statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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subgroup.properly_discontinuous_smul_opposite_of_tendsto_cofinite
(S : subgroup G) (hS : tendsto S.subtype cofinite (cocompact G)) :
properly_discontinuous_smul S.opposite G | { finite_disjoint_inter_image := begin
intros K L hK hL,
have : continuous (λ p : G × G, (p.1⁻¹, p.2)) := continuous_inv.prod_map continuous_id,
have H : set.finite _ :=
hS ((hK.prod hL).image (continuous_mul.comp this)).compl_mem_cocompact,
rw [preimage_compl, compl_compl] at H,
convert H,
... | lemma | subgroup.properly_discontinuous_smul_opposite_of_tendsto_cofinite | 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"
] | [
"compl_compl",
"continuous",
"continuous_id",
"properly_discontinuous_smul",
"set.finite",
"set.op_smul_inter_ne_empty_iff",
"subgroup"
] | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the right, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`discrete_topology`.)
If `G` is Hausdorff, this can be combined with `t2_space_of_properly_discontinuous_smul_of_t2_space`
to show that the q... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_open_separated_mul_right {K U : set G} (hK : is_compact K) (hU : is_open U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), K * V ⊆ U | begin
apply hK.induction_on,
{ exact ⟨univ, by simp⟩ },
{ rintros s t hst ⟨V, hV, hV'⟩,
exact ⟨V, hV, (mul_subset_mul_right hst).trans hV'⟩ },
{ rintros s t ⟨V, V_in, hV'⟩ ⟨W, W_in, hW'⟩,
use [V ∩ W, inter_mem V_in W_in],
rw union_mul,
exact union_subset ((mul_subset_mul_left (V.inter_subset_le... | lemma | compact_open_separated_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"
] | [
"is_compact",
"is_open",
"mem_map",
"mem_nhds_within_of_mem_nhds",
"nhds_prod_eq",
"tendsto_mul"
] | Given a compact set `K` inside an open set `U`, there is a open neighborhood `V` of `1`
such that `K * V ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_open_separated_mul_left {K U : set G} (hK : is_compact K) (hU : is_open U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U | begin
rcases compact_open_separated_mul_right (hK.image continuous_op) (op_homeomorph.is_open_map U hU)
(image_subset op hKU) with ⟨V, (hV : V ∈ 𝓝 (op (1 : G))), hV' : op '' K * V ⊆ op '' U⟩,
refine ⟨op ⁻¹' V, continuous_op.continuous_at hV, _⟩,
rwa [← image_preimage_eq V op_surjective, ← image_op_mul, image... | lemma | compact_open_separated_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"
] | [
"compact_open_separated_mul_right",
"is_compact",
"is_open"
] | Given a compact set `K` inside an open set `U`, there is a open neighborhood `V` of `1`
such that `V * K ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_covered_by_mul_left_translates {K V : set G} (hK : is_compact K)
(hV : (interior V).nonempty) : ∃ t : finset G, K ⊆ ⋃ g ∈ t, (λ h, g * h) ⁻¹' V | begin
obtain ⟨t, ht⟩ : ∃ t : finset G, K ⊆ ⋃ x ∈ t, interior (((*) x) ⁻¹' V),
{ refine hK.elim_finite_subcover (λ x, interior $ ((*) x) ⁻¹' V) (λ x, is_open_interior) _,
cases hV with g₀ hg₀,
refine λ g hg, mem_Union.2 ⟨g₀ * g⁻¹, _⟩,
refine preimage_interior_subset_interior_preimage (continuous_const.mu... | lemma | compact_covered_by_mul_left_translates | 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",
"finset",
"interior",
"inv_mul_cancel_right",
"is_compact",
"is_open_interior",
"preimage_interior_subset_interior_preimage"
] | A compact set is covered by finitely many left multiplicative translates of a set
with non-empty interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separable_locally_compact_group.sigma_compact_space
[separable_space G] [locally_compact_space G] : sigma_compact_space G | begin
obtain ⟨L, hLc, hL1⟩ := exists_compact_mem_nhds (1 : G),
refine ⟨⟨λ n, (λ x, x * dense_seq G n) ⁻¹' L, _, _⟩⟩,
{ intro n, exact (homeomorph.mul_right _).is_compact_preimage.mpr hLc },
{ refine Union_eq_univ_iff.2 (λ x, _),
obtain ⟨_, ⟨n, rfl⟩, hn⟩ : (range (dense_seq G) ∩ (λ y, x * y) ⁻¹' L).nonempty,... | instance | separable_locally_compact_group.sigma_compact_space | 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_at",
"exists_compact_mem_nhds",
"homeomorph.mul_left",
"homeomorph.mul_right",
"locally_compact_space",
"sigma_compact_space"
] | Every locally compact separable topological group is σ-compact.
Note: this is not true if we drop the topological group hypothesis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_disjoint_smul_of_is_compact [noncompact_space G] {K L : set G}
(hK : is_compact K) (hL : is_compact L) : ∃ (g : G), disjoint K (g • L) | begin
have A : ¬ (K * L⁻¹ = univ), from (hK.mul hL.inv).ne_univ,
obtain ⟨g, hg⟩ : ∃ g, g ∉ K * L⁻¹,
{ contrapose! A, exact eq_univ_iff_forall.2 A },
refine ⟨g, _⟩,
apply disjoint_left.2 (λ a ha h'a, hg _),
rcases h'a with ⟨b, bL, rfl⟩,
refine ⟨g * b, b⁻¹, ha, by simpa only [set.mem_inv, inv_inv] using bL,... | lemma | exists_disjoint_smul_of_is_compact | 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"
] | [
"disjoint",
"inv_inv",
"is_compact",
"mul_inv_cancel_right",
"noncompact_space",
"set.mem_inv",
"smul_eq_mul"
] | Given two compact sets in a noncompact topological group, there is a translate of the second
one that is disjoint from the first one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_is_compact_is_closed_nhds_of_group [locally_compact_space G]
{U : set G} (hU : U ∈ 𝓝 (1 : G)) :
∃ (K : set G), is_compact K ∧ is_closed K ∧ K ⊆ U ∧ (1 : G) ∈ interior K | begin
obtain ⟨L, Lint, LU, Lcomp⟩ : ∃ (L : set G) (H : L ∈ 𝓝 (1 : G)), L ⊆ U ∧ is_compact L,
from local_compact_nhds hU,
obtain ⟨V, Vnhds, hV⟩ : ∃ V ∈ 𝓝 (1 : G), ∀ (v ∈ V) (w ∈ V), v * w ∈ L,
{ have : ((λ p : G × G, p.1 * p.2) ⁻¹' L) ∈ 𝓝 ((1, 1) : G × G),
{ refine continuous_at_fst.mul continuous_at_sn... | lemma | local_is_compact_is_closed_nhds_of_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"
] | [
"closure",
"continuous_at_snd",
"div_eq_mul_inv",
"interior",
"interior_mono",
"interior_subset",
"is_closed",
"is_closed_closure",
"is_compact",
"is_compact_of_is_closed_subset",
"local_compact_nhds",
"locally_compact_space",
"mul_one",
"nhds_prod_eq",
"one_mul",
"subset_closure"
] | In a locally compact group, any neighborhood of the identity contains a compact closed
neighborhood of the identity, even without separation assumptions on the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y | calc 𝓝 (x * y) = map ((*) x) (map (λ a, a * y) (𝓝 1 * 𝓝 1)) : by simp
... = map₂ (λ a b, x * (a * b * y)) (𝓝 1) (𝓝 1) : by rw [← map₂_mul, map_map₂, map_map₂]
... = map₂ (λ a b, x * a * (b * y)) (𝓝 1) (𝓝 1) : by simp only [mul_assoc]
... = 𝓝 x * 𝓝 y : by rw [← map_mul_left_nhds_one x, ← map_mul_right_nhds_one ... | lemma | nhds_mul | 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"
] | [
"map_mul_left_nhds_one",
"map_mul_right_nhds_one",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mul_hom : G →ₙ* (filter G) | { to_fun := 𝓝,
map_mul' := λ_ _, nhds_mul _ _ } | def | nhds_mul_hom | 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",
"nhds_mul"
] | On a topological group, `𝓝 : G → filter G` can be promoted to a `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.has_continuous_const_smul : has_continuous_const_smul G (G ⧸ Γ) | { continuous_const_smul := λ g,
by convert ((@continuous_const _ _ _ _ g).mul continuous_id).quotient_map' _ } | instance | quotient_group.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"
] | [
"continuous_const",
"continuous_id",
"has_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_group.continuous_smul₁ (x : G ⧸ Γ) : continuous (λ g : G, g • x) | begin
induction x using quotient_group.induction_on,
exact continuous_quotient_mk.comp (continuous_mul_right x)
end | lemma | quotient_group.continuous_smul₁ | 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",
"quotient_group.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_group.second_countable_topology [second_countable_topology G] :
second_countable_topology (G ⧸ Γ) | has_continuous_const_smul.second_countable_topology | instance | quotient_group.second_countable_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"
] | [
"has_continuous_const_smul.second_countable_topology"
] | The quotient of a second countable topological group by a subgroup is second countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_units_homeomorph [group G] [topological_space G] [has_continuous_inv G] : G ≃ₜ Gˣ | { to_equiv := to_units.to_equiv,
continuous_to_fun := units.continuous_iff.2 ⟨continuous_id, continuous_inv⟩,
continuous_inv_fun := units.continuous_coe } | def | to_units_homeomorph | 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",
"topological_space",
"units.continuous_coe"
] | If `G` is a group with topological `⁻¹`, then it is homeomorphic to its units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.prod_units : (α × β)ˣ ≃ₜ (αˣ × βˣ) | { continuous_to_fun := (continuous_fst.units_map (monoid_hom.fst α β)).prod_mk
(continuous_snd.units_map (monoid_hom.snd α β)),
continuous_inv_fun := units.continuous_iff.2 ⟨continuous_coe.fst'.prod_mk continuous_coe.snd',
continuous_coe_inv.fst'.prod_mk continuous_coe_inv.snd'⟩,
to_equiv := mul_equiv.prod... | def | units.homeomorph.prod_units | 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"
] | [
"monoid_hom.fst",
"monoid_hom.snd"
] | The topological group isomorphism between the units of a product of two monoids, and the product
of the units of each monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_group_Inf {ts : set (topological_space G)}
(h : ∀ t ∈ ts, @topological_group G t _) :
@topological_group G (Inf ts) _ | { to_has_continuous_inv := @has_continuous_inv_Inf _ _ _ $
λ t ht, @topological_group.to_has_continuous_inv G t _ $ h t ht,
to_has_continuous_mul := @has_continuous_mul_Inf _ _ _ $
λ t ht, @topological_group.to_has_continuous_mul G t _ $ h t ht } | lemma | topological_group_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_Inf",
"has_continuous_mul_Inf",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group_infi {ts' : ι → topological_space G}
(h' : ∀ i, @topological_group G (ts' i) _) :
@topological_group G (⨅ i, ts' i) _ | by { rw ← Inf_range, exact topological_group_Inf (set.forall_range_iff.mpr h') } | lemma | topological_group_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",
"topological_group",
"topological_group_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_group_inf {t₁ t₂ : topological_space G}
(h₁ : @topological_group G t₁ _) (h₂ : @topological_group G t₂ _) :
@topological_group G (t₁ ⊓ t₂) _ | by { rw inf_eq_infi, refine topological_group_infi (λ b, _), cases b; assumption } | lemma | topological_group_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"
] | [
"inf_eq_infi",
"topological_group",
"topological_group_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_topology (α : Type u) [group α]
extends topological_space α, topological_group α : Type u | structure | group_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"
] | [
"group",
"topological_group",
"topological_space"
] | A group topology on a group `α` is a topology for which multiplication and inversion
are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_topology (α : Type u) [add_group α]
extends topological_space α, topological_add_group α : Type u | structure | add_group_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"
] | [
"add_group",
"topological_add_group",
"topological_space"
] | An additive group topology on an additive group `α` is a topology for which addition and
negation are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mul' (g : group_topology α) :
by haveI | g.to_topological_space; exact continuous (λ p : α × α, p.1 * p.2) :=
begin
letI := g.to_topological_space,
haveI := g.to_topological_group,
exact continuous_mul,
end | lemma | group_topology.continuous_mul' | 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",
"group_topology"
] | A version of the global `continuous_mul` suitable for dot notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inv' (g : group_topology α) :
by haveI | g.to_topological_space; exact continuous (has_inv.inv : α → α) :=
begin
letI := g.to_topological_space,
haveI := g.to_topological_group,
exact continuous_inv,
end | lemma | group_topology.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",
"group_topology"
] | A version of the global `continuous_inv` suitable for dot notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_topological_space_injective :
function.injective (to_topological_space : group_topology α → topological_space α) | λ f g h, by { cases f, cases g, congr' } | lemma | group_topology.to_topological_space_injective | 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_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext' {f g : group_topology α} (h : f.is_open = g.is_open) : f = g | to_topological_space_injective $ topological_space_eq h | lemma | group_topology.ext' | 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_topology",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_le {x y : group_topology α} :
x.to_topological_space ≤ y.to_topological_space ↔ x ≤ y | iff.rfl | lemma | group_topology.to_topological_space_le | 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_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_top :
(⊤ : group_topology α).to_topological_space = ⊤ | rfl | lemma | group_topology.to_topological_space_top | 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_topology",
"to_topological_space_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_bot :
(⊥ : group_topology α).to_topological_space = ⊥ | rfl | lemma | group_topology.to_topological_space_bot | 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_topology",
"to_topological_space_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_inf (x y : group_topology α) :
(x ⊓ y).to_topological_space = x.to_topological_space ⊓ y.to_topological_space | rfl | lemma | group_topology.to_topological_space_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"
] | [
"group_topology",
"to_topological_space_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_Inf (s : set (group_topology α)) :
(Inf s).to_topological_space = Inf (to_topological_space '' s) | rfl | lemma | group_topology.to_topological_space_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"
] | [
"group_topology",
"to_topological_space_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_infi {ι} (s : ι → group_topology α) :
(⨅ i, s i).to_topological_space = ⨅ i, (s i).to_topological_space | congr_arg Inf (range_comp _ _).symm | lemma | group_topology.to_topological_space_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"
] | [
"group_topology",
"to_topological_space_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced {α β : Type*} [t : topological_space α] [group β] (f : α → β) :
group_topology β | Inf {b : group_topology β | (topological_space.coinduced f t) ≤ b.to_topological_space} | def | group_topology.coinduced | 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",
"group_topology",
"topological_space",
"topological_space.coinduced"
] | Given `f : α → β` and a topology on `α`, the coinduced group topology on `β` is the finest
topology such that `f` is continuous and `β` is a topological group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coinduced_continuous {α β : Type*} [t : topological_space α] [group β]
(f : α → β) : cont t (coinduced f).to_topological_space f | begin
rw [continuous_Inf_rng],
rintros _ ⟨t', ht', rfl⟩,
exact continuous_iff_coinduced_le.2 ht'
end | lemma | group_topology.coinduced_continuous | 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"
] | [
"cont",
"continuous_Inf_rng",
"group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.positive_compacts.locally_compact_space_of_group
[t2_space G] (K : positive_compacts G) :
locally_compact_space G | begin
refine locally_compact_of_compact_nhds (λ x, _),
obtain ⟨y, hy⟩ := K.interior_nonempty,
let F := homeomorph.mul_left (x * y⁻¹),
refine ⟨F '' K, _, K.is_compact.image F.continuous⟩,
suffices : F.symm ⁻¹' K ∈ 𝓝 x, by { convert this, apply equiv.image_eq_preimage },
apply continuous_at.preimage_mem_nhds... | lemma | topological_space.positive_compacts.locally_compact_space_of_group | topology.algebra.group | src/topology/algebra/group/compact.lean | [
"topology.algebra.group.basic",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous_at.preimage_mem_nhds",
"equiv.image_eq_preimage",
"homeomorph.mul_left",
"homeomorph.mul_left_symm",
"locally_compact_of_compact_nhds",
"locally_compact_space",
"t2_space"
] | Every separated topological group in which there exists a compact set with nonempty interior
is locally compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.has_continuous_smul [locally_compact_space G] :
has_continuous_smul G (G ⧸ Γ) | { continuous_smul := begin
let F : G × G ⧸ Γ → G ⧸ Γ := λ p, p.1 • p.2,
change continuous F,
have H : continuous (F ∘ (λ p : G × G, (p.1, quotient_group.mk p.2))),
{ change continuous (λ p : G × G, quotient_group.mk (p.1 * p.2)),
refine continuous_coinduced_rng.comp continuous_mul },
exact quo... | instance | quotient_group.has_continuous_smul | topology.algebra.group | src/topology/algebra/group/compact.lean | [
"topology.algebra.group.basic",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous",
"continuous_mul",
"has_continuous_smul",
"locally_compact_space",
"quotient_group.mk",
"quotient_map.continuous_lift_prod_right",
"quotient_map_quotient_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum (f : β → α) (a : α) : Prop | tendsto (λs:finset β, ∑ b in s, f b) at_top (𝓝 a) | def | has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset"
] | Infinite sum on a topological monoid
The `at_top` filter on `finset β` is the limit of all finite sets towards the entire type. So we sum
up bigger and bigger sets. This sum operation is invariant under reordering. In particular,
the function `ℕ → ℝ` sending `n` to `(-1)^n / (n+1)` does not have a
sum for this definit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable (f : β → α) : Prop | ∃a, has_sum f a | def | summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | `summable f` means that `f` has some (infinite) sum. Use `tsum` to get the value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum {β} (f : β → α) | if h : summable f then classical.some h else 0 | def | tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | `∑' i, f i` is the sum of `f` it exists, or 0 otherwise | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable.has_sum (ha : summable f) : has_sum f (∑'b, f b) | by simp [ha, tsum]; exact some_spec ha | lemma | summable.has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.summable (h : has_sum f a) : summable f | ⟨a, h⟩ | lemma | has_sum.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_zero : has_sum (λb, 0 : β → α) 0 | by simp [has_sum, tendsto_const_nhds] | lemma | has_sum_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"tendsto_const_nhds"
] | Constant zero function has sum `0` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_empty [is_empty β] : has_sum f 0 | by convert has_sum_zero | lemma | has_sum_empty | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_zero",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_zero : summable (λb, 0 : β → α) | has_sum_zero.summable | lemma | summable_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_empty [is_empty β] : summable f | has_sum_empty.summable | lemma | summable_empty | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"is_empty",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_zero_of_not_summable (h : ¬ summable f) : ∑'b, f b = 0 | by simp [tsum, h] | lemma | tsum_eq_zero_of_not_summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_congr (hfg : ∀b, f b = g b) :
summable f ↔ summable g | iff_of_eq (congr_arg summable $ funext hfg) | lemma | summable_congr | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"iff_of_eq",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.congr (hf : summable f) (hfg : ∀b, f b = g b) :
summable g | (summable_congr hfg).mp hf | lemma | summable.congr | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"summable_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.has_sum_of_sum_eq {g : γ → α}
(h_eq : ∀u:finset γ, ∃v:finset β, ∀v', v ⊆ v' → ∃u', u ⊆ u' ∧ ∑ x in u', g x = ∑ b in v', f b)
(hf : has_sum g a) :
has_sum f a | le_trans (map_at_top_finset_sum_le_of_sum_eq h_eq) hf | lemma | has_sum.has_sum_of_sum_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff_has_sum {g : γ → α}
(h₁ : ∀u:finset γ, ∃v:finset β, ∀v', v ⊆ v' → ∃u', u ⊆ u' ∧ ∑ x in u', g x = ∑ b in v', f b)
(h₂ : ∀v:finset β, ∃u:finset γ, ∀u', u ⊆ u' → ∃v', v ⊆ v' ∧ ∑ b in v', f b = ∑ x in u', g x) :
has_sum f a ↔ has_sum g a | ⟨has_sum.has_sum_of_sum_eq h₂, has_sum.has_sum_of_sum_eq h₁⟩ | lemma | has_sum_iff_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum",
"has_sum.has_sum_of_sum_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.has_sum_iff {g : γ → β} (hg : injective g)
(hf : ∀ x ∉ set.range g, f x = 0) :
has_sum (f ∘ g) a ↔ has_sum f a | by simp only [has_sum, tendsto, hg.map_at_top_finset_sum_eq hf] | lemma | function.injective.has_sum_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.summable_iff {g : γ → β} (hg : injective g)
(hf : ∀ x ∉ set.range g, f x = 0) :
summable (f ∘ g) ↔ summable f | exists_congr $ λ _, hg.has_sum_iff hf | lemma | function.injective.summable_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"set.range",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_subtype_iff_of_support_subset {s : set β} (hf : support f ⊆ s) :
has_sum (f ∘ coe : s → α) a ↔ has_sum f a | subtype.coe_injective.has_sum_iff $ by simpa using support_subset_iff'.1 hf | lemma | has_sum_subtype_iff_of_support_subset | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_subtype_iff_indicator {s : set β} :
has_sum (f ∘ coe : s → α) a ↔ has_sum (s.indicator f) a | by rw [← set.indicator_range_comp, subtype.range_coe,
has_sum_subtype_iff_of_support_subset set.support_indicator_subset] | lemma | has_sum_subtype_iff_indicator | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_subtype_iff_of_support_subset",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_subtype_iff_indicator {s : set β} :
summable (f ∘ coe : s → α) ↔ summable (s.indicator f) | exists_congr (λ _, has_sum_subtype_iff_indicator) | lemma | summable_subtype_iff_indicator | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_subtype_iff_indicator",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_subtype_support : has_sum (f ∘ coe : support f → α) a ↔ has_sum f a | has_sum_subtype_iff_of_support_subset $ set.subset.refl _ | lemma | has_sum_subtype_support | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_subtype_iff_of_support_subset",
"set.subset.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_fintype [fintype β] (f : β → α) : has_sum f (∑ b, f b) | order_top.tendsto_at_top_nhds _ | lemma | has_sum_fintype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"fintype",
"has_sum",
"order_top.tendsto_at_top_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.has_sum (s : finset β) (f : β → α) :
has_sum (f ∘ coe : (↑s : set β) → α) (∑ b in s, f b) | by { rw ← sum_attach, exact has_sum_fintype _ } | lemma | finset.has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum",
"has_sum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.summable (s : finset β) (f : β → α) :
summable (f ∘ coe : (↑s : set β) → α) | (s.has_sum f).summable | lemma | finset.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.summable {s : set β} (hs : s.finite) (f : β → α) :
summable (f ∘ coe : s → α) | by convert hs.to_finset.summable f; simp only [hs.coe_to_finset] | lemma | set.finite.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_sum_of_ne_finset_zero (hf : ∀b∉s, f b = 0) : has_sum f (∑ b in s, f b) | (has_sum_subtype_iff_of_support_subset $ support_subset_iff'.2 hf).1 $ s.has_sum f | lemma | has_sum_sum_of_ne_finset_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_subtype_iff_of_support_subset"
] | If a function `f` vanishes outside of a finite set `s`, then it `has_sum` `∑ b in s, f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_ne_finset_zero (hf : ∀b∉s, f b = 0) : summable f | (has_sum_sum_of_ne_finset_zero hf).summable | lemma | summable_of_ne_finset_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_sum_of_ne_finset_zero",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_single {f : β → α} (b : β) (hf : ∀b' ≠ b, f b' = 0) :
has_sum f (f b) | suffices has_sum f (∑ b' in {b}, f b'),
by simpa using this,
has_sum_sum_of_ne_finset_zero $ by simpa [hf] | lemma | has_sum_single | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_sum_of_ne_finset_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_ite_eq (b : β) [decidable_pred (= b)] (a : α) :
has_sum (λb', if b' = b then a else 0) a | begin
convert has_sum_single b _,
{ exact (if_pos rfl).symm },
assume b' hb',
exact if_neg hb'
end | lemma | has_sum_ite_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_pi_single [decidable_eq β] (b : β) (a : α) :
has_sum (pi.single b a) a | show has_sum (λ x, pi.single b a x) a, by simpa only [pi.single_apply] using has_sum_ite_eq b a | lemma | has_sum_pi_single | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_ite_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.has_sum_iff (e : γ ≃ β) :
has_sum (f ∘ e) a ↔ has_sum f a | e.injective.has_sum_iff $ by simp | lemma | equiv.has_sum_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.has_sum_range_iff {g : γ → β} (hg : injective g) :
has_sum (λ x : set.range g, f x) a ↔ has_sum (f ∘ g) a | (equiv.of_injective g hg).has_sum_iff.symm | lemma | function.injective.has_sum_range_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.of_injective",
"has_sum",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.summable_iff (e : γ ≃ β) :
summable (f ∘ e) ↔ summable f | exists_congr $ λ a, e.has_sum_iff | lemma | equiv.summable_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.prod_symm {f : β × γ → α} (hf : summable f) : summable (λ p : γ × β, f p.swap) | (equiv.prod_comm γ β).summable_iff.2 hf | lemma | summable.prod_symm | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.prod_comm",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.has_sum_iff_of_support {g : γ → α} (e : support f ≃ support g)
(he : ∀ x : support f, g (e x) = f x) :
has_sum f a ↔ has_sum g a | have (g ∘ coe) ∘ e = f ∘ coe, from funext he,
by rw [← has_sum_subtype_support, ← this, e.has_sum_iff, has_sum_subtype_support] | lemma | equiv.has_sum_iff_of_support | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_subtype_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff_has_sum_of_ne_zero_bij {g : γ → α} (i : support g → β)
(hi : ∀ ⦃x y⦄, i x = i y → (x : γ) = y)
(hf : support f ⊆ set.range i) (hfg : ∀ x, f (i x) = g x) :
has_sum f a ↔ has_sum g a | iff.symm $ equiv.has_sum_iff_of_support
(equiv.of_bijective (λ x, ⟨i x, λ hx, x.coe_prop $ hfg x ▸ hx⟩)
⟨λ x y h, subtype.ext $ hi $ subtype.ext_iff.1 h,
λ y, (hf y.coe_prop).imp $ λ x hx, subtype.ext hx⟩)
hfg | lemma | has_sum_iff_has_sum_of_ne_zero_bij | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.has_sum_iff_of_support",
"equiv.of_bijective",
"has_sum",
"set.range",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.summable_iff_of_support {g : γ → α} (e : support f ≃ support g)
(he : ∀ x : support f, g (e x) = f x) :
summable f ↔ summable g | exists_congr $ λ _, e.has_sum_iff_of_support he | lemma | equiv.summable_iff_of_support | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.map [add_comm_monoid γ] [topological_space γ] (hf : has_sum f a)
{G} [add_monoid_hom_class G α γ] (g : G) (hg : continuous g) :
has_sum (g ∘ f) (g a) | have g ∘ (λs:finset β, ∑ b in s, f b) = (λs:finset β, ∑ b in s, g (f b)),
from funext $ map_sum g _,
show tendsto (λs:finset β, ∑ b in s, g (f b)) at_top (𝓝 (g a)),
from this ▸ (hg.tendsto a).comp hf | lemma | has_sum.map | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"add_monoid_hom_class",
"continuous",
"finset",
"has_sum",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.map [add_comm_monoid γ] [topological_space γ] (hf : summable f)
{G} [add_monoid_hom_class G α γ] (g : G) (hg : continuous g) :
summable (g ∘ f) | (hf.has_sum.map g hg).summable | lemma | summable.map | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"add_monoid_hom_class",
"continuous",
"summable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.map_iff_of_left_inverse [add_comm_monoid γ] [topological_space γ]
{G G'} [add_monoid_hom_class G α γ] [add_monoid_hom_class G' γ α] (g : G) (g' : G')
(hg : continuous g) (hg' : continuous g') (hinv : function.left_inverse g' g) :
summable (g ∘ f) ↔ summable f | ⟨λ h, begin
have := h.map _ hg',
rwa [←function.comp.assoc, hinv.id] at this,
end, λ h, h.map _ hg⟩ | lemma | summable.map_iff_of_left_inverse | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"add_monoid_hom_class",
"continuous",
"summable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.map_iff_of_equiv [add_comm_monoid γ] [topological_space γ]
{G} [add_equiv_class G α γ] (g : G)
(hg : continuous g) (hg' : continuous (add_equiv_class.inv g : γ → α)) :
summable (g ∘ f) ↔ summable f | summable.map_iff_of_left_inverse g (g : α ≃+ γ).symm hg hg' (add_equiv_class.left_inv g) | lemma | summable.map_iff_of_equiv | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"add_equiv_class",
"continuous",
"summable",
"summable.map_iff_of_left_inverse",
"topological_space"
] | A special case of `summable.map_iff_of_left_inverse` for convenience | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.tendsto_sum_nat {f : ℕ → α} (h : has_sum f a) :
tendsto (λn:ℕ, ∑ i in range n, f i) at_top (𝓝 a) | h.comp tendsto_finset_range | lemma | has_sum.tendsto_sum_nat | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | If `f : ℕ → α` has sum `a`, then the partial sums `∑_{i=0}^{n-1} f i` converge to `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.unique {a₁ a₂ : α} [t2_space α] : has_sum f a₁ → has_sum f a₂ → a₁ = a₂ | tendsto_nhds_unique | lemma | has_sum.unique | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"t2_space",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.has_sum_iff_tendsto_nat [t2_space α] {f : ℕ → α} {a : α} (hf : summable f) :
has_sum f a ↔ tendsto (λn:ℕ, ∑ i in range n, f i) at_top (𝓝 a) | begin
refine ⟨λ h, h.tendsto_sum_nat, λ h, _⟩,
rw tendsto_nhds_unique h hf.has_sum.tendsto_sum_nat,
exact hf.has_sum
end | lemma | summable.has_sum_iff_tendsto_nat | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable",
"t2_space",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.summable_iff_of_has_sum_iff {α' : Type*} [add_comm_monoid α']
[topological_space α'] {e : α' → α} (hes : function.surjective e) {f : β → α} {g : γ → α'}
(he : ∀ {a}, has_sum f (e a) ↔ has_sum g a) :
summable f ↔ summable g | hes.exists.trans $ exists_congr $ @he | lemma | function.surjective.summable_iff_of_has_sum_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"has_sum",
"summable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.add (hf : has_sum f a) (hg : has_sum g b) : has_sum (λb, f b + g b) (a + b) | by simp only [has_sum, sum_add_distrib]; exact hf.add hg | lemma | has_sum.add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.add (hf : summable f) (hg : summable g) : summable (λb, f b + g b) | (hf.has_sum.add hg.has_sum).summable | lemma | summable.add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_sum {f : γ → β → α} {a : γ → α} {s : finset γ} :
(∀i∈s, has_sum (f i) (a i)) → has_sum (λb, ∑ i in s, f i b) (∑ i in s, a i) | finset.induction_on s (by simp only [has_sum_zero, sum_empty, forall_true_iff])
(by simp only [has_sum.add, sum_insert, mem_insert, forall_eq_or_imp,
forall_2_true_iff, not_false_iff, forall_true_iff] {contextual := tt}) | lemma | has_sum_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"finset.induction_on",
"forall_2_true_iff",
"forall_eq_or_imp",
"forall_true_iff",
"has_sum",
"has_sum.add",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_sum {f : γ → β → α} {s : finset γ} (hf : ∀i∈s, summable (f i)) :
summable (λb, ∑ i in s, f i b) | (has_sum_sum $ assume i hi, (hf i hi).has_sum).summable | lemma | summable_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum",
"has_sum_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.add_disjoint {s t : set β} (hs : disjoint s t)
(ha : has_sum (f ∘ coe : s → α) a) (hb : has_sum (f ∘ coe : t → α) b) :
has_sum (f ∘ coe : s ∪ t → α) (a + b) | begin
rw has_sum_subtype_iff_indicator at *,
rw set.indicator_union_of_disjoint hs,
exact ha.add hb
end | lemma | has_sum.add_disjoint | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"has_sum",
"has_sum_subtype_iff_indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_sum_disjoint {ι} (s : finset ι) {t : ι → set β} {a : ι → α}
(hs : (s : set ι).pairwise (disjoint on t))
(hf : ∀ i ∈ s, has_sum (f ∘ coe : t i → α) (a i)) :
has_sum (f ∘ coe : (⋃ i ∈ s, t i) → α) (∑ i in s, a i) | begin
simp_rw has_sum_subtype_iff_indicator at *,
rw set.indicator_finset_bUnion _ _ hs,
exact has_sum_sum hf,
end | lemma | has_sum_sum_disjoint | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"finset",
"has_sum",
"has_sum_subtype_iff_indicator",
"has_sum_sum",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.add_is_compl {s t : set β} (hs : is_compl s t)
(ha : has_sum (f ∘ coe : s → α) a) (hb : has_sum (f ∘ coe : t → α) b) :
has_sum f (a + b) | by simpa [← hs.compl_eq]
using (has_sum_subtype_iff_indicator.1 ha).add (has_sum_subtype_iff_indicator.1 hb) | lemma | has_sum.add_is_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.add_compl {s : set β} (ha : has_sum (f ∘ coe : s → α) a)
(hb : has_sum (f ∘ coe : sᶜ → α) b) :
has_sum f (a + b) | ha.add_is_compl is_compl_compl hb | lemma | has_sum.add_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"is_compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.add_compl {s : set β} (hs : summable (f ∘ coe : s → α))
(hsc : summable (f ∘ coe : sᶜ → α)) :
summable f | (hs.has_sum.add_compl hsc.has_sum).summable | lemma | summable.add_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.compl_add {s : set β} (ha : has_sum (f ∘ coe : sᶜ → α) a)
(hb : has_sum (f ∘ coe : s → α) b) :
has_sum f (a + b) | ha.add_is_compl is_compl_compl.symm hb | lemma | has_sum.compl_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.even_add_odd {f : ℕ → α} (he : has_sum (λ k, f (2 * k)) a)
(ho : has_sum (λ k, f (2 * k + 1)) b) :
has_sum f (a + b) | begin
have := mul_right_injective₀ (two_ne_zero' ℕ),
replace he := this.has_sum_range_iff.2 he,
replace ho := ((add_left_injective 1).comp this).has_sum_range_iff.2 ho,
refine he.add_is_compl _ ho,
simpa [(∘)] using nat.is_compl_even_odd
end | lemma | has_sum.even_add_odd | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"mul_right_injective₀",
"nat.is_compl_even_odd",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.compl_add {s : set β} (hs : summable (f ∘ coe : sᶜ → α))
(hsc : summable (f ∘ coe : s → α)) :
summable f | (hs.has_sum.compl_add hsc.has_sum).summable | lemma | summable.compl_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.even_add_odd {f : ℕ → α} (he : summable (λ k, f (2 * k)))
(ho : summable (λ k, f (2 * k + 1))) :
summable f | (he.has_sum.even_add_odd ho.has_sum).summable | lemma | summable.even_add_odd | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.sigma [regular_space α] {γ : β → Type*} {f : (Σ b:β, γ b) → α} {g : β → α} {a : α}
(ha : has_sum f a) (hf : ∀b, has_sum (λc, f ⟨b, c⟩) (g b)) : has_sum g a | begin
refine (at_top_basis.tendsto_iff (closed_nhds_basis a)).mpr _,
rintros s ⟨hs, hsc⟩,
rcases mem_at_top_sets.mp (ha hs) with ⟨u, hu⟩,
use [u.image sigma.fst, trivial],
intros bs hbs,
simp only [set.mem_preimage, ge_iff_le, finset.le_iff_subset] at hu,
have : tendsto (λ t : finset (Σ b, γ b), ∑ p in t.... | lemma | has_sum.sigma | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"closed_nhds_basis",
"finset",
"finset.le_iff_subset",
"ge_iff_le",
"has_sum",
"regular_space",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.prod_fiberwise [regular_space α] {f : β × γ → α} {g : β → α} {a : α}
(ha : has_sum f a) (hf : ∀b, has_sum (λc, f (b, c)) (g b)) :
has_sum g a | has_sum.sigma ((equiv.sigma_equiv_prod β γ).has_sum_iff.2 ha) hf | lemma | has_sum.prod_fiberwise | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.sigma_equiv_prod",
"has_sum",
"has_sum.sigma",
"regular_space"
] | If a series `f` on `β × γ` has sum `a` and for each `b` the restriction of `f` to `{b} × γ`
has sum `g b`, then the series `g` has sum `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable.sigma' [regular_space α] {γ : β → Type*} {f : (Σb:β, γ b) → α}
(ha : summable f) (hf : ∀b, summable (λc, f ⟨b, c⟩)) :
summable (λb, ∑'c, f ⟨b, c⟩) | (ha.has_sum.sigma (assume b, (hf b).has_sum)).summable | lemma | summable.sigma' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"regular_space",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.sigma_of_has_sum [t3_space α] {γ : β → Type*} {f : (Σ b:β, γ b) → α} {g : β → α}
{a : α} (ha : has_sum g a) (hf : ∀b, has_sum (λc, f ⟨b, c⟩) (g b)) (hf' : summable f) :
has_sum f a | by simpa [(hf'.has_sum.sigma hf).unique ha] using hf'.has_sum | lemma | has_sum.sigma_of_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable",
"t3_space",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.update' {α β : Type*} [topological_space α] [add_comm_monoid α] [t2_space α]
[has_continuous_add α] {f : β → α} {a a' : α} (hf : has_sum f a)
(b : β) (x : α) (hf' : has_sum (f.update b x) a') : a + x = a' + f b | begin
have : ∀ b', f b' + ite (b' = b) x 0 = f.update b x b' + ite (b' = b) (f b) 0,
{ intro b',
split_ifs with hb',
{ simpa only [function.update_apply, hb', eq_self_iff_true] using add_comm (f b) x },
{ simp only [function.update_apply, hb', if_false] } },
have h := hf.add ((has_sum_ite_eq b x)),
... | lemma | has_sum.update' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"has_continuous_add",
"has_sum",
"has_sum.unique",
"has_sum_ite_eq",
"t2_space",
"topological_space"
] | Version of `has_sum.update` for `add_comm_monoid` rather than `add_comm_group`.
Rather than showing that `f.update` has a specific sum in terms of `has_sum`,
it gives a relationship between the sums of `f` and `f.update` given that both exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_add_of_has_sum_ite {α β : Type*} [topological_space α] [add_comm_monoid α]
[t2_space α] [has_continuous_add α] {f : β → α} {a : α} (hf : has_sum f a) (b : β) (a' : α)
(hf' : has_sum (λ n, ite (n = b) 0 (f n)) a') : a = a' + f b | begin
refine (add_zero a).symm.trans (hf.update' b 0 _),
convert hf',
exact funext (f.update_apply b 0),
end | lemma | eq_add_of_has_sum_ite | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"has_continuous_add",
"has_sum",
"t2_space",
"topological_space"
] | Version of `has_sum_ite_sub_has_sum` for `add_comm_monoid` rather than `add_comm_group`.
Rather than showing that the `ite` expression has a specific sum in terms of `has_sum`,
it gives a relationship between the sums of `f` and `ite (n = b) 0 (f n)` given that both exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_congr_subtype (f : β → α) {s t : set β} (h : s = t) :
∑' (x : s), f x = ∑' (x : t), f x | by rw h | lemma | tsum_congr_subtype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_zero' (hz : is_closed ({0} : set α)) : ∑' b : β, (0 : α) = 0 | begin
classical,
rw [tsum, dif_pos summable_zero],
suffices : ∀ (x : α), has_sum (λ (b : β), (0 : α)) x → x = 0,
{ exact this _ (classical.some_spec _) },
intros x hx,
contrapose! hx,
simp only [has_sum, tendsto_nhds, finset.sum_const_zero, filter.mem_at_top_sets, ge_iff_le,
finset.le_eq_sub... | lemma | tsum_zero' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"exists_and_distrib_right",
"exists_prop",
"filter.mem_at_top_sets",
"finset.le_eq_subset",
"ge_iff_le",
"has_sum",
"is_closed",
"not_exists",
"not_forall",
"set.mem_preimage",
"subset_refl",
"summable_zero",
"tendsto_nhds",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_zero [t1_space α] : ∑' b : β, (0 : α) = 0 | tsum_zero' is_closed_singleton | lemma | tsum_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"is_closed_singleton",
"t1_space",
"tsum_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.tsum_eq (ha : has_sum f a) : ∑'b, f b = a | (summable.has_sum ⟨a, ha⟩).unique ha | lemma | has_sum.tsum_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable.has_sum",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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