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 ⌀ |
|---|---|---|---|---|---|---|
equivIoc : AddCircle p ≃ Ioc a (a + p) :=
QuotientAddGroup.equivIocMod hp.out a | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivIoc | The equivalence between `AddCircle p` and the half-open interval `(a, a + p]`, whose inverse
is the natural quotient map. |
liftIco (f : 𝕜 → B) : AddCircle p → B :=
restrict _ f ∘ AddCircle.equivIco p a | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco | Given a function on `𝕜`, return the unique function on `AddCircle p` agreeing with `f` on
`[a, a + p)`. |
liftIoc (f : 𝕜 → B) : AddCircle p → B :=
restrict _ f ∘ AddCircle.equivIoc p a
variable {p a} | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc | Given a function on `𝕜`, return the unique function on `AddCircle p` agreeing with `f` on
`(a, a + p]`. |
coe_eq_coe_iff_of_mem_Ico {x y : 𝕜} (hx : x ∈ Ico a (a + p)) (hy : y ∈ Ico a (a + p)) :
(x : AddCircle p) = y ↔ x = y := by
refine ⟨fun h => ?_, by tauto⟩
suffices (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this
apply_fun equivIco p a at h
rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIc... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_eq_coe_iff_of_mem_Ico | null |
liftIco_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico a (a + p)) :
liftIco p a f ↑x = f x := by
have : (equivIco p a) x = ⟨x, hx⟩ := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIco, comp_apply, this]
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco_coe_apply | null |
liftIoc_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ioc a (a + p)) :
liftIoc p a f ↑x = f x := by
have : (equivIoc p a) x = ⟨x, hx⟩ := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIoc, comp_apply, this]
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_coe_apply | null |
eq_coe_Ico (a : AddCircle p) : ∃ b, b ∈ Ico 0 p ∧ ↑b = a := by
let b := QuotientAddGroup.equivIcoMod hp.out 0 a
exact ⟨b.1, by simpa only [zero_add] using b.2,
(QuotientAddGroup.equivIcoMod hp.out 0).symm_apply_apply a⟩ | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | eq_coe_Ico | null |
coe_eq_zero_iff_of_mem_Ico (ha : a ∈ Ico 0 p) :
(a : AddCircle p) = 0 ↔ a = 0 := by
have h0 : 0 ∈ Ico 0 (0 + p) := by simpa [zero_add, left_mem_Ico] using hp.out
have ha' : a ∈ Ico 0 (0 + p) := by rwa [zero_add]
rw [← AddCircle.coe_eq_coe_iff_of_mem_Ico ha' h0, QuotientAddGroup.mk_zero]
variable (p a) | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_eq_zero_iff_of_mem_Ico | null |
@[continuity]
continuous_equivIco_symm : Continuous (equivIco p a).symm :=
continuous_quotient_mk'.comp continuous_subtype_val
@[continuity] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuous_equivIco_symm | null |
continuous_equivIoc_symm : Continuous (equivIoc p a).symm :=
continuous_quotient_mk'.comp continuous_subtype_val
variable [OrderTopology 𝕜] {x : AddCircle p} | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuous_equivIoc_symm | null |
continuousAt_equivIco (hx : x ≠ a) : ContinuousAt (equivIco p a) x := by
induction x using QuotientAddGroup.induction_on
rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map]
exact (continuousAt_toIcoMod hp.out a hx).codRestrict _ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuousAt_equivIco | null |
continuousAt_equivIoc (hx : x ≠ a) : ContinuousAt (equivIoc p a) x := by
induction x using QuotientAddGroup.induction_on
rw [ContinuousAt, Filter.Tendsto, QuotientAddGroup.nhds_eq, Filter.map_map]
exact (continuousAt_toIocMod hp.out a hx).codRestrict _ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuousAt_equivIoc | null |
@[simps] openPartialHomeomorphCoe [DiscreteTopology (zmultiples p)] :
OpenPartialHomeomorph 𝕜 (AddCircle p) where
toFun := (↑)
invFun := fun x ↦ equivIco p a x
source := Ioo a (a + p)
target := {↑a}ᶜ
map_source' := by
intro x hx hx'
exact hx.1.ne' ((coe_eq_coe_iff_of_mem_Ico (Ioo_subset_Ico_self ... | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | openPartialHomeomorphCoe | The quotient map `𝕜 → AddCircle p` as an open partial homeomorphism. |
isLocalHomeomorph_coe [DiscreteTopology (zmultiples p)] [DenselyOrdered 𝕜] :
IsLocalHomeomorph ((↑) : 𝕜 → AddCircle p) := by
intro a
obtain ⟨b, hb1, hb2⟩ := exists_between (sub_lt_self a hp.out)
exact ⟨openPartialHomeomorphCoe p b, ⟨hb2, lt_add_of_sub_right_lt hb1⟩, rfl⟩ | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | isLocalHomeomorph_coe | null |
@[simp]
coe_image_Ico_eq : ((↑) : 𝕜 → AddCircle p) '' Ico a (a + p) = univ := by
rw [image_eq_range]
exact (equivIco p a).symm.range_eq_univ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_image_Ico_eq | The image of the closed-open interval `[a, a + p)` under the quotient map `𝕜 → AddCircle p` is
the entire space. |
@[simp]
coe_image_Ioc_eq : ((↑) : 𝕜 → AddCircle p) '' Ioc a (a + p) = univ := by
rw [image_eq_range]
exact (equivIoc p a).symm.range_eq_univ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_image_Ioc_eq | The image of the closed-open interval `[a, a + p)` under the quotient map `𝕜 → AddCircle p` is
the entire space. |
@[simp]
coe_image_Icc_eq : ((↑) : 𝕜 → AddCircle p) '' Icc a (a + p) = univ :=
eq_top_mono (image_mono Ico_subset_Icc_self) <| coe_image_Ico_eq _ _ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_image_Icc_eq | The image of the closed interval `[0, p]` under the quotient map `𝕜 → AddCircle p` is the
entire space. |
equivAddCircle (hp : p ≠ 0) (hq : q ≠ 0) : AddCircle p ≃+ AddCircle q :=
QuotientAddGroup.congr _ _ (AddAut.mulRight <| (Units.mk0 p hp)⁻¹ * Units.mk0 q hq) <| by
rw [AddMonoidHom.map_zmultiples, AddMonoidHom.coe_coe, AddAut.mulRight_apply, Units.val_mul,
Units.val_mk0, Units.val_inv_eq_inv_val, Units.val_m... | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivAddCircle | The rescaling equivalence between additive circles with different periods. |
equivAddCircle_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
equivAddCircle p q hp hq (x : 𝕜) = (x * (p⁻¹ * q) : 𝕜) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivAddCircle_apply_mk | null |
equivAddCircle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
(equivAddCircle p q hp hq).symm (x : 𝕜) = (x * (q⁻¹ * p) : 𝕜) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivAddCircle_symm_apply_mk | null |
homeomorphAddCircle (hp : p ≠ 0) (hq : q ≠ 0) : AddCircle p ≃ₜ AddCircle q :=
⟨equivAddCircle p q hp hq,
(continuous_quotient_mk'.comp (continuous_mul_right (p⁻¹ * q))).quotient_lift _,
(continuous_quotient_mk'.comp (continuous_mul_right (q⁻¹ * p))).quotient_lift _⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | homeomorphAddCircle | The rescaling homeomorphism between additive circles with different periods. |
homeomorphAddCircle_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
homeomorphAddCircle p q hp hq (x : 𝕜) = (x * (p⁻¹ * q) : 𝕜) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | homeomorphAddCircle_apply_mk | null |
homeomorphAddCircle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
(homeomorphAddCircle p q hp hq).symm (x : 𝕜) = (x * (q⁻¹ * p) : 𝕜) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | homeomorphAddCircle_symm_apply_mk | null |
natCast_div_mul_eq_nsmul (r : 𝕜) (m : ℕ) :
(↑(↑m / q * r) : AddCircle p) = m • (r / q : AddCircle p) := by
rw [mul_comm_div, ← nsmul_eq_mul, coe_nsmul] | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | natCast_div_mul_eq_nsmul | null |
intCast_div_mul_eq_zsmul (r : 𝕜) (m : ℤ) :
(↑(↑m / q * r) : AddCircle p) = m • (r / q : AddCircle p) := by
rw [mul_comm_div, ← zsmul_eq_mul, coe_zsmul]
variable [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [hp : Fact (0 < p)] | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | intCast_div_mul_eq_zsmul | null |
@[simp]
coe_equivIco_mk_apply (x : 𝕜) :
(equivIco p 0 <| QuotientAddGroup.mk x : 𝕜) = Int.fract (x / p) * p :=
toIcoMod_eq_fract_mul _ x | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_equivIco_mk_apply | null |
@[simp] coe_fract (x : 𝕜) : (↑(Int.fract x) : AddCircle (1 : 𝕜)) = x := by
simp [← Int.self_sub_floor] | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_fract | null |
addOrderOf_period_div {n : ℕ} (h : 0 < n) : addOrderOf ((p / n : 𝕜) : AddCircle p) = n := by
rw [addOrderOf_eq_iff h]
replace h : 0 < (n : 𝕜) := Nat.cast_pos.2 h
refine ⟨?_, fun m hn h0 => ?_⟩ <;> simp only [Ne, ← coe_nsmul, nsmul_eq_mul]
· rw [mul_div_cancel₀ _ h.ne', coe_period]
rw [coe_eq_zero_of_pos_iff... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_period_div | null |
gcd_mul_addOrderOf_div_eq {n : ℕ} (m : ℕ) (hn : 0 < n) :
m.gcd n * addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n := by
rw [natCast_div_mul_eq_nsmul, IsOfFinAddOrder.addOrderOf_nsmul]
· rw [addOrderOf_period_div hn, Nat.gcd_comm, Nat.mul_div_cancel']
exact n.gcd_dvd_left m
· rwa [← addOrderOf_pos_iff, addO... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | gcd_mul_addOrderOf_div_eq | null |
addOrderOf_div_of_gcd_eq_one {m n : ℕ} (hn : 0 < n) (h : m.gcd n = 1) :
addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n := by
convert gcd_mul_addOrderOf_div_eq p m hn
rw [h, one_mul] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_div_of_gcd_eq_one | null |
addOrderOf_div_of_gcd_eq_one' {m : ℤ} {n : ℕ} (hn : 0 < n) (h : m.natAbs.gcd n = 1) :
addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n := by
cases m
· simp only [Int.ofNat_eq_coe, Int.cast_natCast, Int.natAbs_natCast] at h ⊢
exact addOrderOf_div_of_gcd_eq_one hn h
· simp only [Int.cast_negSucc, neg_div, neg_... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_div_of_gcd_eq_one' | null |
addOrderOf_coe_rat {q : ℚ} : addOrderOf (↑(↑q * p) : AddCircle p) = q.den := by
have : (↑(q.den : ℤ) : 𝕜) ≠ 0 := by
norm_cast
exact q.pos.ne.symm
rw [← q.num_divInt_den, Rat.cast_divInt_of_ne_zero _ this, Int.cast_natCast, Rat.num_divInt_den,
addOrderOf_div_of_gcd_eq_one' q.pos q.reduced] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_coe_rat | null |
nsmul_eq_zero_iff {u : AddCircle p} {n : ℕ} (h : 0 < n) :
n • u = 0 ↔ ∃ m < n, ↑(↑m / ↑n * p) = u := by
refine ⟨QuotientAddGroup.induction_on u fun k hk ↦ ?_, ?_⟩
· rw [← addOrderOf_dvd_iff_nsmul_eq_zero]
rintro ⟨m, -, rfl⟩
constructor; rw [mul_comm, eq_comm]
exact gcd_mul_addOrderOf_div_eq p m h
... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | nsmul_eq_zero_iff | null |
addOrderOf_eq_pos_iff {u : AddCircle p} {n : ℕ} (h : 0 < n) :
addOrderOf u = n ↔ ∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n * p) = u := by
refine ⟨QuotientAddGroup.induction_on u ?_, ?_⟩
· rintro ⟨m, -, h₁, rfl⟩
exact addOrderOf_div_of_gcd_eq_one h h₁
rintro k rfl
obtain ⟨m, hm, hk⟩ := (nsmul_eq_zero_iff h).mp (a... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_eq_pos_iff | null |
exists_gcd_eq_one_of_isOfFinAddOrder {u : AddCircle p} (h : IsOfFinAddOrder u) :
∃ m : ℕ, m.gcd (addOrderOf u) = 1 ∧ m < addOrderOf u ∧ ↑((m : 𝕜) / addOrderOf u * p) = u :=
let ⟨m, hl, hg, he⟩ := (addOrderOf_eq_pos_iff h.addOrderOf_pos).1 rfl
⟨m, hg, hl, he⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | exists_gcd_eq_one_of_isOfFinAddOrder | null |
not_isOfFinAddOrder_iff_forall_rat_ne_div {a : 𝕜} :
¬ IsOfFinAddOrder (a : AddCircle p) ↔ ∀ q : ℚ, (q : 𝕜) ≠ a / p := by
simp +contextual [← QuotientAddGroup.mk_zsmul, mul_comm (Int.cast _), mem_zmultiples_iff,
eq_div_iff (Fact.out : 0 < p).ne', isOfFinAddOrder_iff_zsmul_eq_zero, Rat.forall, div_eq_iff,
... | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | not_isOfFinAddOrder_iff_forall_rat_ne_div | null |
isOfFinAddOrder_iff_exists_rat_eq_div {a : 𝕜} :
IsOfFinAddOrder (a : AddCircle p) ↔ ∃ q : ℚ, (q : 𝕜) = a / p := by
simpa using not_isOfFinAddOrder_iff_forall_rat_ne_div.not_right
@[deprecated not_isOfFinAddOrder_iff_forall_rat_ne_div (since := "2025-08-13")] | lemma | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | isOfFinAddOrder_iff_exists_rat_eq_div | null |
addOrderOf_coe_eq_zero_iff_forall_rat_ne_div {a : 𝕜} :
addOrderOf (a : AddCircle p) = 0 ↔ ∀ q : ℚ, (q : 𝕜) ≠ a / p := by
simp [not_isOfFinAddOrder_iff_forall_rat_ne_div]
variable (p) | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | addOrderOf_coe_eq_zero_iff_forall_rat_ne_div | null |
setAddOrderOfEquiv {n : ℕ} (hn : 0 < n) :
{ u : AddCircle p | addOrderOf u = n } ≃ { m | m < n ∧ m.gcd n = 1 } :=
Equiv.symm <|
Equiv.ofBijective (fun m => ⟨↑((m : 𝕜) / n * p), addOrderOf_div_of_gcd_eq_one hn m.prop.2⟩)
(by
refine ⟨fun m₁ m₂ h => Subtype.ext ?_, fun u => ?_⟩
· simp_rw [... | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | setAddOrderOfEquiv | The natural bijection between points of order `n` and natural numbers less than and coprime to
`n`. The inverse of the map sends `m ↦ (m/n * p : AddCircle p)` where `m` is coprime to `n` and
satisfies `0 ≤ m < n`. |
card_addOrderOf_eq_totient {n : ℕ} :
Nat.card { u : AddCircle p // addOrderOf u = n } = n.totient := by
rcases n.eq_zero_or_pos with (rfl | hn)
· simp only [Nat.totient_zero, addOrderOf_eq_zero_iff]
rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ | h)
· have : Infinite { u : AddCircle p ... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | card_addOrderOf_eq_totient | null |
finite_setOf_addOrderOf_eq {n : ℕ} (hn : 0 < n) :
{u : AddCircle p | addOrderOf u = n}.Finite :=
finite_coe_iff.mp <| Nat.finite_of_card_ne_zero <| by simp [hn.ne']
@[deprecated (since := "2025-03-26")]
alias finite_setOf_add_order_eq := finite_setOf_addOrderOf_eq | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | finite_setOf_addOrderOf_eq | null |
finite_torsion {n : ℕ} (hn : 0 < n) :
{ u : AddCircle p | n • u = 0 }.Finite := by
convert Set.finite_range (fun m : Fin n ↦ (↑(↑m / ↑n * p) : AddCircle p))
simp_rw [nsmul_eq_zero_iff hn, range, Fin.exists_iff, exists_prop] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | finite_torsion | null |
EndpointIdent : Icc a (a + p) → Icc a (a + p) → Prop
| mk :
EndpointIdent ⟨a, left_mem_Icc.mpr <| le_add_of_nonneg_right hp.out.le⟩
⟨a + p, right_mem_Icc.mpr <| le_add_of_nonneg_right hp.out.le⟩
variable [Archimedean 𝕜] | inductive | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | EndpointIdent | The relation identifying the endpoints of `Icc a (a + p)`. |
equivIccQuot : 𝕋 ≃ Quot (EndpointIdent p a) where
toFun x := Quot.mk _ <| inclusion Ico_subset_Icc_self (equivIco _ _ x)
invFun x :=
Quot.liftOn x (↑) <| by
rintro _ _ ⟨_⟩
exact (coe_add_period p a).symm
left_inv := (equivIco p a).symm_apply_apply
right_inv :=
Quot.ind <| by
rintro ⟨x... | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivIccQuot | The equivalence between `AddCircle p` and the quotient of `[a, a + p]` by the relation
identifying the endpoints. |
equivIccQuot_comp_mk_eq_toIcoMod :
equivIccQuot p a ∘ Quotient.mk'' = fun x =>
Quot.mk _ ⟨toIcoMod hp.out a x, Ico_subset_Icc_self <| toIcoMod_mem_Ico _ _ x⟩ :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivIccQuot_comp_mk_eq_toIcoMod | null |
equivIccQuot_comp_mk_eq_toIocMod :
equivIccQuot p a ∘ Quotient.mk'' = fun x =>
Quot.mk _ ⟨toIocMod hp.out a x, Ioc_subset_Icc_self <| toIocMod_mem_Ioc _ _ x⟩ := by
rw [equivIccQuot_comp_mk_eq_toIcoMod]
funext x
by_cases h : a ≡ x [PMOD p]
· simp_rw [(modEq_iff_toIcoMod_eq_left hp.out).1 h, (modEq_iff_... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivIccQuot_comp_mk_eq_toIocMod | null |
homeoIccQuot [TopologicalSpace 𝕜] [OrderTopology 𝕜] : 𝕋 ≃ₜ Quot (EndpointIdent p a) where
toEquiv := equivIccQuot p a
continuous_toFun := by
simp_rw [isQuotientMap_quotient_mk'.continuous_iff, continuous_iff_continuousAt,
continuousAt_iff_continuous_left_right]
intro x; constructor
on_goal 1 =>... | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | homeoIccQuot | The natural map from `[a, a + p] ⊂ 𝕜` with endpoints identified to `𝕜 / ℤ • p`, as a
homeomorphism of topological spaces. |
liftIoc_eq_liftIco {f : 𝕜 → B} (hf : f a = f (a + p)) :
liftIoc p a f = liftIco p a f := by
ext q
obtain ⟨x, hx, rfl⟩ := by simpa only [mem_image] using coe_image_Ico_eq p a ▸ mem_univ q
rw [liftIco_coe_apply hx]
obtain (⟨rfl, -⟩ | h) := by rwa [mem_Ico, le_iff_eq_or_lt, or_and_right] at hx
· rw [← coe_a... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_eq_liftIco | null |
liftIco_eq_lift_Icc {f : 𝕜 → B} (h : f a = f (a + p)) :
liftIco p a f =
Quot.lift (restrict (Icc a <| a + p) f)
(by
rintro _ _ ⟨_⟩
exact h) ∘
equivIccQuot p a :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco_eq_lift_Icc | null |
liftIoc_eq_lift_Icc {f : 𝕜 → B} (h : f a = f (a + p)) :
liftIoc p a f =
Quot.lift (restrict (Icc a <| a + p) f)
(by
rintro _ _ ⟨_⟩
exact h) ∘
equivIccQuot p a := by
rw [← liftIco_eq_lift_Icc h]
exact liftIoc_eq_liftIco h | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_eq_lift_Icc | null |
liftIco_zero_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico 0 p) : liftIco p 0 f ↑x = f x :=
liftIco_coe_apply (by rwa [zero_add]) | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco_zero_coe_apply | null |
liftIoc_zero_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ioc 0 p) : liftIoc p 0 f ↑x = f x :=
liftIoc_coe_apply (by rwa [zero_add])
variable [TopologicalSpace 𝕜] [OrderTopology 𝕜] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_zero_coe_apply | null |
liftIco_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f a = f (a + p))
(hc : ContinuousOn f <| Icc a (a + p)) : Continuous (liftIco p a f) := by
rw [liftIco_eq_lift_Icc hf]
refine Continuous.comp ?_ (homeoIccQuot p a).continuous_toFun
exact continuous_coinduced_dom.mpr (continuousOn_iff_continuous_restri... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco_continuous | null |
liftIco_zero_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f 0 = f p)
(hc : ContinuousOn f <| Icc 0 p) : Continuous (liftIco p 0 f) :=
liftIco_continuous (by rwa [zero_add] : f 0 = f (0 + p)) (by rwa [zero_add]) | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIco_zero_continuous | null |
liftIoc_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f a = f (a + p))
(hc : ContinuousOn f <| Icc a (a + p)) : Continuous (liftIoc p a f) := by
rw [liftIoc_eq_lift_Icc hf]
refine Continuous.comp ?_ (homeoIccQuot p a).continuous_toFun
exact continuous_coinduced_dom.mpr (continuousOn_iff_continuous_restri... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_continuous | null |
liftIoc_zero_continuous [TopologicalSpace B] {f : 𝕜 → B} (hf : f 0 = f p)
(hc : ContinuousOn f <| Icc 0 p) : Continuous (liftIoc p 0 f) :=
liftIoc_continuous (by rwa [zero_add] : f 0 = f (0 + p)) (by rwa [zero_add]) | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | liftIoc_zero_continuous | null |
dense_addSubgroupClosure_pair_iff {a b : ℝ} :
Dense (AddSubgroup.closure {a, b} : Set ℝ) ↔ Irrational (a / b) := by
rcases eq_or_ne b 0 with rfl | hb
· rw [pair_comm]
simpa [← AddSubgroup.zmultiples_eq_closure] using not_denseRange_zsmul
constructor
· rintro hd ⟨r, hr⟩
refine not_denseRange_zsmul (a... | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Algebra.Order.Archimedean"
] | Mathlib/Topology/Instances/AddCircle/DenseSubgroup.lean | dense_addSubgroupClosure_pair_iff | The additive subgroup of real numbers generated by `a` and `b` is dense
iff `a / b` is an irrational number.
Here we rely on the fact that `a / 0 = 0` in Mathlib,
so we don't need to add `b ≠ 0` as an assumption. |
denseRange_zsmul_coe_iff {a p : ℝ} :
DenseRange (· • a : ℤ → AddCircle p) ↔ Irrational (a / p) := by
rw [← dense_addSubgroupClosure_pair_iff, DenseRange,
← QuotientAddGroup.dense_preimage_mk, ← QuotientAddGroup.coe_mk',
← AddSubgroup.coe_zmultiples, ← AddSubgroup.coe_comap, ← AddMonoidHom.map_zmultiples,
... | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Algebra.Order.Archimedean"
] | Mathlib/Topology/Instances/AddCircle/DenseSubgroup.lean | denseRange_zsmul_coe_iff | The multiples of a number `a` are dense on a circle of length `p` iff `a / p` is irrational. |
denseRange_zsmul_iff {p : ℝ} [Fact (0 < p)] {a : AddCircle p} :
DenseRange (· • a : ℤ → AddCircle p) ↔ addOrderOf a = 0 := by
rcases QuotientAddGroup.mk_surjective a with ⟨a, rfl⟩
simp [denseRange_zsmul_coe_iff, isOfFinAddOrder_iff_exists_rat_eq_div, Irrational] | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Algebra.Order.Archimedean"
] | Mathlib/Topology/Instances/AddCircle/DenseSubgroup.lean | denseRange_zsmul_iff | The multiples of a number `a` are dense on a circle of length `p > 0`
iff `a` has infinite additive order. |
dense_addSubgroup_iff_ne_zmultiples {p : ℝ} [Fact (0 < p)] {s : AddSubgroup (AddCircle p)} :
Dense (s : Set (AddCircle p)) ↔ ∀ a, addOrderOf a ≠ 0 → s ≠ .zmultiples a := by
constructor
· rintro hd a ha rfl
rw [AddSubgroup.coe_zmultiples, ← DenseRange, denseRange_zsmul_iff] at hd
exact ha hd
· intro h
... | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Algebra.Order.Archimedean"
] | Mathlib/Topology/Instances/AddCircle/DenseSubgroup.lean | dense_addSubgroup_iff_ne_zmultiples | A subgroup of the circle `ℝ⧸pℤ`, `p > 0`, is dense
iff it is not generated by a single element of finite additive order. |
pathConnectedSpace : PathConnectedSpace <| AddCircle p :=
(inferInstance : PathConnectedSpace (Quotient _)) | instance | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | pathConnectedSpace | null |
compactSpace [Fact (0 < p)] : CompactSpace <| AddCircle p := by
rw [← isCompact_univ_iff, ← coe_image_Icc_eq p 0]
exact isCompact_Icc.image (AddCircle.continuous_mk' p) | instance | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | compactSpace | The "additive circle" `ℝ ⧸ (ℤ ∙ p)` is compact. |
UnitAddCircle :=
AddCircle (1 : ℝ) | abbrev | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | UnitAddCircle | The action on `ℝ` by right multiplication of its the subgroup `zmultiples p` (the multiples of
`p:ℝ`) is properly discontinuous. -/
instance : ProperlyDiscontinuousVAdd (zmultiples p).op ℝ :=
(zmultiples p).properlyDiscontinuousVAdd_opposite_of_tendsto_cofinite
(AddSubgroup.tendsto_zmultiples_subtype_cofinite p)
... |
noncomputable toAddCircle : ZMod N →+ UnitAddCircle :=
lift N ⟨AddMonoidHom.mk' (fun j ↦ ↑(j / N : ℝ)) (by simp [add_div]),
by simp [div_self (NeZero.ne _)]⟩ | def | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | toAddCircle | The `AddMonoidHom` from `ZMod N` to `ℝ / ℤ` sending `j mod N` to `j / N mod 1`. |
toAddCircle_intCast (j : ℤ) :
toAddCircle (j : ZMod N) = ↑(j / N : ℝ) := by
simp [toAddCircle] | lemma | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | toAddCircle_intCast | null |
toAddCircle_natCast (j : ℕ) :
toAddCircle (j : ZMod N) = ↑(j / N : ℝ) := by
simpa using toAddCircle_intCast (N := N) j | lemma | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | toAddCircle_natCast | null |
toAddCircle_apply (j : ZMod N) :
toAddCircle j = ↑(j.val / N : ℝ) := by
rw [← toAddCircle_natCast, natCast_zmod_val]
variable (N) in | lemma | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | toAddCircle_apply | Explicit formula for `toCircle j`. Note that this is "evil" because it uses `ZMod.val`. Where
possible, it is recommended to lift `j` to `ℤ` and use `toAddCircle_intCast` instead. |
toAddCircle_injective : Function.Injective (toAddCircle : ZMod N → _) := by
intro x y hxy
have : (0 : ℝ) < N := Nat.cast_pos.mpr (NeZero.pos _)
rwa [toAddCircle_apply, toAddCircle_apply, AddCircle.coe_eq_coe_iff_of_mem_Ico,
div_left_inj' this.ne', Nat.cast_inj, (val_injective N).eq_iff] at hxy <;>
exact ⟨by... | lemma | Topology | [
"Mathlib.Topology.Connected.PathConnected",
"Mathlib.Topology.Instances.AddCircle.Defs",
"Mathlib.Topology.Instances.ZMultiples"
] | Mathlib/Topology/Instances/AddCircle/Real.lean | toAddCircle_injective | null |
isOpen_ne_top : IsOpen { a : ℝ≥0∞ | a ≠ ∞ } := isOpen_ne | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | isOpen_ne_top | null |
isOpen_Ico_zero : IsOpen (Ico 0 b) := by
rw [ENNReal.Ico_eq_Iio]
exact isOpen_Iio | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | isOpen_Ico_zero | null |
coe_range_mem_nhds : range ((↑) : ℝ≥0 → ℝ≥0∞) ∈ 𝓝 (r : ℝ≥0∞) :=
IsOpen.mem_nhds isOpenEmbedding_coe.isOpen_range <| mem_range_self _
@[fun_prop] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | coe_range_mem_nhds | null |
continuous_coe : Continuous ((↑) : ℝ≥0 → ℝ≥0∞) :=
isEmbedding_coe.continuous | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_coe | null |
tendsto_coe_toNNReal {a : ℝ≥0∞} (ha : a ≠ ⊤) : Tendsto (↑) (𝓝 a.toNNReal) (𝓝 a) := by
nth_rewrite 2 [← coe_toNNReal ha]
exact continuous_coe.tendsto _ | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_coe_toNNReal | null |
continuous_coe_iff {α} [TopologicalSpace α] {f : α → ℝ≥0} :
(Continuous fun a => (f a : ℝ≥0∞)) ↔ Continuous f :=
isEmbedding_coe.continuous_iff.symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_coe_iff | null |
nhds_coe {r : ℝ≥0} : 𝓝 (r : ℝ≥0∞) = (𝓝 r).map (↑) :=
(isOpenEmbedding_coe.map_nhds_eq r).symm | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_coe | null |
tendsto_nhds_coe_iff {α : Type*} {l : Filter α} {x : ℝ≥0} {f : ℝ≥0∞ → α} :
Tendsto f (𝓝 ↑x) l ↔ Tendsto (f ∘ (↑) : ℝ≥0 → α) (𝓝 x) l := by
rw [nhds_coe, tendsto_map'_iff] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_coe_iff | null |
continuousAt_coe_iff {α : Type*} [TopologicalSpace α] {x : ℝ≥0} {f : ℝ≥0∞ → α} :
ContinuousAt f ↑x ↔ ContinuousAt (f ∘ (↑) : ℝ≥0 → α) x :=
tendsto_nhds_coe_iff | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousAt_coe_iff | null |
continuous_ofReal : Continuous ENNReal.ofReal :=
(continuous_coe_iff.2 continuous_id).comp continuous_real_toNNReal | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuous_ofReal | null |
tendsto_ofReal {f : Filter α} {m : α → ℝ} {a : ℝ} (h : Tendsto m f (𝓝 a)) :
Tendsto (fun a => ENNReal.ofReal (m a)) f (𝓝 (ENNReal.ofReal a)) :=
(continuous_ofReal.tendsto a).comp h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_ofReal | null |
tendsto_toNNReal {a : ℝ≥0∞} (ha : a ≠ ∞) :
Tendsto ENNReal.toNNReal (𝓝 a) (𝓝 a.toNNReal) := by
lift a to ℝ≥0 using ha
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_toNNReal | null |
tendsto_toNNReal_iff {f : α → ℝ≥0∞} {u : Filter α} (ha : a ≠ ∞) (hf : ∀ x, f x ≠ ∞) :
Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 (a.toNNReal)) ↔ Tendsto f u (𝓝 a) := by
refine ⟨fun h => ?_, fun h => (ENNReal.tendsto_toNNReal ha).comp h⟩
rw [← coe_comp_toNNReal_comp hf]
exact (tendsto_coe_toNNReal ha).comp h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_toNNReal_iff | null |
tendsto_toNNReal_iff' {f : α → ℝ≥0∞} {u : Filter α} {a : ℝ≥0} (hf : ∀ x, f x ≠ ∞) :
Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 a) ↔ Tendsto f u (𝓝 a) := by
rw [← toNNReal_coe a]
exact tendsto_toNNReal_iff coe_ne_top hf | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_toNNReal_iff' | null |
eventuallyEq_of_toReal_eventuallyEq {l : Filter α} {f g : α → ℝ≥0∞}
(hfi : ∀ᶠ x in l, f x ≠ ∞) (hgi : ∀ᶠ x in l, g x ≠ ∞)
(hfg : (fun x => (f x).toReal) =ᶠ[l] fun x => (g x).toReal) : f =ᶠ[l] g := by
filter_upwards [hfi, hgi, hfg] with _ hfx hgx _
rwa [← ENNReal.toReal_eq_toReal hfx hgx] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | eventuallyEq_of_toReal_eventuallyEq | null |
continuousOn_toNNReal : ContinuousOn ENNReal.toNNReal { a | a ≠ ∞ } := fun _a ha =>
ContinuousAt.continuousWithinAt (tendsto_toNNReal ha) | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousOn_toNNReal | null |
tendsto_toReal {a : ℝ≥0∞} (ha : a ≠ ∞) : Tendsto ENNReal.toReal (𝓝 a) (𝓝 a.toReal) :=
NNReal.tendsto_coe.2 <| tendsto_toNNReal ha | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_toReal | null |
continuousOn_toReal : ContinuousOn ENNReal.toReal { a | a ≠ ∞ } :=
NNReal.continuous_coe.comp_continuousOn continuousOn_toNNReal | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousOn_toReal | null |
continuousAt_toReal (hx : x ≠ ∞) : ContinuousAt ENNReal.toReal x :=
continuousOn_toReal.continuousAt (isOpen_ne_top.mem_nhds_iff.mpr hx) | lemma | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | continuousAt_toReal | null |
neTopHomeomorphNNReal : { a | a ≠ ∞ } ≃ₜ ℝ≥0 where
toEquiv := neTopEquivNNReal
continuous_toFun := continuousOn_iff_continuous_restrict.1 continuousOn_toNNReal
continuous_invFun := continuous_coe.subtype_mk _ | def | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | neTopHomeomorphNNReal | The set of finite `ℝ≥0∞` numbers is homeomorphic to `ℝ≥0`. |
ltTopHomeomorphNNReal : { a | a < ∞ } ≃ₜ ℝ≥0 := by
refine (Homeomorph.setCongr ?_).trans neTopHomeomorphNNReal
simp only [lt_top_iff_ne_top] | def | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | ltTopHomeomorphNNReal | The set of finite `ℝ≥0∞` numbers is homeomorphic to `ℝ≥0`. |
nhds_top : 𝓝 ∞ = ⨅ (a) (_ : a ≠ ∞), 𝓟 (Ioi a) :=
nhds_top_order.trans <| by simp [lt_top_iff_ne_top, Ioi] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_top | null |
nhds_top' : 𝓝 ∞ = ⨅ r : ℝ≥0, 𝓟 (Ioi ↑r) :=
nhds_top.trans <| iInf_ne_top _ | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_top' | null |
nhds_top_basis : (𝓝 ∞).HasBasis (fun a => a < ∞) fun a => Ioi a :=
_root_.nhds_top_basis | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_top_basis | null |
tendsto_nhds_top_iff_nnreal {m : α → ℝ≥0∞} {f : Filter α} :
Tendsto m f (𝓝 ∞) ↔ ∀ x : ℝ≥0, ∀ᶠ a in f, ↑x < m a := by
simp only [nhds_top', tendsto_iInf, tendsto_principal, mem_Ioi] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_top_iff_nnreal | null |
tendsto_nhds_top_iff_nat {m : α → ℝ≥0∞} {f : Filter α} :
Tendsto m f (𝓝 ∞) ↔ ∀ n : ℕ, ∀ᶠ a in f, ↑n < m a :=
tendsto_nhds_top_iff_nnreal.trans
⟨fun h n => by simpa only [ENNReal.coe_natCast] using h n, fun h x =>
let ⟨n, hn⟩ := exists_nat_gt x
(h n).mono fun _ => lt_trans <| by rwa [← ENNReal.coe... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_top_iff_nat | null |
tendsto_nhds_top {m : α → ℝ≥0∞} {f : Filter α} (h : ∀ n : ℕ, ∀ᶠ a in f, ↑n < m a) :
Tendsto m f (𝓝 ∞) :=
tendsto_nhds_top_iff_nat.2 h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nhds_top | null |
tendsto_nat_nhds_top : Tendsto (fun n : ℕ => ↑n) atTop (𝓝 ∞) :=
tendsto_nhds_top fun n =>
mem_atTop_sets.2 ⟨n + 1, fun _m hm => mem_setOf.2 <| Nat.cast_lt.2 <| Nat.lt_of_succ_le hm⟩
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_nat_nhds_top | null |
tendsto_coe_nhds_top {f : α → ℝ≥0} {l : Filter α} :
Tendsto (fun x => (f x : ℝ≥0∞)) l (𝓝 ∞) ↔ Tendsto f l atTop := by
rw [tendsto_nhds_top_iff_nnreal, atTop_basis_Ioi.tendsto_right_iff]; simp
@[simp] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_coe_nhds_top | null |
tendsto_ofReal_nhds_top {f : α → ℝ} {l : Filter α} :
Tendsto (fun x ↦ ENNReal.ofReal (f x)) l (𝓝 ∞) ↔ Tendsto f l atTop :=
tendsto_coe_nhds_top.trans Real.tendsto_toNNReal_atTop_iff | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_ofReal_nhds_top | null |
tendsto_ofReal_atTop : Tendsto ENNReal.ofReal atTop (𝓝 ∞) :=
tendsto_ofReal_nhds_top.2 tendsto_id | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | tendsto_ofReal_atTop | null |
nhds_zero : 𝓝 (0 : ℝ≥0∞) = ⨅ (a) (_ : a ≠ 0), 𝓟 (Iio a) :=
nhds_bot_order.trans <| by simp [pos_iff_ne_zero, Iio] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Data.ENNReal.BigOperators",
"Mathlib.Tactic.Bound",
"Mathlib.Topology.Order.LiminfLimsup",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Instances.NNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Real",
"Mathlib.Topology.MetricSpace... | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | nhds_zero | null |
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