.topological_space,
have ha := continuous_Inf_dom ha continuous_id,
have hb := continuous_Inf_dom hb continuous_id,
have h_continuous_id := @continuous.prod_map _ _ _ _ ta tb (Inf tas... | lemma | continuous_Inf_dom₂ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous.comp",
"continuous.prod_map",
"continuous_Inf_dom",
"continuous_id",
"topological_space"
] | A version of `continuous_Inf_dom` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.eventually.prod_inl_nhds {p : α → Prop} {a : α} (h : ∀ᶠ x in 𝓝 a, p x) (b : β) :
∀ᶠ x in 𝓝 (a, b), p (x : α × β).1 | continuous_at_fst h | lemma | filter.eventually.prod_inl_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.prod_inr_nhds {p : β → Prop} {b : β} (h : ∀ᶠ x in 𝓝 b, p x) (a : α) :
∀ᶠ x in 𝓝 (a, b), p (x : α × β).2 | continuous_at_snd h | lemma | filter.eventually.prod_inr_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.prod_mk_nhds {pa : α → Prop} {a} (ha : ∀ᶠ x in 𝓝 a, pa x)
{pb : β → Prop} {b} (hb : ∀ᶠ y in 𝓝 b, pb y) :
∀ᶠ p in 𝓝 (a, b), pa (p : α × β).1 ∧ pb p.2 | (ha.prod_inl_nhds b).and (hb.prod_inr_nhds a) | lemma | filter.eventually.prod_mk_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_swap : continuous (prod.swap : α × β → β × α) | continuous_snd.prod_mk continuous_fst | lemma | continuous_swap | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_fst",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_left {f : α → β → γ} (a : α)
(h : continuous (uncurry f)) : continuous (f a) | show continuous (uncurry f ∘ (λ b, (a, b))), from h.comp (by continuity) | lemma | continuous_uncurry_left | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuity",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_right {f : α → β → γ} (b : β)
(h : continuous (uncurry f)) : continuous (λ a, f a b) | show continuous (uncurry f ∘ (λ a, (a, b))), from h.comp (by continuity) | lemma | continuous_uncurry_right | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuity",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_curry {g : α × β → γ} (a : α)
(h : continuous g) : continuous (curry g a) | show continuous (g ∘ (λ b, (a, b))), from h.comp (by continuity) | lemma | continuous_curry | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuity",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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