statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
affine_homeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 | { to_fun := λ x, a * x + b,
inv_fun := λ y, (y - b) / a,
left_inv := λ x, by { simp only [add_sub_cancel], exact mul_div_cancel_left x h, },
right_inv := λ y, by { simp [mul_div_cancel' _ h], }, } | def | affine_homeomorph | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"inv_fun",
"mul_div_cancel'",
"mul_div_cancel_left"
] | The map `λ x, a * x + b`, as a homeomorphism from `𝕜` (a topological field) to itself, when `a ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_min.inv {f : α → β} {a : α} (h1 : is_local_min f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
is_local_max f⁻¹ a | by filter_upwards [h1, h2] with z h3 h4 using (inv_le_inv h4 h2.self_of_nhds).mpr h3 | lemma | is_local_min.inv | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"inv_le_inv",
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.eq_one_or_eq_neg_one_of_sq_eq [ring 𝕜] [no_zero_divisors 𝕜]
(hS : is_preconnected S) (hf : continuous_on f S) (hsq : eq_on (f ^ 2) 1 S) :
(eq_on f 1 S) ∨ (eq_on f (-1) S) | begin
simp_rw [eq_on, pi.one_apply, pi.pow_apply, sq_eq_one_iff] at hsq,
-- First deal with crazy case where `S` is empty.
by_cases hSe : ∀ (x:α), x ∉ S,
{ left, intros x hx,
exfalso, exact hSe x hx, },
push_neg at hSe,
choose y hy using hSe,
suffices : ∀ (x:α), x ∈ S → f x = f y,
{ rcases (hsq hy),... | lemma | is_preconnected.eq_one_or_eq_neg_one_of_sq_eq | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"continuous_on",
"discrete_of_t1_of_finite",
"is_preconnected",
"no_zero_divisors",
"pi.one_apply",
"pi.pow_apply",
"ring",
"sq_eq_one_iff"
] | If `f` is a function `α → 𝕜` which is continuous on a preconnected set `S`, and
`f ^ 2 = 1` on `S`, then either `f = 1` on `S`, or `f = -1` on `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.eq_or_eq_neg_of_sq_eq [field 𝕜] [has_continuous_inv₀ 𝕜] [has_continuous_mul 𝕜]
(hS : is_preconnected S) (hf : continuous_on f S) (hg : continuous_on g S)
(hsq : eq_on (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x:α}, x ∈ S → g x ≠ 0) :
(eq_on f g S) ∨ (eq_on f (-g) S) | begin
rcases hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg (λ z hz, hg_ne hz)) (λ x hx, _) with h | h,
{ refine or.inl (λ x hx, _),
rw ←div_eq_one_iff_eq (hg_ne hx),
exact h hx },
{ refine or.inr (λ x hx, _),
specialize h hx,
rwa [pi.div_apply, pi.neg_apply, pi.one_apply, div_eq_iff (hg_ne hx), neg_... | lemma | is_preconnected.eq_or_eq_neg_of_sq_eq | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"continuous_on",
"div_eq_iff",
"div_pow",
"div_self",
"field",
"has_continuous_inv₀",
"has_continuous_mul",
"is_preconnected",
"neg_one_mul",
"pi.div_apply",
"pi.one_apply",
"pow_ne_zero"
] | If `f, g` are functions `α → 𝕜`, both continuous on a preconnected set `S`, with
`f ^ 2 = g ^ 2` on `S`, and `g z ≠ 0` all `z ∈ S`, then either `f = g` or `f = -g` on
`S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.eq_of_sq_eq [field 𝕜] [has_continuous_inv₀ 𝕜] [has_continuous_mul 𝕜]
(hS : is_preconnected S) (hf : continuous_on f S) (hg : continuous_on g S)
(hsq : eq_on (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x:α}, x ∈ S → g x ≠ 0)
{y : α} (hy : y ∈ S) (hy' : f y = g y) : eq_on f g S | λ x hx, begin
rcases hS.eq_or_eq_neg_of_sq_eq hf hg @hsq @hg_ne with h | h,
{ exact h hx },
{ rw [h hy, eq_comm, ←sub_eq_zero, sub_eq_add_neg, pi.neg_apply,
neg_neg, ←mul_two, mul_eq_zero] at hy',
cases hy', -- need to handle case of `char 𝕜 = 2` separately
{ exfalso, exact hg_ne hy hy' },
{ rw... | lemma | is_preconnected.eq_of_sq_eq | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"continuous_on",
"field",
"has_continuous_inv₀",
"has_continuous_mul",
"is_preconnected",
"mul_eq_zero",
"mul_zero"
] | If `f, g` are functions `α → 𝕜`, both continuous on a preconnected set `S`, with
`f ^ 2 = g ^ 2` on `S`, and `g z ≠ 0` all `z ∈ S`, then as soon as `f = g` holds at
one point of `S` it holds for all points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_filter_basis (G : Type u) [group G] extends filter_basis G | (one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U)
(mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U)
(inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x⁻¹) ⁻¹' U)
(conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x₀*x*x₀⁻¹) ⁻¹' U) | class | group_filter_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"filter_basis",
"group"
] | A `group_filter_basis` on a group is a `filter_basis` satisfying some additional axioms.
Example : if `G` is a topological group then the neighbourhoods of the identity are a
`group_filter_basis`. Conversely given a `group_filter_basis` one can define a topology
compatible with the group structure on `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_filter_basis (A : Type u) [add_group A] extends filter_basis A | (zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U)
(add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U)
(neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, -x) ⁻¹' U)
(conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x₀+x+-x₀) ⁻¹' U) | class | add_group_filter_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"add_group",
"filter_basis"
] | A `add_group_filter_basis` on an additive group is a `filter_basis` satisfying some additional
axioms. Example : if `G` is a topological group then the neighbourhoods of the identity are a
`add_group_filter_basis`. Conversely given a `add_group_filter_basis` one can define a topology
compatible with the group str... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_filter_basis_of_comm {G : Type*} [comm_group G]
(sets : set (set G))
(nonempty : sets.nonempty)
(inter_sets : ∀ (x y), x ∈ sets → y ∈ sets → ∃ z ∈ sets, z ⊆ x ∩ y)
(one : ∀ U ∈ sets, (1 : G) ∈ U)
(mul : ∀ U ∈ sets, ∃ V ∈ sets, V * V ⊆ U)
(inv : ∀ U ∈ sets, ∃ V ∈ sets, ... | { sets := sets,
nonempty := nonempty,
inter_sets := inter_sets,
one' := one,
mul' := mul,
inv' := inv,
conj' := λ x U U_in, ⟨U, U_in, by simp⟩ } | def | group_filter_basis_of_comm | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"comm_group",
"group_filter_basis"
] | `group_filter_basis` constructor in the commutative group case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one {U : set G} : U ∈ B → (1 : G) ∈ U | group_filter_basis.one' | lemma | group_filter_basis.one | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul {U : set G} : U ∈ B → ∃ V ∈ B, V*V ⊆ U | group_filter_basis.mul' | lemma | group_filter_basis.mul | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv {U : set G} : U ∈ B → ∃ V ∈ B, V ⊆ (λ x, x⁻¹) ⁻¹' U | group_filter_basis.inv' | lemma | group_filter_basis.inv | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj : ∀ x₀, ∀ {U}, U ∈ B → ∃ V ∈ B, V ⊆ (λ x, x₀*x*x₀⁻¹) ⁻¹' U | group_filter_basis.conj' | lemma | group_filter_basis.conj | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_subset_self (B : group_filter_basis G) {U : set G} (h : U ∈ B) : U ⊆ U * U | λ x x_in, ⟨1, x, one h, x_in, one_mul x⟩ | lemma | group_filter_basis.prod_subset_self | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N (B : group_filter_basis G) : G → filter G | λ x, map (λ y, x*y) B.to_filter_basis.filter | def | group_filter_basis.N | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"filter",
"group_filter_basis"
] | The neighborhood function of a `group_filter_basis` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N_one (B : group_filter_basis G) : B.N 1 = B.to_filter_basis.filter | by simp only [N, one_mul, map_id'] | lemma | group_filter_basis.N_one | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis (B : group_filter_basis G) (x : G) :
has_basis (B.N x) (λ V : set G, V ∈ B) (λ V, (λ y, x*y) '' V) | has_basis.map (λ y, x * y) to_filter_basis.has_basis | lemma | group_filter_basis.has_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology (B : group_filter_basis G) : topological_space G | topological_space.mk_of_nhds B.N | def | group_filter_basis.topology | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"topological_space",
"topological_space.mk_of_nhds"
] | The topological space structure coming from a group filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_eq (B : group_filter_basis G) {x₀ : G} :
@nhds G (B.topology) x₀ = B.N x₀ | begin
rw [topological_space.nhds_mk_of_nhds],
{ intros x U U_in,
rw (B.has_basis x).mem_iff at U_in,
rcases U_in with ⟨V, V_in, H⟩,
simpa [mem_pure] using H (mem_image_of_mem _ (group_filter_basis.one V_in)), },
{ intros x U U_in,
rw (B.has_basis x).mem_iff at U_in,
rcases U_in with ⟨V, V_in, ... | lemma | group_filter_basis.nhds_eq | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"filter_basis.mem_filter_of_mem",
"group_filter_basis",
"group_filter_basis.mul",
"group_filter_basis.one",
"mul_assoc",
"nhds",
"topological_space.nhds_mk_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_one_eq (B : group_filter_basis G) :
@nhds G (B.topology) (1 : G) = B.to_filter_basis.filter | by { rw B.nhds_eq, simp only [N, one_mul], exact map_id } | lemma | group_filter_basis.nhds_one_eq | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"map_id",
"nhds",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_has_basis (B : group_filter_basis G) (x₀ : G) :
has_basis (@nhds G B.topology x₀) (λ V : set G, V ∈ B) (λ V, (λ y, x₀ * y) '' V) | by { rw B.nhds_eq, apply B.has_basis } | lemma | group_filter_basis.nhds_has_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_one_has_basis (B : group_filter_basis G) :
has_basis (@nhds G B.topology 1) (λ V : set G, V ∈ B) id | by { rw B.nhds_one_eq, exact B.to_filter_basis.has_basis } | lemma | group_filter_basis.nhds_one_has_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_one (B : group_filter_basis G) {U : set G} (hU : U ∈ B) : U ∈ @nhds G B.topology 1 | begin
rw B.nhds_one_has_basis.mem_iff,
exact ⟨U, hU, rfl.subset⟩
end | lemma | group_filter_basis.mem_nhds_one | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"group_filter_basis",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_group (B : group_filter_basis G) :
@topological_group G B.topology _ | begin
letI := B.topology,
have basis := B.nhds_one_has_basis,
have basis' := basis.prod basis,
refine topological_group.of_nhds_one _ _ _ _,
{ rw basis'.tendsto_iff basis,
suffices : ∀ U ∈ B, ∃ V W, (V ∈ B ∧ W ∈ B) ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U, by simpa,
intros U U_in,
rcases mul U_in with ⟨... | instance | group_filter_basis.is_topological_group | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"basis",
"basis.prod",
"group_filter_basis",
"topological_group",
"topological_group.of_nhds_one"
] | If a group is endowed with a topological structure coming from a group filter basis then it's a
topological group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_filter_basis (R : Type u) [ring R] extends add_group_filter_basis R | (mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U)
(mul_left' : ∀ (x₀ : R) {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x₀*x) ⁻¹' U)
(mul_right' : ∀ (x₀ : R) {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x*x₀) ⁻¹' U) | class | ring_filter_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"add_group_filter_basis",
"ring"
] | A `ring_filter_basis` on a ring is a `filter_basis` satisfying some additional axioms.
Example : if `R` is a topological ring then the neighbourhoods of the identity are a
`ring_filter_basis`. Conversely given a `ring_filter_basis` on a ring `R`, one can define a
topology on `R` which is compatible with the ring ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul {U : set R} (hU : U ∈ B) : ∃ V ∈ B, V * V ⊆ U | mul' hU | lemma | ring_filter_basis.mul | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left (x₀ : R) {U : set R} (hU : U ∈ B) :
∃ V ∈ B, V ⊆ (λ x, x₀*x) ⁻¹' U | mul_left' x₀ hU | lemma | ring_filter_basis.mul_left | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right (x₀ : R) {U : set R} (hU : U ∈ B) :
∃ V ∈ B, V ⊆ (λ x, x*x₀) ⁻¹' U | mul_right' x₀ hU | lemma | ring_filter_basis.mul_right | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology : topological_space R | B.to_add_group_filter_basis.topology | def | ring_filter_basis.topology | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"topological_space"
] | The topology associated to a ring filter basis.
It has the given basis as a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_ring {R : Type u} [ring R] (B : ring_filter_basis R) :
@topological_ring R B.topology _ | begin
let B' := B.to_add_group_filter_basis,
letI := B'.topology,
have basis := B'.nhds_zero_has_basis,
have basis' := basis.prod basis,
haveI := B'.is_topological_add_group,
apply topological_ring.of_add_group_of_nhds_zero,
{ rw basis'.tendsto_iff basis,
suffices : ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ... | instance | ring_filter_basis.is_topological_ring | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"basis",
"basis.prod",
"ring",
"ring_filter_basis",
"topological_ring",
"topological_ring.of_add_group_of_nhds_zero"
] | If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_filter_basis (R M : Type*) [comm_ring R] [topological_space R]
[add_comm_group M] [module R M] extends add_group_filter_basis M | (smul' : ∀ {U}, U ∈ sets → ∃ (V ∈ 𝓝 (0 : R)) (W ∈ sets), V • W ⊆ U)
(smul_left' : ∀ (x₀ : R) {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (λ x, x₀ • x) ⁻¹' U)
(smul_right' : ∀ (m₀ : M) {U}, U ∈ sets → ∀ᶠ x in 𝓝 (0 : R), x • m₀ ∈ U) | structure | module_filter_basis | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"add_group_filter_basis",
"comm_ring",
"module",
"topological_space"
] | A `module_filter_basis` on a module is a `filter_basis` satisfying some additional axioms.
Example : if `M` is a topological module then the neighbourhoods of zero are a
`module_filter_basis`. Conversely given a `module_filter_basis` one can define a topology
compatible with the module structure on `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_filter_basis.has_mem : has_mem (set M) (module_filter_basis R M) | ⟨λ s B, s ∈ B.sets⟩ | instance | module_filter_basis.group_filter_basis.has_mem | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"module_filter_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul {U : set M} (hU : U ∈ B) : ∃ (V ∈ 𝓝 (0 : R)) (W ∈ B), V • W ⊆ U | B.smul' hU | lemma | module_filter_basis.smul | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_left (x₀ : R) {U : set M} (hU : U ∈ B) : ∃ V ∈ B, V ⊆ (λ x, x₀ • x) ⁻¹' U | B.smul_left' x₀ hU | lemma | module_filter_basis.smul_left | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right (m₀ : M) {U : set M} (hU : U ∈ B) : ∀ᶠ x in 𝓝 (0 : R), x • m₀ ∈ U | B.smul_right' m₀ hU | lemma | module_filter_basis.smul_right | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology : topological_space M | B.to_add_group_filter_basis.topology | def | module_filter_basis.topology | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"topological_space"
] | The topology associated to a module filter basis on a module over a topological ring.
It has the given basis as a basis of neighborhoods of zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topology' {R M : Type*} [comm_ring R] {tR : topological_space R}
[add_comm_group M] [module R M] (B : module_filter_basis R M) : topological_space M | B.to_add_group_filter_basis.topology | def | module_filter_basis.topology' | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"comm_ring",
"module",
"module_filter_basis",
"topological_space"
] | The topology associated to a module filter basis on a module over a topological ring.
It has the given basis as a basis of neighborhoods of zero. This version gets the ring
topology by unification instead of type class inference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.has_continuous_smul.of_basis_zero {ι : Type*} [topological_ring R]
[topological_space M] [topological_add_group M] {p : ι → Prop} {b : ι → set M}
(h : has_basis (𝓝 0) p b) (hsmul : ∀ {i}, p i → ∃ (V ∈ 𝓝 (0 : R)) j (hj : p j), V • (b j) ⊆ b i)
(hsmul_left : ∀ (x₀ : R) {i}, p i → ∃ j (hj : p j), (b j) ⊆ (λ... | begin
apply has_continuous_smul.of_nhds_zero,
{ rw h.tendsto_right_iff,
intros i hi,
rcases hsmul hi with ⟨V, V_in, j, hj, hVj⟩,
apply mem_of_superset (prod_mem_prod V_in $ h.mem_of_mem hj),
rintros ⟨v, w⟩ ⟨v_in : v ∈ V, w_in : w ∈ (b j)⟩,
exact hVj (set.smul_mem_smul v_in w_in) },
{ intro m₀,... | lemma | has_continuous_smul.of_basis_zero | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"has_continuous_smul",
"has_continuous_smul.of_nhds_zero",
"set.smul_mem_smul",
"topological_add_group",
"topological_ring",
"topological_space"
] | A topological add group whith a basis of `𝓝 0` satisfying the axioms of `module_filter_basis`
is a topological module.
This lemma is mathematically useless because one could obtain such a result by applying
`module_filter_basis.has_continuous_smul` and use the fact that group topologies are characterized
by their nei... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_smul [topological_ring R] :
@has_continuous_smul R M _ _ B.topology | begin
let B' := B.to_add_group_filter_basis,
letI := B'.topology,
haveI := B'.is_topological_add_group,
exact has_continuous_smul.of_basis_zero B'.nhds_zero_has_basis (λ _, B.smul) B.smul_left
B.smul_right,
end | instance | module_filter_basis.has_continuous_smul | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"has_continuous_smul",
"has_continuous_smul.of_basis_zero",
"topological_ring"
] | If a module is endowed with a topological structure coming from
a module filter basis then it's a topological module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_bases {R M : Type*} [comm_ring R]
[add_comm_group M] [module R M] (BR : ring_filter_basis R) (BM : add_group_filter_basis M)
(smul : ∀ {U}, U ∈ BM → ∃ (V ∈ BR) (W ∈ BM), V • W ⊆ U)
(smul_left : ∀ (x₀ : R) {U}, U ∈ BM → ∃ V ∈ BM, V ⊆ (λ x, x₀ • x) ⁻¹' U)
(smul_right : ∀ (m₀ : M) {U}, U ∈ BM → ∃ V ∈ BR, V ⊆ (λ... | { smul' := begin
intros U U_in,
rcases smul U_in with ⟨V, V_in, W, W_in, H⟩,
exact ⟨V, BR.to_add_group_filter_basis.mem_nhds_zero V_in, W, W_in, H⟩
end,
smul_left' := smul_left,
smul_right' := begin
intros m₀ U U_in,
rcases smul_right m₀ U_in with ⟨V, V_in, H⟩,
exact mem_of_superset (BR.to... | def | module_filter_basis.of_bases | topology.algebra | src/topology/algebra/filter_basis.lean | [
"order.filter.bases",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"add_group_filter_basis",
"comm_ring",
"module",
"module_filter_basis",
"ring_filter_basis"
] | Build a module filter basis from compatible ring and additive group filter bases. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.completion.coe_zero [has_zero α] : ((0 : α) : completion α) = 0 | rfl | lemma | uniform_space.completion.coe_zero | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (a : α) : ((- a : α) : completion α) = - a | (map_coe uniform_continuous_neg a).symm | lemma | uniform_space.completion.coe_neg | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (a b : α) : ((a - b : α) : completion α) = a - b | (map₂_coe_coe a b has_sub.sub uniform_continuous_sub).symm | lemma | uniform_space.completion.coe_sub | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (a b : α) : ((a + b : α) : completion α) = a + b | (map₂_coe_coe a b (+) uniform_continuous_add).symm | lemma | uniform_space.completion.coe_add | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_compl : α →+ completion α | { to_fun := coe,
map_add' := coe_add,
map_zero' := coe_zero } | def | uniform_space.completion.to_compl | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [] | The map from a group to its completion as a group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_to_compl : continuous (to_compl : α → completion α) | continuous_coe α | lemma | uniform_space.completion.continuous_to_compl | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_inducing_to_compl : dense_inducing (to_compl : α → completion α) | dense_inducing_coe | lemma | uniform_space.completion.dense_inducing_to_compl | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.extension [complete_space β] [separated_space β] (f : α →+ β)
(hf : continuous f) : completion α →+ β | have hf : uniform_continuous f, from uniform_continuous_add_monoid_hom_of_continuous hf,
{ to_fun := completion.extension f,
map_zero' := by rw [← coe_zero, extension_coe hf, f.map_zero],
map_add' := assume a b, completion.induction_on₂ a b
(is_closed_eq
(continuous_extension.comp continuous_add)
((contin... | def | add_monoid_hom.extension | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"complete_space",
"continuous",
"continuous_fst",
"continuous_snd",
"is_closed_eq",
"separated_space",
"uniform_continuous"
] | Extension to the completion of a continuous group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.extension_coe [complete_space β] [separated_space β] (f : α →+ β)
(hf : continuous f) (a : α) : f.extension hf a = f a | extension_coe (uniform_continuous_add_monoid_hom_of_continuous hf) a | lemma | add_monoid_hom.extension_coe | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"complete_space",
"continuous",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.continuous_extension [complete_space β] [separated_space β] (f : α →+ β)
(hf : continuous f) : continuous (f.extension hf) | continuous_extension | lemma | add_monoid_hom.continuous_extension | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"complete_space",
"continuous",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.completion (f : α →+ β) (hf : continuous f) : completion α →+ completion β | (to_compl.comp f).extension (continuous_to_compl.comp hf) | def | add_monoid_hom.completion | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"continuous"
] | Completion of a continuous group hom, as a group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.continuous_completion (f : α →+ β)
(hf : continuous f) : continuous (f.completion hf : completion α → completion β) | continuous_map | lemma | add_monoid_hom.continuous_completion | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"continuous",
"continuous_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.completion_coe (f : α →+ β)
(hf : continuous f) (a : α) : f.completion hf a = f a | map_coe (uniform_continuous_add_monoid_hom_of_continuous hf) a | lemma | add_monoid_hom.completion_coe | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.completion_zero : (0 : α →+ β).completion continuous_const = 0 | begin
ext x,
apply completion.induction_on x,
{ apply is_closed_eq ((0 : α →+ β).continuous_completion continuous_const),
simp [continuous_const] },
{ intro a,
simp [(0 : α →+ β).completion_coe continuous_const, coe_zero] }
end | lemma | add_monoid_hom.completion_zero | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"continuous_const",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.completion_add {γ : Type*} [add_comm_group γ] [uniform_space γ]
[uniform_add_group γ] (f g : α →+ γ) (hf : continuous f) (hg : continuous g) :
(f + g).completion (hf.add hg) = f.completion hf + g.completion hg | begin
have hfg := hf.add hg,
ext x,
apply completion.induction_on x,
{ exact is_closed_eq ((f+g).continuous_completion hfg)
((f.continuous_completion hf).add (g.continuous_completion hg)) },
{ intro a,
simp [(f+g).completion_coe hfg, coe_add, f.completion_coe hf, g.completion_coe hg] }
end | lemma | add_monoid_hom.completion_add | topology.algebra | src/topology/algebra/group_completion.lean | [
"topology.algebra.uniform_group",
"topology.algebra.uniform_mul_action",
"topology.uniform_space.completion"
] | [
"add_comm_group",
"continuous",
"is_closed_eq",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.div_const {x : G₀} (hf : tendsto f l (𝓝 x)) (y : G₀) :
tendsto (λa, f a / y) l (𝓝 (x / y)) | by simpa only [div_eq_mul_inv] using hf.mul tendsto_const_nhds | lemma | filter.tendsto.div_const | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"div_eq_mul_inv",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.div_const {a : α} (hf : continuous_at f a) (y : G₀) :
continuous_at (λ x, f x / y) a | by simpa only [div_eq_mul_inv] using hf.mul continuous_at_const | lemma | continuous_at.div_const | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at",
"continuous_at_const",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.div_const {a} (hf : continuous_within_at f s a) (y : G₀) :
continuous_within_at (λ x, f x / y) s a | hf.div_const _ | lemma | continuous_within_at.div_const | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.div_const (hf : continuous_on f s) (y : G₀) : continuous_on (λ x, f x / y) s | by simpa only [div_eq_mul_inv] using hf.mul continuous_on_const | lemma | continuous_on.div_const | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on",
"continuous_on_const",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.div_const (hf : continuous f) (y : G₀) :
continuous (λ x, f x / y) | by simpa only [div_eq_mul_inv] using hf.mul continuous_const | lemma | continuous.div_const | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous",
"continuous_const",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inv₀ (G₀ : Type*) [has_zero G₀] [has_inv G₀] [topological_space G₀] : Prop | (continuous_at_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → continuous_at has_inv.inv x) | class | has_continuous_inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at",
"topological_space"
] | A type with `0` and `has_inv` such that `λ x, x⁻¹` is continuous at all nonzero points. Any
normed (semi)field has this property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_inv₀ {x : G₀} (hx : x ≠ 0) : tendsto has_inv.inv (𝓝 x) (𝓝 x⁻¹) | continuous_at_inv₀ hx | lemma | tendsto_inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_inv₀ : continuous_on (has_inv.inv : G₀ → G₀) {0}ᶜ | λ x hx, (continuous_at_inv₀ hx).continuous_within_at | lemma | continuous_on_inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inv₀ {a : G₀} (hf : tendsto f l (𝓝 a))
(ha : a ≠ 0) :
tendsto (λ x, (f x)⁻¹) l (𝓝 a⁻¹) | (tendsto_inv₀ ha).comp hf | lemma | filter.tendsto.inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"tendsto_inv₀"
] | If a function converges to a nonzero value, its inverse converges to the inverse of this value.
We use the name `tendsto.inv₀` as `tendsto.inv` is already used in multiplicative topological
groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at.inv₀ (hf : continuous_within_at f s a) (ha : f a ≠ 0) :
continuous_within_at (λ x, (f x)⁻¹) s a | hf.inv₀ ha | lemma | continuous_within_at.inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.inv₀ (hf : continuous_at f a) (ha : f a ≠ 0) :
continuous_at (λ x, (f x)⁻¹) a | hf.inv₀ ha | lemma | continuous_at.inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.inv₀ (hf : continuous f) (h0 : ∀ x, f x ≠ 0) :
continuous (λ x, (f x)⁻¹) | continuous_iff_continuous_at.2 $ λ x, (hf.tendsto x).inv₀ (h0 x) | lemma | continuous.inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.inv₀ (hf : continuous_on f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
continuous_on (λ x, (f x)⁻¹) s | λ x hx, (hf x hx).inv₀ (h0 x hx) | lemma | continuous_on.inv₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.embedding_coe₀ [group_with_zero G₀] [topological_space G₀] [has_continuous_inv₀ G₀] :
embedding (coe : G₀ˣ → G₀) | units.embedding_coe_mk $ continuous_on_inv₀.mono $ λ x, is_unit.ne_zero | theorem | units.embedding_coe₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"embedding",
"group_with_zero",
"has_continuous_inv₀",
"is_unit.ne_zero",
"topological_space",
"units.embedding_coe_mk"
] | If `G₀` is a group with zero with topology such that `x ↦ x⁻¹` is continuous at all nonzero
points. Then the coercion `Mˣ → M` is a topological embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.div {l : filter α} {a b : G₀} (hf : tendsto f l (𝓝 a))
(hg : tendsto g l (𝓝 b)) (hy : b ≠ 0) :
tendsto (f / g) l (𝓝 (a / b)) | by simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ hy) | lemma | filter.tendsto.div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"div_eq_mul_inv",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_mul_iff_of_ne_zero [t1_space G₀]
{f g : α → G₀} {l : filter α} {x y : G₀}
(hg : tendsto g l (𝓝 y)) (hy : y ≠ 0) :
tendsto (λ n, f n * g n) l (𝓝 $ x * y) ↔ tendsto f l (𝓝 x) | begin
refine ⟨λ hfg, _, λ hf, hf.mul hg⟩,
rw ←mul_div_cancel x hy,
refine tendsto.congr' _ (hfg.div hg hy),
refine eventually.mp (hg.eventually_ne hy) (eventually_of_forall (λ n hn, mul_div_cancel _ hn)),
end | lemma | filter.tendsto_mul_iff_of_ne_zero | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"filter",
"mul_div_cancel",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.div (hf : continuous_within_at f s a) (hg : continuous_within_at g s a)
(h₀ : g a ≠ 0) :
continuous_within_at (f / g) s a | hf.div hg h₀ | lemma | continuous_within_at.div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.div (hf : continuous_on f s) (hg : continuous_on g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
continuous_on (f / g) s | λ x hx, (hf x hx).div (hg x hx) (h₀ x hx) | lemma | continuous_on.div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.div (hf : continuous_at f a) (hg : continuous_at g a) (h₀ : g a ≠ 0) :
continuous_at (f / g) a | hf.div hg h₀ | lemma | continuous_at.div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at"
] | Continuity at a point of the result of dividing two functions continuous at that point, where
the denominator is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.div (hf : continuous f) (hg : continuous g) (h₀ : ∀ x, g x ≠ 0) :
continuous (f / g) | by simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀) | lemma | continuous.div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_div : continuous_on (λ p : G₀ × G₀, p.1 / p.2) {p | p.2 ≠ 0} | continuous_on_fst.div continuous_on_snd $ λ _, id | lemma | continuous_on_div | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on",
"continuous_on_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_div_cases {f g : α → G₀} (h : α → G₀ → β)
(hf : continuous_at f a) (hg : continuous_at g a)
(hh : g a ≠ 0 → continuous_at ↿h (a, f a / g a))
(h2h : g a = 0 → tendsto ↿h (𝓝 a ×ᶠ ⊤) (𝓝 (h a 0))) :
continuous_at (λ x, h x (f x / g x)) a | begin
show continuous_at (↿h ∘ (λ x, (x, f x / g x))) a,
by_cases hga : g a = 0,
{ rw [continuous_at], simp_rw [comp_app, hga, div_zero],
exact (h2h hga).comp (continuous_at_id.prod_mk tendsto_top) },
{ exact continuous_at.comp (hh hga) (continuous_at_id.prod (hf.div hg hga)) }
end | lemma | continuous_at.comp_div_cases | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at",
"continuous_at.comp",
"div_zero"
] | The function `f x / g x` is discontinuous when `g x = 0`.
However, under appropriate conditions, `h x (f x / g x)` is still continuous.
The condition is that if `g a = 0` then `h x y` must tend to `h a 0` when `x` tends to `a`,
with no information about `y`. This is represented by the `⊤` filter.
Note: `filter.tendsto_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.comp_div_cases {f g : α → G₀} (h : α → G₀ → β)
(hf : continuous f) (hg : continuous g)
(hh : ∀ a, g a ≠ 0 → continuous_at ↿h (a, f a / g a))
(h2h : ∀ a, g a = 0 → tendsto ↿h (𝓝 a ×ᶠ ⊤) (𝓝 (h a 0))) :
continuous (λ x, h x (f x / g x)) | continuous_iff_continuous_at.mpr $
λ a, hf.continuous_at.comp_div_cases _ hg.continuous_at (hh a) (h2h a) | lemma | continuous.comp_div_cases | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous",
"continuous_at"
] | `h x (f x / g x)` is continuous under certain conditions, even if the denominator is sometimes
`0`. See docstring of `continuous_at.comp_div_cases`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left₀ (c : α) (hc : c ≠ 0) : α ≃ₜ α | { continuous_to_fun := continuous_mul_left _,
continuous_inv_fun := continuous_mul_left _,
.. equiv.mul_left₀ c hc } | def | homeomorph.mul_left₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_mul_left",
"equiv.mul_left₀"
] | Left multiplication by a nonzero element in a `group_with_zero` with continuous multiplication
is a homeomorphism of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right₀ (c : α) (hc : c ≠ 0) : α ≃ₜ α | { continuous_to_fun := continuous_mul_right _,
continuous_inv_fun := continuous_mul_right _,
.. equiv.mul_right₀ c hc } | def | homeomorph.mul_right₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_mul_right",
"equiv.mul_right₀"
] | Right multiplication by a nonzero element in a `group_with_zero` with continuous multiplication
is a homeomorphism of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_left₀ (c : α) (hc : c ≠ 0) : ⇑(homeomorph.mul_left₀ c hc) = (*) c | rfl | lemma | homeomorph.coe_mul_left₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"homeomorph.mul_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left₀_symm_apply (c : α) (hc : c ≠ 0) :
((homeomorph.mul_left₀ c hc).symm : α → α) = (*) c⁻¹ | rfl | lemma | homeomorph.mul_left₀_symm_apply | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"homeomorph.mul_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_right₀ (c : α) (hc : c ≠ 0) :
⇑(homeomorph.mul_right₀ c hc) = λ x, x * c | rfl | lemma | homeomorph.coe_mul_right₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"homeomorph.mul_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right₀_symm_apply (c : α) (hc : c ≠ 0) :
((homeomorph.mul_right₀ c hc).symm : α → α) = λ x, x * c⁻¹ | rfl | lemma | homeomorph.mul_right₀_symm_apply | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"homeomorph.mul_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_zpow₀ (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) : continuous_at (λ x, x ^ m) x | begin
cases m,
{ simpa only [zpow_of_nat] using continuous_at_pow x m },
{ simp only [zpow_neg_succ_of_nat],
have hx : x ≠ 0, from h.resolve_right (int.neg_succ_of_nat_lt_zero m).not_le,
exact (continuous_at_pow x (m + 1)).inv₀ (pow_ne_zero _ hx) }
end | lemma | continuous_at_zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at",
"continuous_at_pow",
"pow_ne_zero",
"zpow_neg_succ_of_nat",
"zpow_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_zpow₀ (m : ℤ) : continuous_on (λ x : G₀, x ^ m) {0}ᶜ | λ x hx, (continuous_at_zpow₀ _ _ (or.inl hx)).continuous_within_at | lemma | continuous_on_zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at_zpow₀",
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.zpow₀ {f : α → G₀} {l : filter α} {a : G₀} (hf : tendsto f l (𝓝 a)) (m : ℤ)
(h : a ≠ 0 ∨ 0 ≤ m) :
tendsto (λ x, (f x) ^ m) l (𝓝 (a ^ m)) | (continuous_at_zpow₀ _ m h).tendsto.comp hf | lemma | filter.tendsto.zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at_zpow₀",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.zpow₀ (hf : continuous_at f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) :
continuous_at (λ x, (f x) ^ m) a | hf.zpow₀ m h | lemma | continuous_at.zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.zpow₀ (hf : continuous_within_at f s a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) :
continuous_within_at (λ x, f x ^ m) s a | hf.zpow₀ m h | lemma | continuous_within_at.zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.zpow₀ (hf : continuous_on f s) (m : ℤ) (h : ∀ a ∈ s, f a ≠ 0 ∨ 0 ≤ m) :
continuous_on (λ x, f x ^ m) s | λ a ha, (hf a ha).zpow₀ m (h a ha) | lemma | continuous_on.zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.zpow₀ (hf : continuous f) (m : ℤ) (h0 : ∀ a, f a ≠ 0 ∨ 0 ≤ m) :
continuous (λ x, (f x) ^ m) | continuous_iff_continuous_at.2 $ λ x, (hf.tendsto x).zpow₀ m (h0 x) | lemma | continuous.zpow₀ | topology.algebra | src/topology/algebra/group_with_zero.lean | [
"topology.algebra.monoid",
"algebra.group.pi",
"topology.homeomorph"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization.ring_topology : ring_topology (localization M) | ring_topology.coinduced (localization.monoid_of M).to_fun | def | localization.ring_topology | topology.algebra | src/topology/algebra/localization.lean | [
"ring_theory.localization.basic",
"topology.algebra.ring.basic"
] | [
"localization",
"localization.monoid_of",
"ring_topology",
"ring_topology.coinduced"
] | The ring topology on `localization M` coinduced from the natural homomorphism sending `x : R`
to the equivalence class of `(x, 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_one [topological_space M] [has_one M] : continuous (1 : X → M) | @continuous_const _ _ _ _ 1 | lemma | continuous_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",
"continuous_const",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_add (M : Type u) [topological_space M] [has_add M] : Prop | (continuous_add : continuous (λ p : M × M, p.1 + p.2)) | class | has_continuous_add | 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",
"topological_space"
] | Basic hypothesis to talk about a topological additive monoid or a topological additive
semigroup. A topological additive monoid over `M`, for example, is obtained by requiring both the
instances `add_monoid M` and `has_continuous_add M`.
Continuity in only the left/right argument can be stated using
`has_continuous_co... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_mul (M : Type u) [topological_space M] [has_mul M] : Prop | (continuous_mul : continuous (λ p : M × M, p.1 * p.2)) | class | 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",
"continuous_mul",
"topological_space"
] | Basic hypothesis to talk about a topological monoid or a topological semigroup.
A topological monoid over `M`, for example, is obtained by requiring both the instances `monoid M`
and `has_continuous_mul M`.
Continuity in only the left/right argument can be stated using
`has_continuous_const_smul α α`/`has_continuous_c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_mul : continuous (λp:M×M, p.1 * p.2) | has_continuous_mul.continuous_mul | lemma | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul.to_has_continuous_smul : has_continuous_smul M M | ⟨continuous_mul⟩ | instance | has_continuous_mul.to_has_continuous_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"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_mul.to_has_continuous_smul_op : has_continuous_smul Mᵐᵒᵖ M | ⟨show continuous ((λ p : M × M, p.1 * p.2) ∘ prod.swap ∘ prod.map mul_opposite.unop id), from
continuous_mul.comp $ continuous_swap.comp $ continuous.prod_map mul_opposite.continuous_unop
continuous_id⟩ | instance | has_continuous_mul.to_has_continuous_smul_op | 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.prod_map",
"has_continuous_smul",
"mul_opposite.continuous_unop",
"mul_opposite.unop",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.mul {f g : X → M} (hf : continuous f) (hg : continuous g) :
continuous (λx, f x * g x) | continuous_mul.comp (hf.prod_mk hg : _) | lemma | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul_left (a : M) : continuous (λ b:M, a * b) | continuous_const.mul continuous_id | lemma | continuous_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",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul_right (a : M) : continuous (λ b:M, b * a) | continuous_id.mul continuous_const | lemma | continuous_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",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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