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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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mk' (v : valuation R Γ₀) : valued R Γ₀ | { v := v,
to_uniform_space := @topological_add_group.to_uniform_space R _ v.subgroups_basis.topology _,
to_uniform_add_group := @topological_add_comm_group_is_uniform _ _ v.subgroups_basis.topology _,
is_topological_valuation :=
begin
letI := @topological_add_group.to_uniform_space R _ v.subgroups_basis.top... | def | valued.mk' | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"mk'",
"valuation",
"valued"
] | Alternative `valued` constructor for use when there is no preferred `uniform_space`
structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_zero :
(𝓝 (0 : R)).has_basis (λ _, true) (λ (γ : Γ₀ˣ), { x | v x < (γ : Γ₀) }) | by simp [filter.has_basis_iff, is_topological_valuation] | lemma | valued.has_basis_nhds_zero | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"filter.has_basis_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_uniformity :
(𝓤 R).has_basis (λ _, true) (λ (γ : Γ₀ˣ), { p : R × R | v (p.2 - p.1) < (γ : Γ₀) }) | begin
rw uniformity_eq_comap_nhds_zero,
exact (has_basis_nhds_zero R Γ₀).comap _,
end | lemma | valued.has_basis_uniformity | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_uniform_space_eq :
to_uniform_space = @topological_add_group.to_uniform_space R _ v.subgroups_basis.topology _ | uniform_space_eq
((has_basis_uniformity R Γ₀).eq_of_same_basis $ v.subgroups_basis.has_basis_nhds_zero.comap _) | lemma | valued.to_uniform_space_eq | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"uniform_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds {s : set R} {x : R} :
(s ∈ 𝓝 x) ↔ ∃ (γ : Γ₀ˣ), {y | (v (y - x) : Γ₀) < γ } ⊆ s | by simp only [← nhds_translation_add_neg x, ← sub_eq_add_neg, preimage_set_of_eq, exists_true_left,
((has_basis_nhds_zero R Γ₀).comap (λ y, y - x)).mem_iff] | lemma | valued.mem_nhds | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"exists_true_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_zero {s : set R} :
(s ∈ 𝓝 (0 : R)) ↔ ∃ γ : Γ₀ˣ, {x | v x < (γ : Γ₀) } ⊆ s | by simp only [mem_nhds, sub_zero] | lemma | valued.mem_nhds_zero | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loc_const {x : R} (h : (v x : Γ₀) ≠ 0) : {y : R | v y = v x} ∈ 𝓝 x | begin
rw mem_nhds,
rcases units.exists_iff_ne_zero.mpr h with ⟨γ, hx⟩,
use γ,
rw hx,
intros y y_in,
exact valuation.map_eq_of_sub_lt _ y_in
end | lemma | valued.loc_const | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"valuation.map_eq_of_sub_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_iff {F : filter R} :
cauchy F ↔ F.ne_bot ∧ ∀ γ : Γ₀ˣ, ∃ M ∈ F, ∀ x y ∈ M, (v (y - x) : Γ₀) < γ | begin
rw [to_uniform_space_eq, add_group_filter_basis.cauchy_iff],
apply and_congr iff.rfl,
simp_rw valued.v.subgroups_basis.mem_add_group_filter_basis_iff,
split,
{ intros h γ,
exact h _ (valued.v.subgroups_basis.mem_add_group_filter_basis _) },
{ rintros h - ⟨γ, rfl⟩,
exact h γ }
end | lemma | valued.cauchy_iff | topology.algebra | src/topology/algebra/valuation.lean | [
"topology.algebra.nonarchimedean.bases",
"topology.algebra.uniform_filter_basis",
"ring_theory.valuation.basic"
] | [
"add_group_filter_basis.cauchy_iff",
"cauchy",
"cauchy_iff",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)
(h : v (x - y) < min (γ * ((v y) * (v y))) (v y)) :
v (x⁻¹ - y⁻¹) < γ | begin
have hyp1 : v (x - y) < γ * ((v y) * (v y)),
from lt_of_lt_of_le h (min_le_left _ _),
have hyp1' : v (x - y) * ((v y) * (v y))⁻¹ < γ,
from mul_inv_lt_of_lt_mul₀ hyp1,
have hyp2 : v (x - y) < v y,
from lt_of_lt_of_le h (min_le_right _ _),
have key : v x = v y, from valuation.map_eq_of_sub_lt v ... | lemma | valuation.inversion_estimate | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"inv_mul_cancel",
"map_inv₀",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_inv_lt_of_lt_mul₀",
"mul_inv_rev",
"mul_one",
"mul_sub_left_distrib",
"one_mul",
"valuation.map_eq_of_sub_lt",
"valuation.map_mul",
"valuation.map_sub_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valued.topological_division_ring [valued K Γ₀] : topological_division_ring K | { continuous_at_inv₀ :=
begin
intros x x_ne s s_in,
cases valued.mem_nhds.mp s_in with γ hs, clear s_in,
rw [mem_map, valued.mem_nhds],
change ∃ (γ : Γ₀ˣ), {y : K | (v (y - x) : Γ₀) < γ} ⊆ {x : K | x⁻¹ ∈ s},
have vx_ne := (valuation.ne_zero_iff $ v).mpr x_ne,
let γ' := units.mk0 ... | instance | valued.topological_division_ring | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"mem_map",
"topological_division_ring",
"topological_ring",
"units.coe_mul",
"units.min_coe",
"units.mk0",
"valuation.inversion_estimate",
"valuation.ne_zero_iff",
"valued",
"valued.mem_nhds"
] | The topology coming from a valuation on a division ring makes it a topological division ring
[BouAC, VI.5.1 middle of Proposition 1] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valued_ring.separated [valued K Γ₀] : separated_space K | begin
rw separated_iff_t2,
apply topological_add_group.t2_space_of_zero_sep,
intros x x_ne,
refine ⟨{k | v k < v x}, _, λ h, lt_irrefl _ h⟩,
rw valued.mem_nhds,
have vx_ne := (valuation.ne_zero_iff $ v).mpr x_ne,
let γ' := units.mk0 _ vx_ne,
exact ⟨γ', λ y hy, by simpa using hy⟩,
end | instance | valued_ring.separated | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"separated_iff_t2",
"separated_space",
"units.mk0",
"valuation.ne_zero_iff",
"valued",
"valued.mem_nhds"
] | A valued division ring is separated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valued.continuous_valuation [valued K Γ₀] : continuous (v : K → Γ₀) | begin
rw continuous_iff_continuous_at,
intro x,
rcases eq_or_ne x 0 with rfl|h,
{ rw [continuous_at, map_zero, with_zero_topology.tendsto_zero],
intros γ hγ,
rw [filter.eventually, valued.mem_nhds_zero],
use [units.mk0 γ hγ, subset.rfl] },
{ have v_ne : (v x : Γ₀) ≠ 0, from (valuation.ne_zero_iff ... | lemma | valued.continuous_valuation | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"continuous",
"continuous_at",
"continuous_iff_continuous_at",
"eq_or_ne",
"filter.eventually",
"units.mk0",
"valuation.ne_zero_iff",
"valued",
"valued.loc_const",
"valued.mem_nhds_zero",
"with_zero_topology.tendsto_of_ne_zero",
"with_zero_topology.tendsto_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
completable : completable_top_field K | { nice := begin
rintros F hF h0,
have : ∃ (γ₀ : Γ₀ˣ) (M ∈ F), ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x,
{ rcases filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩,
rcases valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩,
existsi [γ₀, M, M_in],
intros x xM,
apply le_of_not_lt _,
intro hyp,
... | instance | valued.completable | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"completable_top_field",
"filter.inter_mem",
"mem_map",
"min_le_min",
"mul_assoc",
"mul_le_mul_left'",
"mul_le_mul_right'",
"units.coe_mul",
"units.min_coe",
"valuation.inversion_estimate",
"valuation.ne_zero_iff",
"valued.cauchy_iff",
"valued_ring.separated"
] | A valued field is completable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension : hat K → Γ₀ | completion.dense_inducing_coe.extend (v : K → Γ₀) | def | valued.extension | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [] | The extension of the valuation of a valued field to the completion of the field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_extension : continuous (valued.extension : hat K → Γ₀) | begin
refine completion.dense_inducing_coe.continuous_extend _,
intro x₀,
rcases eq_or_ne x₀ 0 with rfl|h,
{ refine ⟨0, _⟩,
erw [← completion.dense_inducing_coe.to_inducing.nhds_eq_comap],
exact valued.continuous_valuation.tendsto' 0 0 (map_zero v) },
{ have preimage_one : v ⁻¹' {(1 : Γ₀)} ∈ 𝓝 (1 : K... | lemma | valued.continuous_extension | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"compl_singleton_mem_nhds",
"continuous",
"continuous_const",
"eq_or_ne",
"filter.inter_mem",
"inv_mul_cancel",
"map_inv₀",
"mul_assoc",
"mul_comm",
"mul_inv",
"mul_inv_cancel",
"mul_ne_zero",
"mul_one",
"ne_of_mem_of_not_mem",
"nhds_prod_eq",
"one_mul",
"valuation.map_mul",
"valua... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_extends (x : K) : extension (x : hat K) = v x | begin
refine completion.dense_inducing_coe.extend_eq_of_tendsto _,
rw ← completion.dense_inducing_coe.nhds_eq_comap,
exact valued.continuous_valuation.continuous_at,
end | lemma | valued.extension_extends | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_valuation :
valuation (hat K) Γ₀ | { to_fun := valued.extension,
map_zero' := by { rw [← v.map_zero, ← valued.extension_extends (0 : K)], refl, },
map_one' := by { rw [← completion.coe_one, valued.extension_extends (1 : K)],
exact valuation.map_one _ },
map_mul' := λ x y, begin
apply completion.induction_on₂ x y,
{ have ... | def | valued.extension_valuation | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"cont",
"continuous",
"continuous_fst",
"continuous_snd",
"is_closed_eq",
"is_closed_le",
"le_max_iff",
"valuation",
"valuation.map_mul",
"valuation.map_one",
"valued.continuous_extension",
"valued.extension",
"valued.extension_extends"
] | the extension of a valuation on a division ring to its completion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_coe_completion_v_lt {γ : Γ₀ˣ} :
closure (coe '' { x : K | v x < (γ : Γ₀) }) = { x : hat K | extension_valuation x < (γ : Γ₀) } | begin
ext x,
let γ₀ := extension_valuation x,
suffices : γ₀ ≠ 0 → (x ∈ closure (coe '' { x : K | v x < (γ : Γ₀) }) ↔ γ₀ < (γ : Γ₀)),
{ cases eq_or_ne γ₀ 0,
{ simp only [h, (valuation.zero_iff _).mp h, mem_set_of_eq, valuation.map_zero, units.zero_lt,
iff_true],
apply subset_closure,
exac... | lemma | valued.closure_coe_completion_v_lt | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"closure",
"eq_or_ne",
"mem_closure_iff_nhds'",
"subset_closure",
"units.zero_lt",
"valuation.map_zero",
"valuation.zero_iff",
"with_zero_topology.singleton_mem_nhds_of_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valued_completion : valued (hat K) Γ₀ | { v := extension_valuation,
is_topological_valuation := λ s,
begin
suffices : has_basis (𝓝 (0 : hat K)) (λ _, true) (λ γ : Γ₀ˣ, { x | extension_valuation x < γ }),
{ rw this.mem_iff,
exact exists_congr (λ γ, by simp), },
simp_rw ← closure_coe_completion_v_lt,
exact (has_basis_nhds_zero K Γ₀).... | instance | valued.valued_completion | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [
"valued"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valued_completion_apply (x : K) : valued.v (x : hat K) = v x | extension_extends x | lemma | valued.valued_completion_apply | topology.algebra | src/topology/algebra/valued_field.lean | [
"topology.algebra.valuation",
"topology.algebra.with_zero_topology",
"topology.algebra.uniform_field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space : topological_space Γ₀ | topological_space.mk_of_nhds $ update pure 0 $ ⨅ γ ≠ 0, 𝓟 (Iio γ) | def | with_zero_topology.topological_space | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"topological_space",
"topological_space.mk_of_nhds",
"update"
] | The topology on a linearly ordered commutative group with a zero element adjoined.
A subset U is open if 0 ∉ U or if there is an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_eq_update : (𝓝 : Γ₀ → filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) | funext $ nhds_mk_of_nhds_single $ le_infi₂ $ λ γ h₀, le_principal_iff.2 $ zero_lt_iff.2 h₀ | lemma | with_zero_topology.nhds_eq_update | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"filter",
"le_infi₂",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) | by rw [nhds_eq_update, update_same] | lemma | with_zero_topology.nhds_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"update_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_zero : (𝓝 (0 : Γ₀)).has_basis (λ γ : Γ₀, γ ≠ 0) Iio | begin
rw [nhds_zero],
refine has_basis_binfi_principal _ ⟨1, one_ne_zero⟩,
exact directed_on_iff_directed.2 (directed_of_inf $ λ a b hab, Iio_subset_Iio hab)
end | lemma | with_zero_topology.has_basis_nhds_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"directed_of_inf"
] | In a linearly ordered group with zero element adjoined, `U` is a neighbourhood of `0` if and
only if there exists a nonzero element `γ₀` such that `Iio γ₀ ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Iio_mem_nhds_zero (hγ : γ ≠ 0) : Iio γ ∈ 𝓝 (0 : Γ₀) | has_basis_nhds_zero.mem_of_mem hγ | lemma | with_zero_topology.Iio_mem_nhds_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_zero_of_units (γ : Γ₀ˣ) : Iio ↑γ ∈ 𝓝 (0 : Γ₀) | Iio_mem_nhds_zero γ.ne_zero | lemma | with_zero_topology.nhds_zero_of_units | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`Iio (γ : Γ₀)` is a neighbourhood of `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_zero : tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ γ₀ ≠ 0, ∀ᶠ x in l, f x < γ₀ | by simp [nhds_zero] | lemma | with_zero_topology.tendsto_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_of_ne_zero {γ : Γ₀} (h₀ : γ ≠ 0) : 𝓝 γ = pure γ | by rw [nhds_eq_update, update_noteq h₀] | lemma | with_zero_topology.nhds_of_ne_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"update_noteq"
] | The neighbourhood filter of a nonzero element consists of all sets containing that
element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_coe_units (γ : Γ₀ˣ) : 𝓝 (γ : Γ₀) = pure (γ : Γ₀) | nhds_of_ne_zero γ.ne_zero | lemma | with_zero_topology.nhds_coe_units | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | The neighbourhood filter of an invertible element consists of all sets containing that
element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({γ} : set Γ₀) ∈ 𝓝 (γ : Γ₀) | by simp | lemma | with_zero_topology.singleton_mem_nhds_of_units | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | If `γ` is an invertible element of a linearly ordered group with zero element adjoined, then
`{γ}` is a neighbourhood of `γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : set Γ₀) ∈ 𝓝 (γ : Γ₀) | by simp [h] | lemma | with_zero_topology.singleton_mem_nhds_of_ne_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | If `γ` is a nonzero element of a linearly ordered group with zero element adjoined, then `{γ}`
is a neighbourhood of `γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
has_basis (𝓝 x) (λ i : unit, true) (λ i, {x}) | by { rw [nhds_of_ne_zero h], exact has_basis_pure _ } | lemma | with_zero_topology.has_basis_nhds_of_ne_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_units (γ : Γ₀ˣ) :
has_basis (𝓝 (γ : Γ₀)) (λ i : unit, true) (λ i, {γ}) | has_basis_nhds_of_ne_zero γ.ne_zero | lemma | with_zero_topology.has_basis_nhds_units | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ | by rw [nhds_of_ne_zero h, tendsto_pure] | lemma | with_zero_topology.tendsto_of_ne_zero | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_units {γ₀ : Γ₀ˣ} : tendsto f l (𝓝 (γ₀ : Γ₀)) ↔ ∀ᶠ x in l, f x = γ₀ | tendsto_of_ne_zero γ₀.ne_zero | lemma | with_zero_topology.tendsto_units | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ | by rcases eq_or_ne γ₁ 0 with rfl|h₀; simp [*, h.ne', Iio_mem_nhds_zero] | lemma | with_zero_topology.Iio_mem_nhds | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"Iio_mem_nhds",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff {s : set Γ₀} : is_open s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ ≠ 0, Iio γ ⊆ s | begin
rw [is_open_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)],
simp [nhds_of_ne_zero, imp_iff_not_or, has_basis_nhds_zero.mem_iff] { contextual := tt }
end | lemma | with_zero_topology.is_open_iff | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"and_forall_ne",
"imp_iff_not_or",
"is_open",
"is_open_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff {s : set Γ₀} : is_closed s ↔ (0 : Γ₀) ∈ s ∨ ∃ γ ≠ 0, s ⊆ Ici γ | by simp only [← is_open_compl_iff, is_open_iff, mem_compl_iff, not_not, ← compl_Ici,
compl_subset_compl] | lemma | with_zero_topology.is_closed_iff | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"is_closed",
"is_open_compl_iff",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Iio {a : Γ₀} : is_open (Iio a) | is_open_iff.mpr $ imp_iff_not_or.mp $ λ ha, ⟨a, ne_of_gt ha, subset.rfl⟩ | lemma | with_zero_topology.is_open_Iio | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"is_open",
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_closed_topology : order_closed_topology Γ₀ | { is_closed_le' :=
begin
simp only [← is_open_compl_iff, compl_set_of, not_le, is_open_iff_mem_nhds],
rintros ⟨a, b⟩ (hab : b < a),
rw [nhds_prod_eq, nhds_of_ne_zero (zero_le'.trans_lt hab).ne', pure_prod],
exact Iio_mem_nhds hab
end } | lemma | with_zero_topology.order_closed_topology | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"Iio_mem_nhds",
"is_closed_le'",
"is_open_compl_iff",
"is_open_iff_mem_nhds",
"nhds_prod_eq",
"order_closed_topology"
] | The topology on a linearly ordered group with zero element adjoined is compatible with the order
structure: the set `{p : Γ₀ × Γ₀ | p.1 ≤ p.2}` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t3_space : t3_space Γ₀ | { to_regular_space := regular_space.of_lift'_closure $ λ γ,
begin
rcases ne_or_eq γ 0 with h₀|rfl,
{ rw [nhds_of_ne_zero h₀, lift'_pure (monotone_closure Γ₀), closure_singleton,
principal_singleton] },
{ exact has_basis_nhds_zero.lift'_closure_eq_self
(λ x hx, is_closed_iff.2 $ o... | lemma | with_zero_topology.t3_space | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"closure_singleton",
"monotone_closure",
"ne_or_eq",
"regular_space.of_lift'_closure",
"t3_space"
] | The topology on a linearly ordered group with zero element adjoined is T₃. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_mul : has_continuous_mul Γ₀ | ⟨begin
rw continuous_iff_continuous_at,
rintros ⟨x, y⟩,
wlog hle : x ≤ y generalizing x y,
{ have := tendsto.comp (this y x (le_of_not_le hle)) (continuous_swap.tendsto (x,y)),
simpa only [mul_comm, function.comp, prod.swap], },
rcases eq_or_ne x 0 with rfl|hx; [rcases eq_or_ne y 0 with rfl|hy, skip],
{... | lemma | with_zero_topology.has_continuous_mul | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"continuous_at",
"continuous_iff_continuous_at",
"div_mul_cancel",
"div_ne_zero",
"eq_or_ne",
"has_continuous_mul",
"mul_comm",
"mul_lt_mul₀",
"mul_lt_right₀",
"mul_one",
"nhds_prod_eq",
"prod.swap",
"pure_le_nhds",
"zero_mul"
] | The topology on a linearly ordered group with zero element adjoined makes it a topological
monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_inv₀ : has_continuous_inv₀ Γ₀ | ⟨λ γ h, by { rw [continuous_at, nhds_of_ne_zero h], exact pure_le_nhds γ⁻¹ }⟩ | lemma | with_zero_topology.has_continuous_inv₀ | topology.algebra | src/topology/algebra/with_zero_topology.lean | [
"algebra.order.with_zero",
"topology.algebra.order.field"
] | [
"continuous_at",
"has_continuous_inv₀",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.mul_left (a : G) : G ≃ₜ G | { continuous_to_fun := continuous_const.mul continuous_id,
continuous_inv_fun := continuous_const.mul continuous_id,
.. equiv.mul_left a } | def | homeomorph.mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_id",
"equiv.mul_left"
] | Multiplication from the left in a topological group as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.coe_mul_left (a : G) : ⇑(homeomorph.mul_left a) = (*) a | rfl | lemma | homeomorph.coe_mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.mul_left_symm (a : G) : (homeomorph.mul_left a).symm = homeomorph.mul_left a⁻¹ | by { ext, refl } | lemma | homeomorph.mul_left_symm | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_mul_left (a : G) : is_open_map (λ x, a * x) | (homeomorph.mul_left a).is_open_map | lemma | is_open_map_mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_left",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.left_coset {U : set G} (h : is_open U) (x : G) : is_open (left_coset x U) | is_open_map_mul_left x _ h | lemma | is_open.left_coset | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_map_mul_left",
"left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_mul_left (a : G) : is_closed_map (λ x, a * x) | (homeomorph.mul_left a).is_closed_map | lemma | is_closed_map_mul_left | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_left",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.left_coset {U : set G} (h : is_closed U) (x : G) : is_closed (left_coset x U) | is_closed_map_mul_left x _ h | lemma | is_closed.left_coset | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed",
"is_closed_map_mul_left",
"left_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.mul_right (a : G) :
G ≃ₜ G | { continuous_to_fun := continuous_id.mul continuous_const,
continuous_inv_fun := continuous_id.mul continuous_const,
.. equiv.mul_right a } | def | homeomorph.mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_const",
"equiv.mul_right"
] | Multiplication from the right in a topological group as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.coe_mul_right (a : G) : ⇑(homeomorph.mul_right a) = λ g, g * a | rfl | lemma | homeomorph.coe_mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.mul_right_symm (a : G) :
(homeomorph.mul_right a).symm = homeomorph.mul_right a⁻¹ | by { ext, refl } | lemma | homeomorph.mul_right_symm | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_mul_right (a : G) : is_open_map (λ x, x * a) | (homeomorph.mul_right a).is_open_map | lemma | is_open_map_mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.right_coset {U : set G} (h : is_open U) (x : G) : is_open (right_coset U x) | is_open_map_mul_right x _ h | lemma | is_open.right_coset | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open",
"is_open_map_mul_right",
"right_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_mul_right (a : G) : is_closed_map (λ x, x * a) | (homeomorph.mul_right a).is_closed_map | lemma | is_closed_map_mul_right | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.mul_right",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.right_coset {U : set G} (h : is_closed U) (x : G) : is_closed (right_coset U x) | is_closed_map_mul_right x _ h | lemma | is_closed.right_coset | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed",
"is_closed_map_mul_right",
"right_coset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_of_open_singleton_one (h : is_open ({1} : set G)) : discrete_topology G | begin
rw ← singletons_open_iff_discrete,
intro g,
suffices : {g} = (λ (x : G), g⁻¹ * x) ⁻¹' {1},
{ rw this, exact (continuous_mul_left (g⁻¹)).is_open_preimage _ h, },
simp only [mul_one, set.preimage_mul_left_singleton, eq_self_iff_true,
inv_inv, set.singleton_eq_singleton_iff],
end | lemma | discrete_topology_of_open_singleton_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_mul_left",
"discrete_topology",
"inv_inv",
"is_open",
"mul_one",
"set.preimage_mul_left_singleton",
"set.singleton_eq_singleton_iff",
"singletons_open_iff_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_iff_open_singleton_one : discrete_topology G ↔ is_open ({1} : set G) | ⟨λ h, forall_open_iff_discrete.mpr h {1}, discrete_topology_of_open_singleton_one⟩ | lemma | discrete_topology_iff_open_singleton_one | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"discrete_topology",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_neg (G : Type u) [topological_space G] [has_neg G] : Prop | (continuous_neg : continuous (λ a : G, -a)) | class | has_continuous_neg | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"topological_space"
] | Basic hypothesis to talk about a topological additive group. A topological additive group
over `M`, for example, is obtained by requiring the instances `add_group M` and
`has_continuous_add M` and `has_continuous_neg M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_inv (G : Type u) [topological_space G] [has_inv G] : Prop | (continuous_inv : continuous (λ a : G, a⁻¹)) | class | has_continuous_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"topological_space"
] | Basic hypothesis to talk about a topological group. A topological group over `M`, for example,
is obtained by requiring the instances `group M` and `has_continuous_mul M` and
`has_continuous_inv M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_inv {s : set G} : continuous_on has_inv.inv s | continuous_inv.continuous_on | lemma | continuous_on_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_inv {s : set G} {x : G} : continuous_within_at has_inv.inv s x | continuous_inv.continuous_within_at | lemma | continuous_within_at_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_inv {x : G} : continuous_at has_inv.inv x | continuous_inv.continuous_at | lemma | continuous_at_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inv (a : G) : tendsto has_inv.inv (𝓝 a) (𝓝 (a⁻¹)) | continuous_at_inv | lemma | tendsto_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inv {f : α → G} {l : filter α} {y : G} (h : tendsto f l (𝓝 y)) :
tendsto (λ x, (f x)⁻¹) l (𝓝 y⁻¹) | (continuous_inv.tendsto y).comp h | lemma | filter.tendsto.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"filter"
] | If a function converges to a value in a multiplicative topological group, then its inverse
converges to the inverse of this value. For the version in normed fields assuming additionally
that the limit is nonzero, use `tendsto.inv'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.inv (hf : continuous f) : continuous (λx, (f x)⁻¹) | continuous_inv.comp hf | lemma | continuous.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.inv (hf : continuous_at f x) : continuous_at (λ x, (f x)⁻¹) x | continuous_at_inv.comp hf | lemma | continuous_at.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.inv (hf : continuous_on f s) : continuous_on (λx, (f x)⁻¹) s | continuous_inv.comp_continuous_on hf | lemma | continuous_on.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.inv (hf : continuous_within_at f s x) :
continuous_within_at (λ x, (f x)⁻¹) s x | hf.inv | lemma | continuous_within_at.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_continuous_inv {C : ι → Type*} [∀ i, topological_space (C i)]
[∀ i, has_inv (C i)] [∀ i, has_continuous_inv (C i)] : has_continuous_inv (Π i, C i) | { continuous_inv := continuous_pi (λ i, (continuous_apply i).inv) } | instance | pi.has_continuous_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_apply",
"continuous_pi",
"has_continuous_inv",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_continuous_inv' : has_continuous_inv (ι → G) | pi.has_continuous_inv | instance | pi.has_continuous_inv' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"has_continuous_inv",
"pi.has_continuous_inv"
] | A version of `pi.has_continuous_inv` for non-dependent functions. It is needed because sometimes
Lean fails to use `pi.has_continuous_inv` for non-dependent functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_inv_of_discrete_topology [topological_space H]
[has_inv H] [discrete_topology H] : has_continuous_inv H | ⟨continuous_of_discrete_topology⟩ | instance | has_continuous_inv_of_discrete_topology | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"discrete_topology",
"has_continuous_inv",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_map_inv [has_inv G₁] [has_inv G₂] [has_continuous_inv G₂] :
is_closed {f : G₁ → G₂ | ∀ x, f x⁻¹ = (f x)⁻¹ } | begin
simp only [set_of_forall],
refine is_closed_Inter (λ i, is_closed_eq (continuous_apply _) (continuous_apply _).inv),
end | lemma | is_closed_set_of_map_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_apply",
"has_continuous_inv",
"is_closed",
"is_closed_Inter",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.inv (hs : is_compact s) : is_compact s⁻¹ | by { rw [← image_inv], exact hs.image continuous_inv } | lemma | is_compact.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.inv (G : Type*) [topological_space G] [has_involutive_inv G]
[has_continuous_inv G] : G ≃ₜ G | { continuous_to_fun := continuous_inv,
continuous_inv_fun := continuous_inv,
.. equiv.inv G } | def | homeomorph.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"equiv.inv",
"has_continuous_inv",
"has_involutive_inv",
"topological_space"
] | Inversion in a topological group as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_inv : is_open_map (has_inv.inv : G → G) | (homeomorph.inv _).is_open_map | lemma | is_open_map_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.inv",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_inv : is_closed_map (has_inv.inv : G → G) | (homeomorph.inv _).is_closed_map | lemma | is_closed_map_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"homeomorph.inv",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.inv (hs : is_open s) : is_open s⁻¹ | hs.preimage continuous_inv | lemma | is_open.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.inv (hs : is_closed s) : is_closed s⁻¹ | hs.preimage continuous_inv | lemma | is_closed.inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_closure : ∀ s : set G, (closure s)⁻¹ = closure s⁻¹ | (homeomorph.inv G).preimage_closure | lemma | inv_closure | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"closure",
"homeomorph.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inv_Inf {ts : set (topological_space G)}
(h : Π t ∈ ts, @has_continuous_inv G t _) :
@has_continuous_inv G (Inf ts) _ | { continuous_inv := continuous_Inf_rng.2 (λ t ht, continuous_Inf_dom ht
(@has_continuous_inv.continuous_inv G t _ (h t ht))) } | lemma | has_continuous_inv_Inf | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_Inf_dom",
"has_continuous_inv",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inv_infi {ts' : ι' → topological_space G}
(h' : Π i, @has_continuous_inv G (ts' i) _) :
@has_continuous_inv G (⨅ i, ts' i) _ | by {rw ← Inf_range, exact has_continuous_inv_Inf (set.forall_range_iff.mpr h')} | lemma | has_continuous_inv_infi | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"Inf_range",
"has_continuous_inv",
"has_continuous_inv_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inv_inf {t₁ t₂ : topological_space G}
(h₁ : @has_continuous_inv G t₁ _) (h₂ : @has_continuous_inv G t₂ _) :
@has_continuous_inv G (t₁ ⊓ t₂) _ | by { rw inf_eq_infi, refine has_continuous_inv_infi (λ b, _), cases b; assumption } | lemma | has_continuous_inv_inf | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"has_continuous_inv",
"has_continuous_inv_infi",
"inf_eq_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.has_continuous_inv {G H : Type*} [has_inv G] [has_inv H]
[topological_space G] [topological_space H] [has_continuous_inv H] {f : G → H} (hf : inducing f)
(hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : has_continuous_inv G | ⟨hf.continuous_iff.2 $ by simpa only [(∘), hf_inv] using hf.continuous.inv⟩ | lemma | inducing.has_continuous_inv | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"has_continuous_inv",
"inducing",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_add_group (G : Type u) [topological_space G] [add_group G]
extends has_continuous_add G, has_continuous_neg G : Prop | class | topological_add_group | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"add_group",
"has_continuous_add",
"has_continuous_neg",
"topological_space"
] | A topological (additive) group is a group in which the addition and negation operations are
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group (G : Type*) [topological_space G] [group G]
extends has_continuous_mul G, has_continuous_inv G : Prop | class | topological_group | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"group",
"has_continuous_inv",
"has_continuous_mul",
"topological_space"
] | A topological group is a group in which the multiplication and inversion operations are
continuous.
When you declare an instance that does not already have a `uniform_space` instance,
you should also provide an instance of `uniform_space` and `uniform_group` using
`topological_group.to_uniform_space` and `topological_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_act.units_has_continuous_const_smul {M} [monoid M] [topological_space M]
[has_continuous_mul M] :
has_continuous_const_smul (conj_act Mˣ) M | ⟨λ m, (continuous_const.mul continuous_id).mul continuous_const⟩ | instance | conj_act.units_has_continuous_const_smul | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"conj_act",
"continuous_id",
"has_continuous_const_smul",
"has_continuous_mul",
"monoid",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group.continuous_conj_prod [has_continuous_inv G] :
continuous (λ g : G × G, g.fst * g.snd * g.fst⁻¹) | continuous_mul.mul (continuous_inv.comp continuous_fst) | lemma | topological_group.continuous_conj_prod | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_fst",
"has_continuous_inv"
] | Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.continuous_conj (g : G) : continuous (λ (h : G), g * h * g⁻¹) | (continuous_mul_right g⁻¹).comp (continuous_mul_left g) | lemma | topological_group.continuous_conj | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_mul_left",
"continuous_mul_right"
] | Conjugation by a fixed element is continuous when `mul` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group.continuous_conj' [has_continuous_inv G]
(h : G) : continuous (λ (g : G), g * h * g⁻¹) | (continuous_mul_right h).mul continuous_inv | lemma | topological_group.continuous_conj' | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_mul_right",
"has_continuous_inv"
] | Conjugation acting on fixed element of the group is continuous when both `mul` and
`inv` are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_zpow : ∀ z : ℤ, continuous (λ a : G, a ^ z) | | (int.of_nat n) := by simpa using continuous_pow n
| -[1+n] := by simpa using (continuous_pow (n + 1)).inv | lemma | continuous_zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group.has_continuous_const_smul_int {A} [add_group A] [topological_space A]
[topological_add_group A] : has_continuous_const_smul ℤ A | ⟨continuous_zsmul⟩ | instance | add_group.has_continuous_const_smul_int | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"add_group",
"has_continuous_const_smul",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group.has_continuous_smul_int {A} [add_group A] [topological_space A]
[topological_add_group A] : has_continuous_smul ℤ A | ⟨continuous_uncurry_of_discrete_topology continuous_zsmul⟩ | instance | add_group.has_continuous_smul_int | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"add_group",
"has_continuous_smul",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.zpow {f : α → G} (h : continuous f) (z : ℤ) :
continuous (λ b, (f b) ^ z) | (continuous_zpow z).comp h | lemma | continuous.zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous",
"continuous_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_zpow {s : set G} (z : ℤ) : continuous_on (λ x, x ^ z) s | (continuous_zpow z).continuous_on | lemma | continuous_on_zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_on",
"continuous_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_zpow (x : G) (z : ℤ) : continuous_at (λ x, x ^ z) x | (continuous_zpow z).continuous_at | lemma | continuous_at_zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at",
"continuous_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.zpow {α} {l : filter α} {f : α → G} {x : G} (hf : tendsto f l (𝓝 x)) (z : ℤ) :
tendsto (λ x, f x ^ z) l (𝓝 (x ^ z)) | (continuous_at_zpow _ _).tendsto.comp hf | lemma | filter.tendsto.zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at_zpow",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.zpow {f : α → G} {x : α} {s : set α} (hf : continuous_within_at f s x)
(z : ℤ) : continuous_within_at (λ x, f x ^ z) s x | hf.zpow z | lemma | continuous_within_at.zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.zpow {f : α → G} {x : α} (hf : continuous_at f x) (z : ℤ) :
continuous_at (λ x, f x ^ z) x | hf.zpow z | lemma | continuous_at.zpow | topology.algebra.group | src/topology/algebra/group/basic.lean | [
"group_theory.group_action.conj_act",
"group_theory.group_action.quotient",
"group_theory.quotient_group",
"topology.algebra.monoid",
"topology.algebra.constructions"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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