fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : {a} * s ∈ 𝓝 (a * b) := by
rwa [← smul_eq_mul, ← smul_eq_mul, singleton_smul, smul_mem_nhds_smul_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | singleton_mul_mem_nhds | null |
singleton_mul_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : {a} * s ∈ 𝓝 a := by
simpa only [mul_one] using singleton_mul_mem_nhds a h | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | singleton_mul_mem_nhds_of_nhds_one | null |
@[to_additive]
IsOpen.mul_right (hs : IsOpen s) : IsOpen (s * t) := by
rw [← image_op_smul]
exact hs.smul_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.mul_right | null |
subset_interior_mul_left : interior s * t ⊆ interior (s * t) :=
interior_maximal (Set.mul_subset_mul_right interior_subset) isOpen_interior.mul_right
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul_left | null |
subset_interior_mul' : interior s * interior t ⊆ interior (s * t) :=
(Set.mul_subset_mul_left interior_subset).trans subset_interior_mul_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul' | null |
mul_singleton_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : s * {a} ∈ 𝓝 (b * a) := by
rw [mul_singleton]
exact smul_mem_nhds_smul (op a) h
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | mul_singleton_mem_nhds | null |
mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : s * {a} ∈ 𝓝 a := by
simpa only [one_mul] using mul_singleton_mem_nhds a h | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | mul_singleton_mem_nhds_of_nhds_one | null |
@[to_additive]
IsOpen.div_left (ht : IsOpen t) : IsOpen (s / t) := by
rw [← iUnion_div_left_image]
exact isOpen_biUnion fun a _ => isOpenMap_div_left a t ht
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.div_left | null |
IsOpen.div_right (hs : IsOpen s) : IsOpen (s / t) := by
rw [← iUnion_div_right_image]
exact isOpen_biUnion fun a _ => isOpenMap_div_right a s hs
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.div_right | null |
subset_interior_div_left : interior s / t ⊆ interior (s / t) :=
interior_maximal (div_subset_div_right interior_subset) isOpen_interior.div_right
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_div_left | null |
subset_interior_div_right : s / interior t ⊆ interior (s / t) :=
interior_maximal (div_subset_div_left interior_subset) isOpen_interior.div_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_div_right | null |
subset_interior_div : interior s / interior t ⊆ interior (s / t) :=
(div_subset_div_left interior_subset).trans subset_interior_div_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_div | null |
IsOpen.mul_closure (hs : IsOpen s) (t : Set G) : s * closure t = s * t := by
refine (mul_subset_iff.2 fun a ha b hb => ?_).antisymm (mul_subset_mul_left subset_closure)
rw [mem_closure_iff] at hb
have hbU : b ∈ s⁻¹ * {a * b} := ⟨a⁻¹, Set.inv_mem_inv.2 ha, a * b, rfl, inv_mul_cancel_left _ _⟩
obtain ⟨_, ⟨c, hc, d, rfl : d = _, rfl⟩, hcs⟩ := hb _ hs.inv.mul_right hbU
exact ⟨c⁻¹, hc, _, hcs, inv_mul_cancel_left _ _⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.mul_closure | null |
IsOpen.closure_mul (ht : IsOpen t) (s : Set G) : closure s * t = s * t := by
rw [← inv_inv (closure s * t), mul_inv_rev, inv_closure, ht.inv.mul_closure, mul_inv_rev, inv_inv,
inv_inv]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.closure_mul | null |
IsOpen.div_closure (hs : IsOpen s) (t : Set G) : s / closure t = s / t := by
simp_rw [div_eq_mul_inv, inv_closure, hs.mul_closure]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.div_closure | null |
IsOpen.closure_div (ht : IsOpen t) (s : Set G) : closure s / t = s / t := by
simp_rw [div_eq_mul_inv, ht.inv.closure_mul]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.closure_div | null |
IsClosed.mul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (s * t) :=
ht.smul_left_of_isCompact hs
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsClosed.mul_left_of_isCompact | null |
IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
IsClosed (t * s) := by
rw [← image_op_smul]
exact IsClosed.smul_left_of_isCompact ht (hs.image continuous_op)
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsClosed.mul_right_of_isCompact | null |
subset_mul_closure_one {G} [MulOneClass G] [TopologicalSpace G] (s : Set G) :
s ⊆ s * (closure {1} : Set G) := by
have : s ⊆ s * ({1} : Set G) := by simp
exact this.trans (smul_subset_smul_left subset_closure)
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_mul_closure_one | null |
IsCompact.mul_closure_one_eq_closure {K : Set G} (hK : IsCompact K) :
K * (closure {1} : Set G) = closure K := by
apply Subset.antisymm ?_ ?_
· calc
K * (closure {1} : Set G) ⊆ closure K * (closure {1} : Set G) :=
smul_subset_smul_right subset_closure
_ ⊆ closure (K * ({1} : Set G)) := smul_set_closure_subset _ _
_ = closure K := by simp
· have : IsClosed (K * (closure {1} : Set G)) :=
IsClosed.smul_left_of_isCompact isClosed_closure hK
rw [IsClosed.closure_subset_iff this]
exact subset_mul_closure_one K
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsCompact.mul_closure_one_eq_closure | null |
IsClosed.mul_closure_one_eq {F : Set G} (hF : IsClosed F) :
F * (closure {1} : Set G) = F := by
refine Subset.antisymm ?_ (subset_mul_closure_one F)
calc
F * (closure {1} : Set G) = closure F * closure ({1} : Set G) := by rw [hF.closure_eq]
_ ⊆ closure (F * ({1} : Set G)) := smul_set_closure_subset _ _
_ = F := by simp
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsClosed.mul_closure_one_eq | null |
compl_mul_closure_one_eq {t : Set G} (ht : t * (closure {1} : Set G) = t) :
tᶜ * (closure {1} : Set G) = tᶜ := by
refine Subset.antisymm ?_ (subset_mul_closure_one tᶜ)
rintro - ⟨x, hx, g, hg, rfl⟩
by_contra H
have : x ∈ t * (closure {1} : Set G) := by
rw [← Subgroup.coe_topologicalClosure_bot G] at hg ⊢
simp only [mem_compl_iff, not_not] at H
exact ⟨x * g, H, g⁻¹, Subgroup.inv_mem _ hg, by simp⟩
rw [ht] at this
exact hx this
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | compl_mul_closure_one_eq | null |
compl_mul_closure_one_eq_iff {t : Set G} :
tᶜ * (closure {1} : Set G) = tᶜ ↔ t * (closure {1} : Set G) = t :=
⟨fun h ↦ by simpa using compl_mul_closure_one_eq h, fun h ↦ compl_mul_closure_one_eq h⟩
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | compl_mul_closure_one_eq_iff | null |
IsOpen.mul_closure_one_eq {U : Set G} (hU : IsOpen U) :
U * (closure {1} : Set G) = U :=
compl_mul_closure_one_eq_iff.1 (hU.isClosed_compl.mul_closure_one_eq) | lemma | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.mul_closure_one_eq | null |
@[to_additive]
group_inseparable_iff {x y : G} : Inseparable x y ↔ x / y ∈ closure (1 : Set G) := by
rw [← singleton_one, ← specializes_iff_mem_closure, specializes_comm, specializes_iff_inseparable,
← (Homeomorph.mulRight y⁻¹).isEmbedding.inseparable_iff]
simp [div_eq_mul_inv]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | group_inseparable_iff | null |
IsTopologicalGroup.t2Space_iff_one_closed : T2Space G ↔ IsClosed ({1} : Set G) :=
⟨fun _ ↦ isClosed_singleton, fun h ↦
have := IsTopologicalGroup.t1Space G h; inferInstance⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsTopologicalGroup.t2Space_iff_one_closed | null |
IsTopologicalGroup.t2Space_of_one_sep (H : ∀ x : G, x ≠ 1 → ∃ U ∈ 𝓝 (1 : G), x ∉ U) :
T2Space G := by
suffices T1Space G from inferInstance
refine t1Space_iff_specializes_imp_eq.2 fun x y hspec ↦ by_contra fun hne ↦ ?_
rcases H (x * y⁻¹) (by rwa [Ne, mul_inv_eq_one]) with ⟨U, hU₁, hU⟩
exact hU <| mem_of_mem_nhds <| hspec.map (continuous_mul_right y⁻¹) (by rwa [mul_inv_cancel]) | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsTopologicalGroup.t2Space_of_one_sep | null |
@[to_additive /-- Given a neighborhood `U` of the identity, one may find a neighborhood `V` of the
identity which is closed, symmetric, and satisfies `V + V ⊆ U`. -/]
exists_closed_nhds_one_inv_eq_mul_subset {U : Set G} (hU : U ∈ 𝓝 1) :
∃ V ∈ 𝓝 1, IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U := by
rcases exists_open_nhds_one_mul_subset hU with ⟨V, V_open, V_mem, hV⟩
rcases exists_mem_nhds_isClosed_subset (V_open.mem_nhds V_mem) with ⟨W, W_mem, W_closed, hW⟩
refine ⟨W ∩ W⁻¹, Filter.inter_mem W_mem (inv_mem_nhds_one G W_mem), W_closed.inter W_closed.inv,
by simp [inter_comm], ?_⟩
calc
W ∩ W⁻¹ * (W ∩ W⁻¹)
⊆ W * W := mul_subset_mul inter_subset_left inter_subset_left
_ ⊆ V * V := mul_subset_mul hW hW
_ ⊆ U := hV | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | exists_closed_nhds_one_inv_eq_mul_subset | Given a neighborhood `U` of the identity, one may find a neighborhood `V` of the identity which
is closed, symmetric, and satisfies `V * V ⊆ U`. |
@[to_additive]
IsCompact.locallyCompactSpace_of_mem_nhds_of_group {K : Set G} (hK : IsCompact K) {x : G}
(h : K ∈ 𝓝 x) : LocallyCompactSpace G := by
suffices WeaklyLocallyCompactSpace G from inferInstance
refine ⟨fun y ↦ ⟨(y * x⁻¹) • K, ?_, ?_⟩⟩
· exact hK.smul _
· rw [← preimage_smul_inv]
exact (continuous_const_smul _).continuousAt.preimage_mem_nhds (by simpa using h) | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsCompact.locallyCompactSpace_of_mem_nhds_of_group | If a point in a topological group has a compact neighborhood, then the group is
locally compact. |
@[to_additive
/-- If a function defined on a topological additive group has a support contained in a compact
set, then either the function is trivial or the group is locally compact. -/]
eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group
[TopologicalSpace α] [Zero α] [T1Space α]
{f : G → α} {k : Set G} (hk : IsCompact k) (hf : support f ⊆ k) (h'f : Continuous f) :
f = 0 ∨ LocallyCompactSpace G := by
refine or_iff_not_imp_left.mpr fun h => ?_
simp_rw [funext_iff, Pi.zero_apply] at h
push_neg at h
obtain ⟨x, hx⟩ : ∃ x, f x ≠ 0 := h
have : k ∈ 𝓝 x :=
mem_of_superset (h'f.isOpen_support.mem_nhds hx) hf
exact IsCompact.locallyCompactSpace_of_mem_nhds_of_group hk this | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group | If a function defined on a topological group has a support contained in a
compact set, then either the function is trivial or the group is locally compact. |
@[to_additive
/-- If a function defined on a topological additive group has compact support,
then either the function is trivial or the group is locally compact. -/]
HasCompactSupport.eq_zero_or_locallyCompactSpace_of_group
[TopologicalSpace α] [Zero α] [T1Space α]
{f : G → α} (hf : HasCompactSupport f) (h'f : Continuous f) :
f = 0 ∨ LocallyCompactSpace G :=
eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group hf (subset_tsupport f) h'f | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | HasCompactSupport.eq_zero_or_locallyCompactSpace_of_group | If a function defined on a topological group has compact support, then either
the function is trivial or the group is locally compact. |
@[to_additive]
instTopologicalSpace (N : Subgroup G) : TopologicalSpace (G ⧸ N) :=
instTopologicalSpaceQuotient
@[to_additive] | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instTopologicalSpace | null |
@[to_additive]
isQuotientMap_mk (N : Subgroup G) : IsQuotientMap (mk : G → G ⧸ N) :=
isQuotientMap_quot_mk
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | isQuotientMap_mk | null |
continuous_mk {N : Subgroup G} : Continuous (mk : G → G ⧸ N) :=
continuous_quot_mk | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | continuous_mk | null |
@[to_additive]
isOpenMap_coe : IsOpenMap ((↑) : G → G ⧸ N) := isOpenMap_quotient_mk'_mul
@[to_additive] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | isOpenMap_coe | null |
isOpenQuotientMap_mk : IsOpenQuotientMap (mk : G → G ⧸ N) :=
MulAction.isOpenQuotientMap_quotientMk
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | isOpenQuotientMap_mk | null |
dense_preimage_mk {s : Set (G ⧸ N)} : Dense ((↑) ⁻¹' s : Set G) ↔ Dense s :=
isOpenQuotientMap_mk.dense_preimage_iff
@[to_additive] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | dense_preimage_mk | null |
dense_image_mk {s : Set G} :
Dense (mk '' s : Set (G ⧸ N)) ↔ Dense (s * (N : Set G)) := by
rw [← dense_preimage_mk, preimage_image_mk_eq_mul]
@[to_additive] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | dense_image_mk | null |
instContinuousSMul : ContinuousSMul G (G ⧸ N) where
continuous_smul := by
rw [← (IsOpenQuotientMap.id.prodMap isOpenQuotientMap_mk).continuous_comp_iff]
exact continuous_mk.comp continuous_mul
@[to_additive] | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instContinuousSMul | null |
instContinuousConstSMul : ContinuousConstSMul G (G ⧸ N) := inferInstance | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instContinuousConstSMul | null |
@[to_additive]
instLocallyCompactSpace [LocallyCompactSpace G] (N : Subgroup G) :
LocallyCompactSpace (G ⧸ N) :=
QuotientGroup.isOpenQuotientMap_mk.locallyCompactSpace
variable (N) | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instLocallyCompactSpace | A quotient of a locally compact group is locally compact. |
@[to_additive
/-- Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. -/]
nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = Filter.map (↑) (𝓝 x) :=
(isOpenQuotientMap_mk.map_nhds_eq _).symm
@[to_additive] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | nhds_eq | Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. |
instFirstCountableTopology [FirstCountableTopology G] :
FirstCountableTopology (G ⧸ N) where
nhds_generated_countable := mk_surjective.forall.2 fun x ↦ nhds_eq N x ▸ inferInstance | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instFirstCountableTopology | null |
@[to_additive
/-- The quotient of a second countable additive topological group by a subgroup is second
countable. -/]
instSecondCountableTopology [SecondCountableTopology G] :
SecondCountableTopology (G ⧸ N) :=
ContinuousConstSMul.secondCountableTopology | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instSecondCountableTopology | The quotient of a second countable topological group by a subgroup is second countable. |
@[to_additive]
instIsTopologicalGroup [N.Normal] : IsTopologicalGroup (G ⧸ N) where
continuous_mul := by
rw [← (isOpenQuotientMap_mk.prodMap isOpenQuotientMap_mk).continuous_comp_iff]
exact continuous_mk.comp continuous_mul
continuous_inv := continuous_inv.quotient_map' _
@[to_additive] | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instIsTopologicalGroup | null |
isClosedMap_coe {H : Subgroup G} (hH : IsCompact (H : Set G)) :
IsClosedMap ((↑) : G → G ⧸ H) := by
intro t ht
rw [← (isQuotientMap_mk H).isClosed_preimage, preimage_image_mk_eq_mul]
exact ht.mul_right_of_isCompact hH
@[to_additive] | theorem | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | isClosedMap_coe | null |
instT3Space [N.Normal] [hN : IsClosed (N : Set G)] : T3Space (G ⧸ N) := by
rw [← QuotientGroup.ker_mk' N] at hN
haveI := IsTopologicalGroup.t1Space (G ⧸ N) ((isQuotientMap_mk N).isClosed_preimage.mp hN)
infer_instance | instance | Topology | [
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Defs",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/Group/Quotient.lean | instT3Space | null |
@[to_additive]
mapClusterPt_atTop_zpow_iff_pow [DivInvMonoid G] [TopologicalSpace G] {x y : G} :
MapClusterPt x atTop (y ^ · : ℤ → G) ↔ MapClusterPt x atTop (y ^ · : ℕ → G) := by
simp_rw [MapClusterPt, ← Nat.map_cast_int_atTop, map_map, comp_def, zpow_natCast]
variable [Group G] [TopologicalSpace G] [CompactSpace G] [IsTopologicalGroup G]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_atTop_zpow_iff_pow | null |
mapClusterPt_self_zpow_atTop_pow (x : G) (m : ℤ) :
MapClusterPt (x ^ m) atTop (x ^ · : ℕ → G) := by
obtain ⟨y, hy⟩ : ∃ y, MapClusterPt y atTop (x ^ · : ℤ → G) :=
exists_clusterPt_of_compactSpace _
rw [← mapClusterPt_atTop_zpow_iff_pow]
have H : MapClusterPt (x ^ m) (atTop.curry atTop) ↿(fun a b ↦ x ^ (m + b - a)) := by
have : ContinuousAt (fun yz ↦ x ^ m * yz.2 / yz.1) (y, y) := by fun_prop
simpa only [comp_def, ← zpow_sub, ← zpow_add, div_eq_mul_inv, Prod.map, mul_inv_cancel_right]
using (hy.curry_prodMap hy).continuousAt_comp this
suffices Tendsto ↿(fun a b ↦ m + b - a) (atTop.curry atTop) atTop from H.of_comp this
refine Tendsto.curry <| .of_forall fun a ↦ ?_
simp only [sub_eq_add_neg] -- TODO: add `Tendsto.atTop_sub_const` etc
exact tendsto_atTop_add_const_right _ _ (tendsto_atTop_add_const_left atTop m tendsto_id)
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_self_zpow_atTop_pow | null |
mapClusterPt_one_atTop_pow (x : G) : MapClusterPt 1 atTop (x ^ · : ℕ → G) := by
simpa using mapClusterPt_self_zpow_atTop_pow x 0
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_one_atTop_pow | null |
mapClusterPt_self_atTop_pow (x : G) : MapClusterPt x atTop (x ^ · : ℕ → G) := by
simpa using mapClusterPt_self_zpow_atTop_pow x 1
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_self_atTop_pow | null |
mapClusterPt_atTop_pow_tfae (x y : G) :
List.TFAE [
MapClusterPt x atTop (y ^ · : ℕ → G),
MapClusterPt x atTop (y ^ · : ℤ → G),
x ∈ closure (range (y ^ · : ℕ → G)),
x ∈ closure (range (y ^ · : ℤ → G)),
] := by
tfae_have 2 ↔ 1 := mapClusterPt_atTop_zpow_iff_pow
tfae_have 3 → 4 := by
refine fun h ↦ closure_mono (range_subset_iff.2 fun n ↦ ?_) h
exact ⟨n, zpow_natCast _ _⟩
tfae_have 4 → 1 := by
refine fun h ↦ closure_minimal ?_ isClosed_setOf_clusterPt h
exact range_subset_iff.2 (mapClusterPt_self_zpow_atTop_pow _)
tfae_have 1 → 3 := by
rw [mem_closure_iff_clusterPt]
exact (ClusterPt.mono · (le_principal_iff.2 range_mem_map))
tfae_finish
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_atTop_pow_tfae | null |
mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers {x y : G} :
MapClusterPt x atTop (y ^ · : ℕ → G) ↔ x ∈ (Subgroup.zpowers y).topologicalClosure :=
(mapClusterPt_atTop_pow_tfae x y).out 0 3
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers | null |
mapClusterPt_inv_atTop_pow {x y : G} :
MapClusterPt x⁻¹ atTop (y ^ · : ℕ → G) ↔ MapClusterPt x atTop (y ^ · : ℕ → G) := by
simp only [mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers, inv_mem_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | mapClusterPt_inv_atTop_pow | null |
closure_range_zpow_eq_pow (x : G) :
closure (range (x ^ · : ℤ → G)) = closure (range (x ^ · : ℕ → G)) := by
ext y
exact (mapClusterPt_atTop_pow_tfae y x).out 3 2
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | closure_range_zpow_eq_pow | null |
denseRange_zpow_iff_pow {x : G} :
DenseRange (x ^ · : ℤ → G) ↔ DenseRange (x ^ · : ℕ → G) := by
simp only [DenseRange, dense_iff_closure_eq, closure_range_zpow_eq_pow]
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | denseRange_zpow_iff_pow | null |
topologicalClosure_subgroupClosure_toSubmonoid (s : Set G) :
(Subgroup.closure s).toSubmonoid.topologicalClosure =
(Submonoid.closure s).topologicalClosure := by
refine le_antisymm ?_ (closure_mono <| Subgroup.le_closure_toSubmonoid _)
refine Submonoid.topologicalClosure_minimal _ ?_ isClosed_closure
rw [Subgroup.closure_toSubmonoid, Submonoid.closure_le]
refine union_subset (Submonoid.subset_closure.trans subset_closure) fun x hx ↦ ?_
refine closure_mono (Submonoid.powers_le.2 (Submonoid.subset_closure <| Set.mem_inv.1 hx)) ?_
rw [Submonoid.coe_powers, ← closure_range_zpow_eq_pow, ← Subgroup.coe_zpowers,
← Subgroup.topologicalClosure_coe, SetLike.mem_coe, ← inv_mem_iff]
exact subset_closure <| Subgroup.mem_zpowers _
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | topologicalClosure_subgroupClosure_toSubmonoid | null |
closure_submonoidClosure_eq_closure_subgroupClosure (s : Set G) :
closure (Submonoid.closure s : Set G) = closure (Subgroup.closure s) :=
congrArg SetLike.coe (topologicalClosure_subgroupClosure_toSubmonoid s).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | closure_submonoidClosure_eq_closure_subgroupClosure | null |
dense_submonoidClosure_iff_subgroupClosure {s : Set G} :
Dense (Submonoid.closure s : Set G) ↔ Dense (Subgroup.closure s : Set G) := by
simp only [dense_iff_closure_eq, closure_submonoidClosure_eq_closure_subgroupClosure] | theorem | Topology | [
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | dense_submonoidClosure_iff_subgroupClosure | null |
instNormalCommutatorClosure : (commutator G).topologicalClosure.Normal :=
Subgroup.is_normal_topologicalClosure (commutator G) | instance | Topology | [
"Mathlib.GroupTheory.Commutator.Basic",
"Mathlib.Tactic.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/TopologicalAbelianization.lean | instNormalCommutatorClosure | null |
TopologicalAbelianization := G ⧸ Subgroup.topologicalClosure (commutator G)
local notation "G_ab" => TopologicalAbelianization | abbrev | Topology | [
"Mathlib.GroupTheory.Commutator.Basic",
"Mathlib.Tactic.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/TopologicalAbelianization.lean | TopologicalAbelianization | The topological abelianization of `absoluteGaloisGroup`, that is, the quotient of
`absoluteGaloisGroup` by the topological closure of its commutator subgroup. |
commGroup : CommGroup (G_ab G) where
mul_comm := fun x y =>
Quotient.inductionOn₂' x y fun a b =>
Quotient.sound' <|
QuotientGroup.leftRel_apply.mpr <| by
have h : (a * b)⁻¹ * (b * a) = ⁅b⁻¹, a⁻¹⁆ := by group
rw [h]
exact Subgroup.le_topologicalClosure _ (Subgroup.commutator_mem_commutator
(Subgroup.mem_top b⁻¹) (Subgroup.mem_top a⁻¹))
__ : Group (G_ab G) := inferInstance | instance | Topology | [
"Mathlib.GroupTheory.Commutator.Basic",
"Mathlib.Tactic.Group",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/TopologicalAbelianization.lean | commGroup | null |
@[to_additive /-- Constant zero function has sum `0` -/]
hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 := by simp [HasProd, tendsto_const_nhds]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_one | Constant one function has product `1` |
hasProd_empty [IsEmpty β] : HasProd f 1 := by
convert @hasProd_one α β _ _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_empty | null |
multipliable_one : Multipliable (fun _ ↦ 1 : β → α) :=
hasProd_one.multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_one | null |
multipliable_empty [IsEmpty β] : Multipliable f :=
hasProd_empty.multipliable | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_empty | null |
@[to_additive /-- See `summable_congr_cofinite` for a version allowing the functions to
disagree on a finite set. -/]
multipliable_congr (hfg : ∀ b, f b = g b) : Multipliable f ↔ Multipliable g :=
iff_of_eq (congr_arg Multipliable <| funext hfg) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_congr | See `multipliable_congr_cofinite` for a version allowing the functions to
disagree on a finite set. |
@[to_additive /-- See `Summable.congr_cofinite` for a version allowing the functions to
disagree on a finite set. -/]
Multipliable.congr (hf : Multipliable f) (hfg : ∀ b, f b = g b) : Multipliable g :=
(multipliable_congr hfg).mp hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.congr | See `Multipliable.congr_cofinite` for a version allowing the functions to
disagree on a finite set. |
HasProd.congr_fun (hf : HasProd f a) (h : ∀ x : β, g x = f x) : HasProd g a :=
(funext h : g = f) ▸ hf
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.congr_fun | null |
HasProd.hasProd_of_prod_eq {g : γ → α}
(h_eq : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
(hf : HasProd g a) : HasProd f a :=
le_trans (map_atTop_finset_prod_le_of_prod_eq h_eq) hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.hasProd_of_prod_eq | null |
hasProd_iff_hasProd {g : γ → α}
(h₁ : ∀ u : Finset γ, ∃ v : Finset β, ∀ v', v ⊆ v' →
∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b)
(h₂ : ∀ v : Finset β, ∃ u : Finset γ, ∀ u', u ⊆ u' →
∃ v', v ⊆ v' ∧ ∏ b ∈ v', f b = ∏ x ∈ u', g x) :
HasProd f a ↔ HasProd g a :=
⟨HasProd.hasProd_of_prod_eq h₂, HasProd.hasProd_of_prod_eq h₁⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_iff_hasProd | null |
Function.Injective.multipliable_iff {g : γ → β} (hg : Injective g)
(hf : ∀ x ∉ Set.range g, f x = 1) : Multipliable (f ∘ g) ↔ Multipliable f :=
exists_congr fun _ ↦ hg.hasProd_iff hf
@[to_additive (attr := simp)] theorem hasProd_extend_one {g : β → γ} (hg : Injective g) :
HasProd (extend g f 1) a ↔ HasProd f a := by
rw [← hg.hasProd_iff, extend_comp hg]
exact extend_apply' _ _
@[to_additive (attr := simp)] theorem multipliable_extend_one {g : β → γ} (hg : Injective g) :
Multipliable (extend g f 1) ↔ Multipliable f :=
exists_congr fun _ ↦ hasProd_extend_one hg
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Injective.multipliable_iff | null |
hasProd_subtype_iff_mulIndicator {s : Set β} :
HasProd (f ∘ (↑) : s → α) a ↔ HasProd (s.mulIndicator f) a := by
rw [← Set.mulIndicator_range_comp, Subtype.range_coe,
hasProd_subtype_iff_of_mulSupport_subset Set.mulSupport_mulIndicator_subset]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_subtype_iff_mulIndicator | null |
multipliable_subtype_iff_mulIndicator {s : Set β} :
Multipliable (f ∘ (↑) : s → α) ↔ Multipliable (s.mulIndicator f) :=
exists_congr fun _ ↦ hasProd_subtype_iff_mulIndicator
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_subtype_iff_mulIndicator | null |
hasProd_subtype_mulSupport : HasProd (f ∘ (↑) : mulSupport f → α) a ↔ HasProd f a :=
hasProd_subtype_iff_of_mulSupport_subset <| Set.Subset.refl _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_subtype_mulSupport | null |
protected Finset.multipliable (s : Finset β) (f : β → α) :
Multipliable (f ∘ (↑) : (↑s : Set β) → α) :=
(s.hasProd f).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Finset.multipliable | null |
protected Set.Finite.multipliable {s : Set β} (hs : s.Finite) (f : β → α) :
Multipliable (f ∘ (↑) : s → α) := by
have := hs.toFinset.multipliable f
rwa [hs.coe_toFinset] at this
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Set.Finite.multipliable | null |
multipliable_of_finite_mulSupport (h : (mulSupport f).Finite) : Multipliable f := by
apply multipliable_of_ne_finset_one (s := h.toFinset); simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_of_finite_mulSupport | null |
Multipliable.of_finite [Finite β] {f : β → α} : Multipliable f :=
multipliable_of_finite_mulSupport <| Set.finite_univ.subset (Set.subset_univ _)
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.of_finite | null |
hasProd_single {f : β → α} (b : β) (hf : ∀ (b') (_ : b' ≠ b), f b' = 1) : HasProd f (f b) :=
suffices HasProd f (∏ b' ∈ {b}, f b') by simpa using this
hasProd_prod_of_ne_finset_one <| by simpa [hf]
@[to_additive (attr := simp)] lemma hasProd_unique [Unique β] (f : β → α) : HasProd f (f default) :=
hasProd_single default (fun _ hb ↦ False.elim <| hb <| Unique.uniq ..)
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_single | null |
hasProd_singleton (m : β) (f : β → α) : HasProd (({m} : Set β).restrict f) (f m) :=
hasProd_unique (Set.restrict {m} f)
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_singleton | null |
hasProd_ite_eq (b : β) [DecidablePred (· = b)] (a : α) :
HasProd (fun b' ↦ if b' = b then a else 1) a := by
convert @hasProd_single _ _ _ _ (fun b' ↦ if b' = b then a else 1) b (fun b' hb' ↦ if_neg hb')
exact (if_pos rfl).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_ite_eq | null |
Equiv.hasProd_iff (e : γ ≃ β) : HasProd (f ∘ e) a ↔ HasProd f a :=
e.injective.hasProd_iff <| by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.hasProd_iff | null |
Function.Injective.hasProd_range_iff {g : γ → β} (hg : Injective g) :
HasProd (fun x : Set.range g ↦ f x) a ↔ HasProd (f ∘ g) a :=
(Equiv.ofInjective g hg).hasProd_iff.symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Injective.hasProd_range_iff | null |
Equiv.multipliable_iff (e : γ ≃ β) : Multipliable (f ∘ e) ↔ Multipliable f :=
exists_congr fun _ ↦ e.hasProd_iff
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.multipliable_iff | null |
Equiv.hasProd_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
(he : ∀ x : mulSupport f, g (e x) = f x) : HasProd f a ↔ HasProd g a := by
have : (g ∘ (↑)) ∘ e = f ∘ (↑) := funext he
rw [← hasProd_subtype_mulSupport, ← this, e.hasProd_iff, hasProd_subtype_mulSupport]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.hasProd_iff_of_mulSupport | null |
hasProd_iff_hasProd_of_ne_one_bij {g : γ → α} (i : mulSupport g → β)
(hi : Injective i) (hf : mulSupport f ⊆ Set.range i)
(hfg : ∀ x, f (i x) = g x) : HasProd f a ↔ HasProd g a :=
Iff.symm <|
Equiv.hasProd_iff_of_mulSupport
(Equiv.ofBijective (fun x ↦ ⟨i x, fun hx ↦ x.coe_prop <| hfg x ▸ hx⟩)
⟨fun _ _ h ↦ hi <| Subtype.ext_iff.1 h, fun y ↦
(hf y.coe_prop).imp fun _ hx ↦ Subtype.ext hx⟩)
hfg
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_iff_hasProd_of_ne_one_bij | null |
Equiv.multipliable_iff_of_mulSupport {g : γ → α} (e : mulSupport f ≃ mulSupport g)
(he : ∀ x : mulSupport f, g (e x) = f x) : Multipliable f ↔ Multipliable g :=
exists_congr fun _ ↦ e.hasProd_iff_of_mulSupport he
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Equiv.multipliable_iff_of_mulSupport | null |
protected HasProd.map [CommMonoid γ] [TopologicalSpace γ] (hf : HasProd f a) {G}
[FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
HasProd (g ∘ f) (g a) := by
have : (g ∘ fun s : Finset β ↦ ∏ b ∈ s, f b) = fun s : Finset β ↦ ∏ b ∈ s, (g ∘ f) b :=
funext <| map_prod g _
unfold HasProd
rw [← this]
exact (hg.tendsto a).comp hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.map | null |
protected Topology.IsInducing.hasProd_iff [CommMonoid γ] [TopologicalSpace γ] {G}
[FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : IsInducing g) (f : β → α) (a : α) :
HasProd (g ∘ f) (g a) ↔ HasProd f a := by
simp_rw [HasProd, comp_apply, ← map_prod]
exact hg.tendsto_nhds_iff.symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Topology.IsInducing.hasProd_iff | null |
protected Multipliable.map [CommMonoid γ] [TopologicalSpace γ] (hf : Multipliable f) {G}
[FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) : Multipliable (g ∘ f) :=
(hf.hasProd.map g hg).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.map | null |
protected Multipliable.map_iff_of_leftInverse [CommMonoid γ] [TopologicalSpace γ] {G G'}
[FunLike G α γ] [MonoidHomClass G α γ] [FunLike G' γ α] [MonoidHomClass G' γ α]
(g : G) (g' : G') (hg : Continuous g) (hg' : Continuous g') (hinv : Function.LeftInverse g' g) :
Multipliable (g ∘ f) ↔ Multipliable f :=
⟨fun h ↦ by
have := h.map _ hg'
rwa [← Function.comp_assoc, hinv.id] at this, fun h ↦ h.map _ hg⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.map_iff_of_leftInverse | null |
Multipliable.map_tprod [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Multipliable f)
{G} [FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) :
g (∏' i, f i) = ∏' i, g (f i) := (HasProd.tprod_eq (HasProd.map hf.hasProd g hg)).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.map_tprod | null |
Topology.IsInducing.multipliable_iff_tprod_comp_mem_range [CommMonoid γ] [TopologicalSpace γ]
[T2Space γ] {G} [FunLike G α γ] [MonoidHomClass G α γ] {g : G} (hg : IsInducing g) (f : β → α) :
Multipliable f ↔ Multipliable (g ∘ f) ∧ ∏' i, g (f i) ∈ Set.range g := by
constructor
· intro hf
constructor
· exact hf.map g hg.continuous
· use ∏' i, f i
exact hf.map_tprod g hg.continuous
· rintro ⟨hgf, a, ha⟩
use a
have := hgf.hasProd
simp_rw [comp_apply, ← ha] at this
exact (hg.hasProd_iff f a).mp this | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Topology.IsInducing.multipliable_iff_tprod_comp_mem_range | null |
@[to_additive /-- A special case of `Summable.map_iff_of_leftInverse` for convenience -/]
protected Multipliable.map_iff_of_equiv [CommMonoid γ] [TopologicalSpace γ] {G}
[EquivLike G α γ] [MulEquivClass G α γ] (g : G) (hg : Continuous g)
(hg' : Continuous (EquivLike.inv g : γ → α)) : Multipliable (g ∘ f) ↔ Multipliable f :=
Multipliable.map_iff_of_leftInverse g (g : α ≃* γ).symm hg hg' (EquivLike.left_inv g)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.map_iff_of_equiv | "A special case of `Multipliable.map_iff_of_leftInverse` for convenience" |
Function.Surjective.multipliable_iff_of_hasProd_iff {α' : Type*} [CommMonoid α']
[TopologicalSpace α'] {e : α' → α} (hes : Function.Surjective e) {f : β → α} {g : γ → α'}
(he : ∀ {a}, HasProd f (e a) ↔ HasProd g a) : Multipliable f ↔ Multipliable g :=
hes.exists.trans <| exists_congr <| @he
variable [ContinuousMul α]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Surjective.multipliable_iff_of_hasProd_iff | null |
HasProd.mul (hf : HasProd f a) (hg : HasProd g b) :
HasProd (fun b ↦ f b * g b) (a * b) := by
dsimp only [HasProd] at hf hg ⊢
simp_rw [prod_mul_distrib]
exact hf.mul hg
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasProd.mul | null |
Multipliable.mul (hf : Multipliable f) (hg : Multipliable g) :
Multipliable fun b ↦ f b * g b :=
(hf.hasProd.mul hg.hasProd).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Multipliable.mul | null |
hasProd_prod {f : γ → β → α} {a : γ → α} {s : Finset γ} :
(∀ i ∈ s, HasProd (f i) (a i)) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) := by
classical
exact Finset.induction_on s (by simp only [hasProd_one, prod_empty, forall_true_iff]) <| by
simp +contextual only [mem_insert, forall_eq_or_imp, not_false_iff,
prod_insert, and_imp]
exact fun x s _ IH hx h ↦ hx.mul (IH h)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | hasProd_prod | null |
multipliable_prod {f : γ → β → α} {s : Finset γ} (hf : ∀ i ∈ s, Multipliable (f i)) :
Multipliable fun b ↦ ∏ i ∈ s, f i b :=
(hasProd_prod fun i hi ↦ (hf i hi).hasProd).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Group.Finset.Indicator",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | multipliable_prod | null |
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