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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continuous_on.mul {f g : X → M} {s : set X} (hf : continuous_on f s)
(hg : continuous_on g s) :
continuous_on (λx, f x * g x) s | (continuous_mul.comp_continuous_on (hf.prod hg) : _) | lemma | continuous_on.mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_mul {a b : M} : tendsto (λp:M×M, p.fst * p.snd) (𝓝 (a, b)) (𝓝 (a * b)) | continuous_iff_continuous_at.mp has_continuous_mul.continuous_mul (a, b) | lemma | tendsto_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.mul {f g : α → M} {x : filter α} {a b : M}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, f x * g x) x (𝓝 (a * b)) | tendsto_mul.comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.const_mul (b : M) {c : M} {f : α → M} {l : filter α}
(h : tendsto (λ (k:α), f k) l (𝓝 c)) : tendsto (λ (k:α), b * f k) l (𝓝 (b * c)) | tendsto_const_nhds.mul h | lemma | filter.tendsto.const_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.mul_const (b : M) {c : M} {f : α → M} {l : filter α}
(h : tendsto (λ (k:α), f k) l (𝓝 c)) : tendsto (λ (k:α), f k * b) l (𝓝 (c * b)) | h.mul tendsto_const_nhds | lemma | filter.tendsto.mul_const | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) | by { rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq], exact continuous_mul.tendsto _ } | lemma | le_nhds_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_one_mul_nhds {M} [mul_one_class M] [topological_space M]
[has_continuous_mul M] (a : M) : 𝓝 (1 : M) * 𝓝 a = 𝓝 a | ((le_nhds_mul _ _).trans_eq $ congr_arg _ (one_mul a)).antisymm $
le_mul_of_one_le_left' $ pure_le_nhds 1 | lemma | nhds_one_mul_nhds | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"le_mul_of_one_le_left'",
"le_nhds_mul",
"mul_one_class",
"one_mul",
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mul_nhds_one {M} [mul_one_class M] [topological_space M]
[has_continuous_mul M] (a : M) : 𝓝 a * 𝓝 1 = 𝓝 a | ((le_nhds_mul _ _).trans_eq $ congr_arg _ (mul_one a)).antisymm $
le_mul_of_one_le_right' $ pure_le_nhds 1 | lemma | nhds_mul_nhds_one | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"le_mul_of_one_le_right'",
"le_nhds_mul",
"mul_one",
"mul_one_class",
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_within_Ioi.const_mul [pos_mul_strict_mono 𝕜] [pos_mul_reflect_lt 𝕜]
(h : tendsto f l (𝓝[>] c)) :
tendsto (λ a, b * f a) l (𝓝[>] (b * c)) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhds_within h).const_mul b) $
(tendsto_nhds_within_iff.mp h).2.mono (λ j, (mul_lt_mul_left hb).mpr) | lemma | filter.tendsto_nhds_within_Ioi.const_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"mul_lt_mul_left",
"pos_mul_reflect_lt",
"pos_mul_strict_mono",
"tendsto_nhds_of_tendsto_nhds_within",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_within_Iio.const_mul [pos_mul_strict_mono 𝕜] [pos_mul_reflect_lt 𝕜]
(h : tendsto f l (𝓝[<] c)) :
tendsto (λ a, b * f a) l (𝓝[<] (b * c)) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhds_within h).const_mul b) $
(tendsto_nhds_within_iff.mp h).2.mono (λ j, (mul_lt_mul_left hb).mpr) | lemma | filter.tendsto_nhds_within_Iio.const_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"mul_lt_mul_left",
"pos_mul_reflect_lt",
"pos_mul_strict_mono",
"tendsto_nhds_of_tendsto_nhds_within",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_within_Ioi.mul_const [mul_pos_strict_mono 𝕜] [mul_pos_reflect_lt 𝕜]
(h : tendsto f l (𝓝[>] c)) :
tendsto (λ a, f a * b) l (𝓝[>] (c * b)) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhds_within h).mul_const b) $
(tendsto_nhds_within_iff.mp h).2.mono (λ j, (mul_lt_mul_right hb).mpr) | lemma | filter.tendsto_nhds_within_Ioi.mul_const | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"mul_lt_mul_right",
"mul_pos_reflect_lt",
"mul_pos_strict_mono",
"tendsto_nhds_of_tendsto_nhds_within",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_nhds_within_Iio.mul_const [mul_pos_strict_mono 𝕜] [mul_pos_reflect_lt 𝕜]
(h : tendsto f l (𝓝[<] c)) :
tendsto (λ a, f a * b) l (𝓝[<] (c * b)) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
((tendsto_nhds_of_tendsto_nhds_within h).mul_const b) $
(tendsto_nhds_within_iff.mp h).2.mono (λ j, (mul_lt_mul_right hb).mpr) | lemma | filter.tendsto_nhds_within_Iio.mul_const | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"mul_lt_mul_right",
"mul_pos_reflect_lt",
"mul_pos_strict_mono",
"tendsto_nhds_of_tendsto_nhds_within",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.units [topological_space N] [monoid N] [has_continuous_mul N] [t2_space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : filter ι} [l.ne_bot]
(h₁ : tendsto (λ x, ↑(f x)) l (𝓝 r₁)) (h₂ : tendsto (λ x, ↑(f x)⁻¹) l (𝓝 r₂)) : Nˣ | { val := r₁,
inv := r₂,
val_inv := by { symmetry, simpa using h₁.mul h₂ },
inv_val := by { symmetry, simpa using h₂.mul h₁ } } | def | filter.tendsto.units | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter",
"has_continuous_mul",
"monoid",
"t2_space",
"topological_space"
] | Construct a unit from limits of units and their inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.mul {f g : X → M} {x : X} (hf : continuous_at f x) (hg : continuous_at g x) :
continuous_at (λx, f x * g x) x | hf.mul hg | lemma | continuous_at.mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.mul {f g : X → M} {s : set X} {x : X} (hf : continuous_within_at f s x)
(hg : continuous_within_at g s x) :
continuous_within_at (λx, f x * g x) s x | hf.mul hg | lemma | continuous_within_at.mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_continuous_mul {C : ι → Type*} [∀ i, topological_space (C i)]
[∀ i, has_mul (C i)] [∀ i, has_continuous_mul (C i)] : has_continuous_mul (Π i, C i) | { continuous_mul := continuous_pi (λ i, (continuous_apply i).fst'.mul (continuous_apply i).snd') } | instance | pi.has_continuous_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_apply",
"continuous_mul",
"continuous_pi",
"has_continuous_mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_continuous_mul' : has_continuous_mul (ι → M) | pi.has_continuous_mul | instance | pi.has_continuous_mul' | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"pi.has_continuous_mul"
] | A version of `pi.has_continuous_mul` for non-dependent functions. It is needed because sometimes
Lean fails to use `pi.has_continuous_mul` for non-dependent functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_mul_of_discrete_topology [topological_space N]
[has_mul N] [discrete_topology N] : has_continuous_mul N | ⟨continuous_of_discrete_topology⟩ | instance | has_continuous_mul_of_discrete_topology | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"discrete_topology",
"has_continuous_mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul.of_nhds_one {M : Type u} [monoid M] [topological_space M]
(hmul : tendsto (uncurry ((*) : M → M → M)) (𝓝 1 ×ᶠ 𝓝 1) $ 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (λ x, x*x₀) (𝓝 1)) : has_continuous_mul M | ⟨begin
rw continuous_iff_continuous_at,
rintros ⟨x₀, y₀⟩,
have key : (λ p : M × M, x₀ * p.1 * (p.2 * y₀)) = ((λ x, x₀*x) ∘ (λ x, x*y₀)) ∘ (uncurry (*)),
{ ext p, simp [uncurry, mul_assoc] },
have key₂ : (λ x, x₀*x) ∘ (λ x, y₀*x) = λ x, (x₀ *y₀)*x,
{ ext x, simp },
calc map (uncurry (*)) (𝓝 ... | lemma | has_continuous_mul.of_nhds_one | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_iff_continuous_at",
"filter.map_map",
"has_continuous_mul",
"monoid",
"mul_assoc",
"nhds_prod_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul_of_comm_of_nhds_one (M : Type u) [comm_monoid M] [topological_space M]
(hmul : tendsto (uncurry ((*) : M → M → M)) (𝓝 1 ×ᶠ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1)) : has_continuous_mul M | begin
apply has_continuous_mul.of_nhds_one hmul hleft,
intros x₀,
simp_rw [mul_comm, hleft x₀]
end | lemma | has_continuous_mul_of_comm_of_nhds_one | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"comm_monoid",
"has_continuous_mul",
"has_continuous_mul.of_nhds_one",
"mul_comm",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_map_one [has_one M₁] [has_one M₂] :
is_closed {f : M₁ → M₂ | f 1 = 1} | is_closed_eq (continuous_apply 1) continuous_const | lemma | is_closed_set_of_map_one | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_apply",
"continuous_const",
"is_closed",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_map_mul [has_mul M₁] [has_mul M₂] [has_continuous_mul M₂] :
is_closed {f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y} | begin
simp only [set_of_forall],
exact is_closed_Inter (λ x, is_closed_Inter (λ y, is_closed_eq (continuous_apply _)
((continuous_apply _).mul (continuous_apply _))))
end | lemma | is_closed_set_of_map_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_apply",
"has_continuous_mul",
"is_closed",
"is_closed_Inter",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_of_mem_closure_range_coe (f : M₁ → M₂)
(hf : f ∈ closure (range (λ (f : F) (x : M₁), f x))) : M₁ →* M₂ | { to_fun := f,
map_one' := (is_closed_set_of_map_one M₁ M₂).closure_subset_iff.2 (range_subset_iff.2 map_one) hf,
map_mul' := (is_closed_set_of_map_mul M₁ M₂).closure_subset_iff.2
(range_subset_iff.2 map_mul) hf } | def | monoid_hom_of_mem_closure_range_coe | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"closure",
"is_closed_set_of_map_mul",
"is_closed_set_of_map_one",
"map_mul",
"map_one"
] | Construct a bundled monoid homomorphism `M₁ →* M₂` from a function `f` and a proof that it
belongs to the closure of the range of the coercion from `M₁ →* M₂` (or another type of bundled
homomorphisms that has a `monoid_hom_class` instance) to `M₁ → M₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom_of_tendsto (f : M₁ → M₂) (g : α → F) [l.ne_bot]
(h : tendsto (λ a x, g a x) l (𝓝 f)) : M₁ →* M₂ | monoid_hom_of_mem_closure_range_coe f $ mem_closure_of_tendsto h $
eventually_of_forall $ λ a, mem_range_self _ | def | monoid_hom_of_tendsto | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"mem_closure_of_tendsto",
"monoid_hom_of_mem_closure_range_coe"
] | Construct a bundled monoid homomorphism from a pointwise limit of monoid homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.is_closed_range_coe :
is_closed (range (coe_fn : (M₁ →* M₂) → (M₁ → M₂))) | is_closed_of_closure_subset $ λ f hf, ⟨monoid_hom_of_mem_closure_range_coe f hf, rfl⟩ | lemma | monoid_hom.is_closed_range_coe | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"is_closed",
"is_closed_of_closure_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.has_continuous_mul {M N F : Type*} [has_mul M] [has_mul N]
[mul_hom_class F M N] [topological_space M] [topological_space N] [has_continuous_mul N]
(f : F) (hf : inducing f) :
has_continuous_mul M | ⟨hf.continuous_iff.2 $ by simpa only [(∘), map_mul f]
using (hf.continuous.fst'.mul hf.continuous.snd')⟩ | lemma | inducing.has_continuous_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"inducing",
"map_mul",
"mul_hom_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul_induced {M N F : Type*} [has_mul M] [has_mul N]
[mul_hom_class F M N] [topological_space N] [has_continuous_mul N] (f : F) :
@has_continuous_mul M (induced f ‹_›) _ | by { letI := induced f ‹_›, exact inducing.has_continuous_mul f ⟨rfl⟩ } | lemma | has_continuous_mul_induced | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"inducing.has_continuous_mul",
"mul_hom_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemigroup.has_continuous_mul [topological_space M] [semigroup M]
[has_continuous_mul M] (S : subsemigroup M) :
has_continuous_mul S | inducing.has_continuous_mul (⟨coe, λ _ _, rfl⟩ : mul_hom S M) ⟨rfl⟩ | instance | subsemigroup.has_continuous_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"inducing.has_continuous_mul",
"mul_hom",
"semigroup",
"subsemigroup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.has_continuous_mul [topological_space M] [monoid M]
[has_continuous_mul M] (S : submonoid M) :
has_continuous_mul S | S.to_subsemigroup.has_continuous_mul | instance | submonoid.has_continuous_mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"monoid",
"submonoid",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.top_closure_mul_self_subset (s : submonoid M) :
closure (s : set M) * closure s ⊆ closure s | image2_subset_iff.2 $ λ x hx y hy, map_mem_closure₂ continuous_mul hx hy $
λ a ha b hb, s.mul_mem ha hb | lemma | submonoid.top_closure_mul_self_subset | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"closure",
"continuous_mul",
"map_mem_closure₂",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.top_closure_mul_self_eq (s : submonoid M) :
closure (s : set M) * closure s = closure s | subset.antisymm
s.top_closure_mul_self_subset
(λ x hx, ⟨x, 1, hx, subset_closure s.one_mem, mul_one _⟩) | lemma | submonoid.top_closure_mul_self_eq | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"closure",
"mul_one",
"submonoid",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.topological_closure (s : submonoid M) : submonoid M | { carrier := closure (s : set M),
one_mem' := subset_closure s.one_mem,
mul_mem' := λ a b ha hb, s.top_closure_mul_self_subset ⟨a, b, ha, hb, rfl⟩ } | def | submonoid.topological_closure | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"closure",
"submonoid",
"subset_closure"
] | The (topological-space) closure of a submonoid of a space `M` with `has_continuous_mul` is
itself a submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid.le_topological_closure (s : submonoid M) :
s ≤ s.topological_closure | subset_closure | lemma | submonoid.le_topological_closure | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"submonoid",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.is_closed_topological_closure (s : submonoid M) :
is_closed (s.topological_closure : set M) | by convert is_closed_closure | lemma | submonoid.is_closed_topological_closure | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"is_closed",
"is_closed_closure",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.topological_closure_minimal
(s : submonoid M) {t : submonoid M} (h : s ≤ t) (ht : is_closed (t : set M)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | submonoid.topological_closure_minimal | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"closure_minimal",
"is_closed",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.comm_monoid_topological_closure [t2_space M] (s : submonoid M)
(hs : ∀ (x y : s), x * y = y * x) : comm_monoid s.topological_closure | { mul_comm :=
have ∀ (x ∈ s) (y ∈ s), x * y = y * x,
from λ x hx y hy, congr_arg subtype.val (hs ⟨x, hx⟩ ⟨y, hy⟩),
λ ⟨x, hx⟩ ⟨y, hy⟩, subtype.ext $
eq_on_closure₂ this continuous_mul (continuous_snd.mul continuous_fst) x hx y hy,
..s.topological_closure.to_monoid } | def | submonoid.comm_monoid_topological_closure | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"comm_monoid",
"continuous_fst",
"continuous_mul",
"eq_on_closure₂",
"mul_comm",
"submonoid",
"subtype.ext",
"t2_space"
] | If a submonoid of a topological monoid is commutative, then so is its topological closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_open_nhds_one_split {s : set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : set M, is_open V ∧ (1 : M) ∈ V ∧ ∀ (v ∈ V) (w ∈ V), v * w ∈ s | have ((λa:M×M, a.1 * a.2) ⁻¹' s) ∈ 𝓝 ((1, 1) : M × M),
from tendsto_mul (by simpa only [one_mul] using hs),
by simpa only [prod_subset_iff] using exists_nhds_square this | lemma | exists_open_nhds_one_split | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"exists_nhds_square",
"is_open",
"one_mul",
"tendsto_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nhds_one_split {s : set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ (v ∈ V) (w ∈ V), v * w ∈ s | let ⟨V, Vo, V1, hV⟩ := exists_open_nhds_one_split hs
in ⟨V, is_open.mem_nhds Vo V1, hV⟩ | lemma | exists_nhds_one_split | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"exists_open_nhds_one_split",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nhds_one_split4 {u : set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M),
∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | begin
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩,
rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩,
use [V, V1],
intros v w s t v_in w_in s_in t_in,
simpa only [mul_assoc] using h _ (h' v v_in w w_in) _ (h' s s_in t t_in)
end | lemma | exists_nhds_one_split4 | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"exists_nhds_one_split",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_open_nhds_one_mul_subset {U : set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : set M, is_open V ∧ (1 : M) ∈ V ∧ V * V ⊆ U | begin
rcases exists_open_nhds_one_split hU with ⟨V, Vo, V1, hV⟩,
use [V, Vo, V1],
rintros _ ⟨x, y, hx, hy, rfl⟩,
exact hV _ hx _ hy
end | lemma | exists_open_nhds_one_mul_subset | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"exists_open_nhds_one_split",
"is_open"
] | Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.mul {s t : set M} (hs : is_compact s) (ht : is_compact t) : is_compact (s * t) | by { rw [← image_mul_prod], exact (hs.prod ht).image continuous_mul } | lemma | is_compact.mul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_mul",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_list_prod {f : ι → α → M} {x : filter α} {a : ι → M} :
∀ l:list ι, (∀i∈l, tendsto (f i) x (𝓝 (a i))) →
tendsto (λb, (l.map (λc, f c b)).prod) x (𝓝 ((l.map a).prod)) | | [] _ := by simp [tendsto_const_nhds]
| (f :: l) h :=
begin
simp only [list.map_cons, list.prod_cons],
exact (h f (list.mem_cons_self _ _)).mul
(tendsto_list_prod l (assume c hc, h c (list.mem_cons_of_mem _ hc)))
end | lemma | tendsto_list_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter",
"list.prod_cons",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_list_prod {f : ι → X → M} (l : list ι)
(h : ∀ i ∈ l, continuous (f i)) :
continuous (λ a, (l.map (λ i, f i a)).prod) | continuous_iff_continuous_at.2 $ assume x, tendsto_list_prod l $ assume c hc,
continuous_iff_continuous_at.1 (h c hc) x | lemma | continuous_list_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"tendsto_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_list_prod {f : ι → X → M} (l : list ι) {t : set X}
(h : ∀ i ∈ l, continuous_on (f i) t) :
continuous_on (λ a, (l.map (λ i, f i a)).prod) t | begin
intros x hx,
rw continuous_within_at_iff_continuous_at_restrict _ hx,
refine tendsto_list_prod _ (λ i hi, _),
specialize h i hi x hx,
rw continuous_within_at_iff_continuous_at_restrict _ hx at h,
exact h,
end | lemma | continuous_on_list_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on",
"continuous_within_at_iff_continuous_at_restrict",
"tendsto_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_pow : ∀ n : ℕ, continuous (λ a : M, a ^ n) | | 0 := by simpa using continuous_const
| (k+1) := by { simp only [pow_succ], exact continuous_id.mul (continuous_pow _) } | lemma | continuous_pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_const",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.has_continuous_const_smul_nat {A} [add_monoid A] [topological_space A]
[has_continuous_add A] : has_continuous_const_smul ℕ A | ⟨continuous_nsmul⟩ | instance | add_monoid.has_continuous_const_smul_nat | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"add_monoid",
"has_continuous_add",
"has_continuous_const_smul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.has_continuous_smul_nat {A} [add_monoid A] [topological_space A]
[has_continuous_add A] : has_continuous_smul ℕ A | ⟨continuous_uncurry_of_discrete_topology continuous_nsmul⟩ | instance | add_monoid.has_continuous_smul_nat | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"add_monoid",
"has_continuous_add",
"has_continuous_smul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.pow {f : X → M} (h : continuous f) (n : ℕ) :
continuous (λ b, (f b) ^ n) | (continuous_pow n).comp h | lemma | continuous.pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_pow {s : set M} (n : ℕ) : continuous_on (λ x, x ^ n) s | (continuous_pow n).continuous_on | lemma | continuous_on_pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on",
"continuous_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_pow (x : M) (n : ℕ) : continuous_at (λ x, x ^ n) x | (continuous_pow n).continuous_at | lemma | continuous_at_pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_at",
"continuous_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.pow {l : filter α} {f : α → M} {x : M} (hf : tendsto f l (𝓝 x)) (n : ℕ) :
tendsto (λ x, f x ^ n) l (𝓝 (x ^ n)) | (continuous_at_pow _ _).tendsto.comp hf | lemma | filter.tendsto.pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_at_pow",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.pow {f : X → M} {x : X} {s : set X} (hf : continuous_within_at f s x)
(n : ℕ) : continuous_within_at (λ x, f x ^ n) s x | hf.pow n | lemma | continuous_within_at.pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.pow {f : X → M} {x : X} (hf : continuous_at f x) (n : ℕ) :
continuous_at (λ x, f x ^ n) x | hf.pow n | lemma | continuous_at.pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.pow {f : X → M} {s : set X} (hf : continuous_on f s) (n : ℕ) :
continuous_on (λ x, f x ^ n) s | λ x hx, (hf x hx).pow n | lemma | continuous_on.pow | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
filter.tendsto (λ x : M, a * x) (filter.cocompact M) (filter.cocompact M) | begin
refine filter.tendsto.of_tendsto_comp _ (filter.comap_cocompact_le (continuous_mul_left b)),
convert filter.tendsto_id,
ext x,
simp [ha],
end | lemma | filter.tendsto_cocompact_mul_left | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_mul_left",
"filter.cocompact",
"filter.comap_cocompact_le",
"filter.tendsto",
"filter.tendsto.of_tendsto_comp",
"filter.tendsto_id"
] | Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
filter.tendsto (λ x : M, x * a) (filter.cocompact M) (filter.cocompact M) | begin
refine filter.tendsto.of_tendsto_comp _ (filter.comap_cocompact_le (continuous_mul_right b)),
convert filter.tendsto_id,
ext x,
simp [ha],
end | lemma | filter.tendsto_cocompact_mul_right | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_mul_right",
"filter.cocompact",
"filter.comap_cocompact_le",
"filter.tendsto",
"filter.tendsto.of_tendsto_comp",
"filter.tendsto_id"
] | Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_scalar_tower.has_continuous_const_smul {R A : Type*} [monoid A] [has_smul R A]
[is_scalar_tower R A A] [topological_space A] [has_continuous_mul A] :
has_continuous_const_smul R A | { continuous_const_smul := λ q, begin
simp only [←smul_one_mul q (_ : A)] { single_pass := tt },
exact continuous_const.mul continuous_id,
end } | instance | is_scalar_tower.has_continuous_const_smul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_id",
"has_continuous_const_smul",
"has_continuous_mul",
"has_smul",
"is_scalar_tower",
"monoid",
"topological_space"
] | If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[algebra R A]` is available. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_comm_class.has_continuous_const_smul {R A : Type*} [monoid A] [has_smul R A]
[smul_comm_class R A A] [topological_space A] [has_continuous_mul A] :
has_continuous_const_smul R A | { continuous_const_smul := λ q, begin
simp only [←mul_smul_one q (_ : A)] { single_pass := tt },
exact continuous_id.mul continuous_const,
end } | instance | smul_comm_class.has_continuous_const_smul | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_const",
"has_continuous_const_smul",
"has_continuous_mul",
"has_smul",
"monoid",
"smul_comm_class",
"topological_space"
] | If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.units_map [monoid M] [monoid N] [topological_space M]
[topological_space N] (f : M →* N) (hf : continuous f) : continuous (units.map f) | units.continuous_iff.2 ⟨hf.comp units.continuous_coe, hf.comp units.continuous_coe_inv⟩ | lemma | continuous.units_map | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"monoid",
"topological_space",
"units.continuous_coe",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.mem_nhds_one (S : submonoid M) (oS : is_open (S : set M)) :
(S : set M) ∈ 𝓝 (1 : M) | is_open.mem_nhds oS S.one_mem | lemma | submonoid.mem_nhds_one | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"is_open",
"is_open.mem_nhds",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_multiset_prod {f : ι → α → M} {x : filter α} {a : ι → M} (s : multiset ι) :
(∀ i ∈ s, tendsto (f i) x (𝓝 (a i))) →
tendsto (λb, (s.map (λc, f c b)).prod) x (𝓝 ((s.map a).prod)) | by { rcases s with ⟨l⟩, simpa using tendsto_list_prod l } | lemma | tendsto_multiset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter",
"multiset",
"tendsto_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_finset_prod {f : ι → α → M} {x : filter α} {a : ι → M} (s : finset ι) :
(∀ i ∈ s, tendsto (f i) x (𝓝 (a i))) → tendsto (λb, ∏ c in s, f c b) x (𝓝 (∏ c in s, a c)) | tendsto_multiset_prod _ | lemma | tendsto_finset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"filter",
"finset",
"tendsto_multiset_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multiset_prod {f : ι → X → M} (s : multiset ι) :
(∀ i ∈ s, continuous (f i)) → continuous (λ a, (s.map (λ i, f i a)).prod) | by { rcases s with ⟨l⟩, simpa using continuous_list_prod l } | lemma | continuous_multiset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_list_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_multiset_prod {f : ι → X → M} (s : multiset ι) {t : set X} :
(∀i ∈ s, continuous_on (f i) t) → continuous_on (λ a, (s.map (λ i, f i a)).prod) t | by { rcases s with ⟨l⟩, simpa using continuous_on_list_prod l } | lemma | continuous_on_multiset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on",
"continuous_on_list_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_finset_prod {f : ι → X → M} (s : finset ι) :
(∀ i ∈ s, continuous (f i)) → continuous (λ a, ∏ i in s, f i a) | continuous_multiset_prod _ | lemma | continuous_finset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_multiset_prod",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_finset_prod {f : ι → X → M} (s : finset ι) {t : set X} :
(∀ i ∈ s, continuous_on (f i) t) → continuous_on (λ a, ∏ i in s, f i a) t | continuous_on_multiset_prod _ | lemma | continuous_on_finset_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_on",
"continuous_on_multiset_prod",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_prod {X M : Type*} [comm_monoid M]
{s : finset ι} {l : filter X} {f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) :
∏ i in s, f i =ᶠ[l] ∏ i in s, g i | begin
replace hs: ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x,
{ rwa eventually_all_finset },
filter_upwards [hs] with x hx,
simp only [finset.prod_apply, finset.prod_congr rfl hx],
end | lemma | eventually_eq_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"comm_monoid",
"filter",
"finset",
"finset.prod_apply",
"finset.prod_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite.exists_finset_mul_support {M : Type*} [comm_monoid M] {f : ι → X → M}
(hf : locally_finite (λ i, mul_support $ f i)) (x₀ : X) :
∃ I : finset ι, ∀ᶠ x in 𝓝 x₀, mul_support (λ i, f i x) ⊆ I | begin
rcases hf x₀ with ⟨U, hxU, hUf⟩,
refine ⟨hUf.to_finset, mem_of_superset hxU $ λ y hy i hi, _⟩,
rw [hUf.coe_to_finset],
exact ⟨y, hi, hy⟩
end | lemma | locally_finite.exists_finset_mul_support | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"comm_monoid",
"finset",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eventually_eq_prod {M : Type*} [comm_monoid M]
{f : ι → X → M} (hf : locally_finite (λ i, mul_support (f i))) (x : X) :
∃ s : finset ι, ∀ᶠ y in 𝓝 x, (∏ᶠ i, f i y) = ∏ i in s, f i y | let ⟨I, hI⟩ := hf.exists_finset_mul_support x in
⟨I, hI.mono (λ y hy, finprod_eq_prod_of_mul_support_subset _ $ λ i hi, hy hi)⟩ | lemma | finprod_eventually_eq_prod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"comm_monoid",
"finprod_eq_prod_of_mul_support_subset",
"finset",
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_finprod {f : ι → X → M} (hc : ∀ i, continuous (f i))
(hf : locally_finite (λ i, mul_support (f i))) :
continuous (λ x, ∏ᶠ i, f i x) | begin
refine continuous_iff_continuous_at.2 (λ x, _),
rcases finprod_eventually_eq_prod hf x with ⟨s, hs⟩,
refine continuous_at.congr _ (eventually_eq.symm hs),
exact tendsto_finset_prod _ (λ i hi, (hc i).continuous_at),
end | lemma | continuous_finprod | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_at",
"continuous_at.congr",
"finprod_eventually_eq_prod",
"locally_finite",
"tendsto_finset_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_finprod_cond {f : ι → X → M} {p : ι → Prop}
(hc : ∀ i, p i → continuous (f i)) (hf : locally_finite (λ i, mul_support (f i))) :
continuous (λ x, ∏ᶠ i (hi : p i), f i x) | begin
simp only [← finprod_subtype_eq_finprod_cond],
exact continuous_finprod (λ i, hc i i.2) (hf.comp_injective subtype.coe_injective)
end | lemma | continuous_finprod_cond | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_finprod",
"finprod_subtype_eq_finprod_cond",
"locally_finite",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul_Inf {ts : set (topological_space M)}
(h : Π t ∈ ts, @has_continuous_mul M t _) :
@has_continuous_mul M (Inf ts) _ | { continuous_mul := continuous_Inf_rng.2 (λ t ht, continuous_Inf_dom₂ ht ht
(@has_continuous_mul.continuous_mul M t _ (h t ht))) } | lemma | has_continuous_mul_Inf | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_Inf_dom₂",
"continuous_mul",
"has_continuous_mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul_infi {ts : ι' → topological_space M}
(h' : Π i, @has_continuous_mul M (ts i) _) :
@has_continuous_mul M (⨅ i, ts i) _ | by { rw ← Inf_range, exact has_continuous_mul_Inf (set.forall_range_iff.mpr h') } | lemma | has_continuous_mul_infi | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"Inf_range",
"has_continuous_mul",
"has_continuous_mul_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul_inf {t₁ t₂ : topological_space M}
(h₁ : @has_continuous_mul M t₁ _) (h₂ : @has_continuous_mul M t₂ _) :
@has_continuous_mul M (t₁ ⊓ t₂) _ | by { rw inf_eq_infi, refine has_continuous_mul_infi (λ b, _), cases b; assumption } | lemma | has_continuous_mul_inf | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"has_continuous_mul",
"has_continuous_mul_infi",
"inf_eq_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right (x : X) : C(X, X) | mk _ (continuous_mul_right x) | def | continuous_map.mul_right | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_mul_right"
] | The continuous map `λ y, y * x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_right (x : X) : ⇑(continuous_map.mul_right x) = λ y, y * x | rfl | lemma | continuous_map.coe_mul_right | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_map.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left (x : X) : C(X, X) | mk _ (continuous_mul_left x) | def | continuous_map.mul_left | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_mul_left"
] | The continuous map `λ y, x * y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_left (x : X) : ⇑(continuous_map.mul_left x) = λ y, x * y | rfl | lemma | continuous_map.coe_mul_left | topology.algebra | src/topology/algebra/monoid.lean | [
"algebra.big_operators.finprod",
"order.filter.pointwise",
"topology.algebra.mul_action",
"algebra.big_operators.pi",
"topology.continuous_function.basic"
] | [
"continuous_map.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul (M X : Type*) [has_smul M X]
[topological_space M] [topological_space X] : Prop | (continuous_smul : continuous (λp : M × X, p.1 • p.2)) | class | has_continuous_smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous",
"has_smul",
"topological_space"
] | Class `has_continuous_smul M X` says that the scalar multiplication `(•) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of multiplicative actions,
including (semi)modules and algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_vadd (M X : Type*) [has_vadd M X]
[topological_space M] [topological_space X] : Prop | (continuous_vadd : continuous (λp : M × X, p.1 +ᵥ p.2)) | class | has_continuous_vadd | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous",
"has_vadd",
"topological_space"
] | Class `has_continuous_vadd M X` says that the additive action `(+ᵥ) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of additive actions,
including (semi)modules and algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_smul.has_continuous_const_smul :
has_continuous_const_smul M X | { continuous_const_smul := λ _, continuous_smul.comp (continuous_const.prod_mk continuous_id) } | instance | has_continuous_smul.has_continuous_const_smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous_id",
"has_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.smul {f : α → M} {g : α → X} {l : filter α} {c : M} {a : X}
(hf : tendsto f l (𝓝 c)) (hg : tendsto g l (𝓝 a)) :
tendsto (λ x, f x • g x) l (𝓝 $ c • a) | (continuous_smul.tendsto _).comp (hf.prod_mk_nhds hg) | lemma | filter.tendsto.smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.smul_const {f : α → M} {l : filter α} {c : M}
(hf : tendsto f l (𝓝 c)) (a : X) :
tendsto (λ x, (f x) • a) l (𝓝 (c • a)) | hf.smul tendsto_const_nhds | lemma | filter.tendsto.smul_const | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.smul (hf : continuous_within_at f s b)
(hg : continuous_within_at g s b) :
continuous_within_at (λ x, f x • g x) s b | hf.smul hg | lemma | continuous_within_at.smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.smul (hf : continuous_at f b) (hg : continuous_at g b) :
continuous_at (λ x, f x • g x) b | hf.smul hg | lemma | continuous_at.smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.smul (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λ x, f x • g x) s | λ x hx, (hf x hx).smul (hg x hx) | lemma | continuous_on.smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.smul (hf : continuous f) (hg : continuous g) :
continuous (λ x, f x • g x) | continuous_smul.comp (hf.prod_mk hg) | lemma | continuous.smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul.op [has_smul Mᵐᵒᵖ X] [is_central_scalar M X] :
has_continuous_smul Mᵐᵒᵖ X | ⟨ suffices continuous (λ p : M × X, mul_opposite.op p.fst • p.snd),
from this.comp (mul_opposite.continuous_unop.prod_map continuous_id),
by simpa only [op_smul_eq_smul] using (continuous_smul : continuous (λ p : M × X, _)) ⟩ | instance | has_continuous_smul.op | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous",
"continuous_id",
"has_continuous_smul",
"has_smul",
"is_central_scalar",
"mul_opposite.op"
] | If a scalar action is central, then its right action is continuous when its left action is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.has_continuous_smul : has_continuous_smul M Xᵐᵒᵖ | ⟨mul_opposite.continuous_op.comp $ continuous_smul.comp $
continuous_id.prod_map mul_opposite.continuous_unop⟩ | instance | mul_opposite.has_continuous_smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.has_continuous_smul : has_continuous_smul Mˣ X | { continuous_smul :=
show continuous ((λ p : M × X, p.fst • p.snd) ∘ (λ p : Mˣ × X, (p.1, p.2))),
from continuous_smul.comp ((units.continuous_coe.comp continuous_fst).prod_mk continuous_snd) } | instance | units.has_continuous_smul | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"continuous",
"continuous_fst",
"continuous_snd",
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_Inf {ts : set (topological_space X)}
(h : Π t ∈ ts, @has_continuous_smul M X _ _ t) :
@has_continuous_smul M X _ _ (Inf ts) | { continuous_smul :=
begin
rw ← @Inf_singleton _ _ ‹topological_space M›,
exact continuous_Inf_rng.2 (λ t ht, continuous_Inf_dom₂ (eq.refl _) ht
(@has_continuous_smul.continuous_smul _ _ _ _ t (h t ht)))
end } | lemma | has_continuous_smul_Inf | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"Inf_singleton",
"continuous_Inf_dom₂",
"has_continuous_smul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_infi {ts' : ι → topological_space X}
(h : Π i, @has_continuous_smul M X _ _ (ts' i)) :
@has_continuous_smul M X _ _ (⨅ i, ts' i) | has_continuous_smul_Inf $ set.forall_range_iff.mpr h | lemma | has_continuous_smul_infi | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"has_continuous_smul",
"has_continuous_smul_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_inf {t₁ t₂ : topological_space X}
[@has_continuous_smul M X _ _ t₁] [@has_continuous_smul M X _ _ t₂] :
@has_continuous_smul M X _ _ (t₁ ⊓ t₂) | by { rw inf_eq_infi, refine has_continuous_smul_infi (λ b, _), cases b; assumption } | lemma | has_continuous_smul_inf | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"has_continuous_smul",
"has_continuous_smul_infi",
"inf_eq_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_torsor.connected_space : connected_space P | { is_preconnected_univ :=
begin
convert is_preconnected_univ.image ((equiv.vadd_const (classical.arbitrary P)) : G → P)
(continuous_id.vadd continuous_const).continuous_on,
rw [set.image_univ, equiv.range_eq_univ]
end,
to_nonempty := infer_instance } | lemma | add_torsor.connected_space | topology.algebra | src/topology/algebra/mul_action.lean | [
"algebra.add_torsor",
"topology.algebra.constructions",
"group_theory.group_action.prod",
"topology.algebra.const_mul_action"
] | [
"classical.arbitrary",
"connected_space",
"continuous_const",
"continuous_on",
"equiv.range_eq_univ",
"equiv.vadd_const",
"set.image_univ"
] | An `add_torsor` for a connected space is a connected space. This is not an instance because
it loops for a group as a torsor over itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_add_subgroup (G : Type*) [add_group G] [topological_space G]
extends add_subgroup G | (is_open' : is_open carrier) | structure | open_add_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"add_group",
"add_subgroup",
"is_open",
"topological_space"
] | The type of open subgroups of a topological additive group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_subgroup (G : Type*) [group G] [topological_space G] extends subgroup G | (is_open' : is_open carrier) | structure | open_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"group",
"is_open",
"subgroup",
"topological_space"
] | The type of open subgroups of a topological group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_subgroup : has_coe_t (open_subgroup G) (subgroup G) | ⟨to_subgroup⟩ | instance | open_subgroup.has_coe_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_injective : injective (coe : open_subgroup G → subgroup G) | | ⟨_, _⟩ ⟨_, _⟩ rfl := rfl | lemma | open_subgroup.coe_subgroup_injective | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_opens : has_coe_t (open_subgroup G) (opens G) | ⟨λ U, ⟨U, U.is_open'⟩⟩ | instance | open_subgroup.has_coe_opens | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_opens : ((U : opens G) : set G) = U | rfl | lemma | open_subgroup.coe_coe_opens | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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