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 ⌀ |
|---|---|---|---|---|---|---|
bdd_le_mul_tendsto_zero' {f g : α → 𝕜} (C : 𝕜) (hf : ∀ᶠ x in l, |f x| ≤ C)
(hg : Tendsto g l (𝓝 0)) : Tendsto (fun x ↦ f x * g x) l (𝓝 0) := by
rw [tendsto_zero_iff_abs_tendsto_zero]
have hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0) := by
convert (hg.const_mul C).abs
simp_rw [mul_zero, abs_zero]
appl... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | bdd_le_mul_tendsto_zero' | If `g` tends to zero and there exists a constant `C : 𝕜` such that eventually `|f x| ≤ C`,
then the product `f * g` tends to zero. |
bdd_le_mul_tendsto_zero {f g : α → 𝕜} {b B : 𝕜} (hb : ∀ᶠ x in l, b ≤ f x)
(hB : ∀ᶠ x in l, f x ≤ B) (hg : Tendsto g l (𝓝 0)) :
Tendsto (fun x ↦ f x * g x) l (𝓝 0) := by
set C := max |b| |B|
have hbC : -C ≤ b := neg_le.mpr (le_max_of_le_left (neg_le_abs b))
have hBC : B ≤ C := le_max_of_le_right (le_ab... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | bdd_le_mul_tendsto_zero | If `g` tends to zero and there exist constants `b B : 𝕜` such that eventually `b ≤ f x| ≤ B`,
then the product `f * g` tends to zero. |
tendsto_bdd_div_atTop_nhds_zero {f g : α → 𝕜} {b B : 𝕜}
(hb : ∀ᶠ x in l, b ≤ f x) (hB : ∀ᶠ x in l, f x ≤ B) (hg : Tendsto g l atTop) :
Tendsto (fun x => f x / g x) l (𝓝 0) := by
simp only [div_eq_mul_inv]
exact bdd_le_mul_tendsto_zero hb hB hg.inv_tendsto_atTop | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | tendsto_bdd_div_atTop_nhds_zero | If `g` tends to `atTop` and there exist constants `b B : 𝕜` such that eventually
`b ≤ f x| ≤ B`, then the quotient `f / g` tends to zero. |
tendsto_const_mul_zpow_atTop_zero {n : ℤ} {c : 𝕜} (hn : n < 0) :
Tendsto (fun x => c * x ^ n) atTop (𝓝 0) :=
mul_zero c ▸ Filter.Tendsto.const_mul c (tendsto_zpow_atTop_zero hn) | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | tendsto_const_mul_zpow_atTop_zero | null |
tendsto_const_mul_pow_nhds_iff' {n : ℕ} {c d : 𝕜} :
Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp [tendsto_const_nhds_iff]
rcases lt_trichotomy c 0 with (hc | rfl | hc)
· have := tendsto_const_mul_pow_atBot_iff.2 ⟨hn, hc⟩
simp ... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | tendsto_const_mul_pow_nhds_iff' | null |
tendsto_const_mul_pow_nhds_iff {n : ℕ} {c d : 𝕜} (hc : c ≠ 0) :
Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d := by
simp [tendsto_const_mul_pow_nhds_iff', hc] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | tendsto_const_mul_pow_nhds_iff | null |
tendsto_const_mul_zpow_atTop_nhds_iff {n : ℤ} {c d : 𝕜} (hc : c ≠ 0) :
Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩
· cases n with
| ofNat n =>
left
simpa [tendsto_const_mul_pow_nhds_iff hc] using h
| negSucc n =>
... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | tendsto_const_mul_zpow_atTop_nhds_iff | null |
comap_mulLeft_nhdsGT_zero {x : 𝕜} (hx : 0 < x) : comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0 := by
rw [nhdsWithin, comap_inf, comap_principal, preimage_const_mul_Ioi _ hx, zero_div]
congr 1
refine ((Homeomorph.mulLeft₀ x hx.ne').comap_nhds_eq _).trans ?_
simp | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | comap_mulLeft_nhdsGT_zero | null |
eventually_nhdsGT_zero_mul_left {x : 𝕜} (hx : 0 < x) {p : 𝕜 → Prop}
(h : ∀ᶠ ε in 𝓝[>] 0, p ε) : ∀ᶠ ε in 𝓝[>] 0, p (x * ε) := by
rw [← comap_mulLeft_nhdsGT_zero hx]
exact h.comap fun ε => x * ε | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Field.lean | eventually_nhdsGT_zero_mul_left | null |
tendsto_mul_pow_div_factorial_sub_atTop (a c : K) (d : ℕ) :
Tendsto (fun n ↦ a * c ^ n / (n - d)!) atTop (𝓝 0) := by
rw [tendsto_order]
constructor
all_goals
intro ε hε
filter_upwards [eventually_mul_pow_lt_factorial_sub (a * ε⁻¹) c d] with n h
rw [mul_right_comm, ← div_eq_mul_inv] at h
· rw [d... | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_mul_pow_div_factorial_sub_atTop | null |
tendsto_pow_div_factorial_atTop (c : K) :
Tendsto (fun n ↦ c ^ n / n !) atTop (𝓝 0) := by
convert tendsto_mul_pow_div_factorial_sub_atTop 1 c 0
rw [one_mul] | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_pow_div_factorial_atTop | null |
tendsto_floor_atTop : Tendsto (floor : α → ℤ) atTop atTop :=
floor_mono.tendsto_atTop_atTop fun b =>
⟨(b + 1 : ℤ), by rw [floor_intCast]; exact (lt_add_one _).le⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_atTop | null |
tendsto_floor_atBot : Tendsto (floor : α → ℤ) atBot atBot :=
floor_mono.tendsto_atBot_atBot fun b => ⟨b, (floor_intCast _).le⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_atBot | null |
tendsto_ceil_atTop : Tendsto (ceil : α → ℤ) atTop atTop :=
ceil_mono.tendsto_atTop_atTop fun b => ⟨b, (ceil_intCast _).ge⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_atTop | null |
tendsto_ceil_atBot : Tendsto (ceil : α → ℤ) atBot atBot :=
ceil_mono.tendsto_atBot_atBot fun b =>
⟨(b - 1 : ℤ), by rw [ceil_intCast]; exact (sub_one_lt _).le⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_atBot | null |
continuousOn_floor (n : ℤ) :
ContinuousOn (fun x => floor x : α → α) (Ico n (n + 1) : Set α) :=
(continuousOn_congr <| floor_eq_on_Ico' n).mpr continuousOn_const | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | continuousOn_floor | null |
continuousOn_ceil [IsStrictOrderedRing α] (n : ℤ) :
ContinuousOn (fun x => ceil x : α → α) (Ioc (n - 1) n : Set α) :=
(continuousOn_congr <| ceil_eq_on_Ioc' n).mpr continuousOn_const | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | continuousOn_ceil | null |
tendsto_floor_right_pure_floor (x : α) : Tendsto (floor : α → ℤ) (𝓝[≥] x) (pure ⌊x⌋) :=
tendsto_pure.2 <| mem_of_superset (Ico_mem_nhdsGE <| lt_floor_add_one x) fun _y hy =>
floor_eq_on_Ico _ _ ⟨(floor_le x).trans hy.1, hy.2⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_right_pure_floor | null |
tendsto_floor_right_pure (n : ℤ) : Tendsto (floor : α → ℤ) (𝓝[≥] n) (pure n) := by
simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_right_pure | null |
tendsto_ceil_left_pure_ceil (x : α) : Tendsto (ceil : α → ℤ) (𝓝[≤] x) (pure ⌈x⌉) :=
tendsto_pure.2 <| mem_of_superset
(Ioc_mem_nhdsLE <| sub_lt_iff_lt_add.2 <| ceil_lt_add_one _) fun _y hy =>
ceil_eq_on_Ioc _ _ ⟨hy.1, hy.2.trans (le_ceil _)⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left_pure_ceil | null |
tendsto_ceil_left_pure (n : ℤ) : Tendsto (ceil : α → ℤ) (𝓝[≤] n) (pure n) := by
simpa only [ceil_intCast] using tendsto_ceil_left_pure_ceil (n : α) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left_pure | null |
tendsto_floor_left_pure_ceil_sub_one (x : α) :
Tendsto (floor : α → ℤ) (𝓝[<] x) (pure (⌈x⌉ - 1)) :=
have h₁ : ↑(⌈x⌉ - 1) < x := by rw [cast_sub, cast_one, sub_lt_iff_lt_add]; exact ceil_lt_add_one _
have h₂ : x ≤ ↑(⌈x⌉ - 1) + 1 := by rw [cast_sub, cast_one, sub_add_cancel]; exact le_ceil _
tendsto_pure.2 <| ... | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_left_pure_ceil_sub_one | null |
tendsto_floor_left_pure_sub_one (n : ℤ) :
Tendsto (floor : α → ℤ) (𝓝[<] n) (pure (n - 1)) := by
simpa only [ceil_intCast] using tendsto_floor_left_pure_ceil_sub_one (n : α)
omit [IsStrictOrderedRing α] in | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_left_pure_sub_one | null |
tendsto_ceil_right_pure_floor_add_one (x : α) :
Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) :=
have : ↑(⌊x⌋ + 1) - 1 ≤ x := by rw [cast_add, cast_one, add_sub_cancel_right]; exact floor_le _
tendsto_pure.2 <| mem_of_superset (Ioc_mem_nhdsGT <| lt_succ_floor _) fun _y hy =>
ceil_eq_on_Ioc _ _ ⟨this.tra... | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right_pure_floor_add_one | null |
tendsto_ceil_right_pure_add_one (n : ℤ) :
Tendsto (ceil : α → ℤ) (𝓝[>] n) (pure (n + 1)) := by
simpa only [floor_intCast] using tendsto_ceil_right_pure_floor_add_one (n : α) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right_pure_add_one | null |
tendsto_floor_right (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝[≥] n) :=
((tendsto_pure_pure _ _).comp (tendsto_floor_right_pure n)).mono_right <|
pure_le_nhdsWithin le_rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_right | null |
tendsto_floor_right' (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝 n) :=
(tendsto_floor_right n).mono_right inf_le_left | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_right' | null |
tendsto_ceil_left (n : ℤ) : Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝[≤] n) :=
((tendsto_pure_pure _ _).comp (tendsto_ceil_left_pure n)).mono_right <|
pure_le_nhdsWithin le_rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left | null |
tendsto_ceil_left' (n : ℤ) :
Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝 n) :=
(tendsto_ceil_left n).mono_right inf_le_left | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_left' | null |
tendsto_floor_left (n : ℤ) :
Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝[≤] (n - 1)) :=
((tendsto_pure_pure _ _).comp (tendsto_floor_left_pure_sub_one n)).mono_right <| by
rw [← @cast_one α, ← cast_sub]; exact pure_le_nhdsWithin le_rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_left | null |
tendsto_ceil_right (n : ℤ) :
Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝[≥] (n + 1)) :=
((tendsto_pure_pure _ _).comp (tendsto_ceil_right_pure_add_one n)).mono_right <| by
rw [← @cast_one α, ← cast_add]; exact pure_le_nhdsWithin le_rfl | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right | null |
tendsto_floor_left' (n : ℤ) :
Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝 (n - 1)) :=
(tendsto_floor_left n).mono_right inf_le_left | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_left' | null |
tendsto_ceil_right' (n : ℤ) :
Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝 (n + 1)) :=
(tendsto_ceil_right n).mono_right inf_le_left | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right' | null |
continuousOn_fract [IsTopologicalAddGroup α] (n : ℤ) :
ContinuousOn (fract : α → α) (Ico n (n + 1) : Set α) :=
continuousOn_id.sub (continuousOn_floor n) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | continuousOn_fract | null |
continuousAt_fract [OrderClosedTopology α] [IsTopologicalAddGroup α]
{x : α} (h : x ≠ ⌊x⌋) : ContinuousAt fract x :=
(continuousOn_fract ⌊x⌋).continuousAt <|
Ico_mem_nhds ((floor_le _).lt_of_ne h.symm) (lt_floor_add_one _)
variable [IsStrictOrderedRing α] | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | continuousAt_fract | null |
tendsto_fract_left' [OrderClosedTopology α] [IsTopologicalAddGroup α] (n : ℤ) :
Tendsto (fract : α → α) (𝓝[<] n) (𝓝 1) := by
rw [← sub_sub_cancel (n : α) 1]
refine (tendsto_id.mono_left nhdsWithin_le_nhds).sub ?_
exact tendsto_floor_left' n | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_fract_left' | null |
tendsto_fract_left [OrderClosedTopology α] [IsTopologicalAddGroup α] (n : ℤ) :
Tendsto (fract : α → α) (𝓝[<] n) (𝓝[<] 1) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ (tendsto_fract_left' _)
(Eventually.of_forall fract_lt_one) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_fract_left | null |
tendsto_fract_right' [OrderClosedTopology α] [IsTopologicalAddGroup α] (n : ℤ) :
Tendsto (fract : α → α) (𝓝[≥] n) (𝓝 0) :=
sub_self (n : α) ▸ (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id).sub (tendsto_floor_right' n) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_fract_right' | null |
tendsto_fract_right [OrderClosedTopology α] [IsTopologicalAddGroup α] (n : ℤ) :
Tendsto (fract : α → α) (𝓝[≥] n) (𝓝[≥] 0) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ (tendsto_fract_right' _)
(Eventually.of_forall fract_nonneg)
local notation "I" => (Icc 0 1 : Set α)
variable [OrderTopology ... | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_fract_right | null |
ContinuousOn.comp_fract' {f : β → α → γ} (h : ContinuousOn (uncurry f) <| univ ×ˢ I)
(hf : ∀ s, f s 0 = f s 1) : Continuous fun st : β × α => f st.1 (fract st.2) := by
change Continuous (uncurry f ∘ Prod.map id fract)
rw [continuous_iff_continuousAt]
rintro ⟨s, t⟩
rcases em (∃ n : ℤ, t = n) with (⟨n, rfl⟩ |... | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | ContinuousOn.comp_fract' | Do not use this, use `ContinuousOn.comp_fract` instead. |
ContinuousOn.comp_fract {s : β → α} {f : β → α → γ}
(h : ContinuousOn (uncurry f) <| univ ×ˢ Icc 0 1) (hs : Continuous s)
(hf : ∀ s, f s 0 = f s 1) : Continuous fun x : β => f x <| Int.fract (s x) :=
(h.comp_fract' hf).comp (continuous_id.prodMk hs) | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | ContinuousOn.comp_fract | null |
ContinuousOn.comp_fract'' {f : α → β} (h : ContinuousOn f I) (hf : f 0 = f 1) :
Continuous (f ∘ fract) :=
ContinuousOn.comp_fract (h.comp continuousOn_snd fun _x hx => (mem_prod.mp hx).2) continuous_id
fun _ => hf | theorem | Topology | [
"Mathlib.Algebra.Order.Floor.Ring",
"Mathlib.Order.Filter.AtTopBot.Floor",
"Mathlib.Topology.Algebra.Order.Group"
] | Mathlib/Topology/Algebra/Order/Floor.lean | ContinuousOn.comp_fract'' | A special case of `ContinuousOn.comp_fract`. |
@[to_additive (attr := continuity)]
continuous_mabs : Continuous (mabs : G → G) :=
continuous_id.max continuous_inv | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | continuous_mabs | null |
@[to_additive]
protected Filter.Tendsto.mabs {a : G} (h : Tendsto f l (𝓝 a)) :
Tendsto (fun x => |f x|ₘ) l (𝓝 |a|ₘ) :=
(continuous_mabs.tendsto _).comp h
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | Filter.Tendsto.mabs | null |
comap_mabs_nhds_one : comap mabs (𝓝 (1 : G)) = 𝓝 1 := by
simp [nhds_eq_iInf_mabs_div]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | comap_mabs_nhds_one | null |
tendsto_one_iff_mabs_tendsto_one (f : α → G) :
Tendsto f l (𝓝 1) ↔ Tendsto (mabs ∘ f) l (𝓝 1) := by
rw [← tendsto_comap_iff, comap_mabs_nhds_one] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | tendsto_one_iff_mabs_tendsto_one | null |
@[to_additive (attr := fun_prop)]
protected Continuous.mabs (h : Continuous f) : Continuous fun x => |f x|ₘ :=
continuous_mabs.comp h
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | Continuous.mabs | null |
protected ContinuousAt.mabs (h : ContinuousAt f x) : ContinuousAt (fun x => |f x|ₘ) x :=
Filter.Tendsto.mabs h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | ContinuousAt.mabs | null |
protected ContinuousWithinAt.mabs (h : ContinuousWithinAt f s x) :
ContinuousWithinAt (fun x => |f x|ₘ) s x :=
Filter.Tendsto.mabs h
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | ContinuousWithinAt.mabs | null |
protected ContinuousOn.mabs (h : ContinuousOn f s) : ContinuousOn (fun x => |f x|ₘ) s :=
fun x hx => (h x hx).mabs
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | ContinuousOn.mabs | null |
tendsto_mabs_nhdsNE_one : Tendsto (mabs : G → G) (𝓝[≠] 1) (𝓝[>] 1) :=
(continuous_mabs.tendsto' (1 : G) 1 mabs_one).inf <|
tendsto_principal_principal.2 fun _x => one_lt_mabs.2
@[deprecated (since := "2025-03-18")]
alias tendsto_abs_nhdsWithin_zero := tendsto_abs_nhdsNE_zero | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | tendsto_mabs_nhdsNE_one | null |
@[to_additive /-- In a linearly ordered additive group, the integer multiples of an element are
dense iff they are the whole group. -/]
denseRange_zpow_iff_surjective {a : G} :
DenseRange (a ^ · : ℤ → G) ↔ Surjective (a ^ · : ℤ → G) := by
refine ⟨fun h ↦ ?_, fun h ↦ h.denseRange⟩
wlog ha₀ : 1 < a generalizing a... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | denseRange_zpow_iff_surjective | In a linearly ordered multiplicative group, the integer powers of an element are dense
iff they are the whole group. |
@[to_additive /-- In a nontrivial densely linearly ordered additive group,
the integer multiples of an element can't be dense. -/]
not_denseRange_zpow [Nontrivial G] [DenselyOrdered G] {a : G} :
¬DenseRange (a ^ · : ℤ → G) :=
denseRange_zpow_iff_surjective.not.mpr fun h ↦
not_isCyclic_of_denselyOrdered G ⟨⟨a,... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Basic",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Algebra/Order/Group.lean | not_denseRange_zpow | In a nontrivial densely linearly ordered commutative group,
the integer powers of an element can't be dense. |
le_limsup_add (h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u := by isBoundedDefault)
(h₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u := by isBoundedDefault)
(h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v := by isBoundedDefault)
(h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v := by isBoundedDefault) :
(l... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | le_limsup_add | null |
limsup_add_le (h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u := by isBoundedDefault)
(h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u := by isBoundedDefault)
(h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v := by isBoundedDefault)
(h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v := by isBoundedDefault) :
li... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_add_le | null |
le_liminf_add (h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u := by isBoundedDefault)
(h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u := by isBoundedDefault)
(h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v := by isBoundedDefault)
(h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v := by isBoundedDefault) :
(l... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | le_liminf_add | null |
liminf_add_le (h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u := by isBoundedDefault)
(h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u := by isBoundedDefault)
(h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v := by isBoundedDefault)
(h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v := by isBoundedDefault) :
li... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_add_le | null |
le_limsup_mul (h₁ : ∃ᶠ x in f, 0 ≤ u x) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v) :
(limsup u f) * liminf v f ≤ limsup (u * v) f := by
have h := IsCoboundedUnder.of_frequently_ge (a := 0)
<| (h₁.and_eventually h₃).mono fun x ⟨ux_0, vx_... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | le_limsup_mul | null |
limsup_mul_le (h₁ : ∃ᶠ x in f, 0 ≤ u x) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v) :
limsup (u * v) f ≤ (limsup u f) * limsup v f := by
have h := IsCoboundedUnder.of_frequently_ge (a := 0)
<| (h₁.and_eventually h₃).mono fun x ⟨ux_0, vx_... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_mul_le | null |
le_liminf_mul [f.NeBot] (h₁ : 0 ≤ᶠ[f] u) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v) :
(liminf u f) * liminf v f ≤ liminf (u * v) f := by
have h := isCoboundedUnder_ge_mul_of_nonneg h₁ h₂ h₃ h₄
have h' := isBoundedUnder_of_eventually_ge ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | le_liminf_mul | null |
liminf_mul_le [f.NeBot] (h₁ : 0 ≤ᶠ[f] u) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v) :
liminf (u * v) f ≤ (limsup u f) * liminf v f := by
have h := isCoboundedUnder_ge_mul_of_nonneg h₁ h₂ h₃ h₄
have h' := isBoundedUnder_of_eventually_ge ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_mul_le | null |
limsup_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
[AddLeftMono R] (f : ι → R) (c : R)
(bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
Filter.limsup (fun i ↦ c + f i) F = c + Filter.limsup f F :=
(Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_const_add | `liminf (c + xᵢ) = c + liminf xᵢ`. |
limsup_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
[AddRightMono R] (f : ι → R) (c : R)
(bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
Filter.limsup (fun i ↦ f i + c) F = Filter.limsup f F + c :=
(Monotone.map_limsSup_of_continuousAt (F := F.map f) (f :=... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_add_const | `limsup (xᵢ + c) = (limsup xᵢ) + c`. |
liminf_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
[AddLeftMono R] (f : ι → R) (c : R)
(cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
Filter.liminf (fun i ↦ c + f i) F = c + Filter.liminf f F :=
(Monotone.map_limsInf_of_continuousAt (F := F.map f) (f := ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_const_add | `liminf (c + xᵢ) = c + liminf xᵢ`. |
liminf_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
[AddRightMono R] (f : ι → R) (c : R)
(cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
Filter.liminf (fun i ↦ f i + c) F = Filter.liminf f F + c :=
(Monotone.map_limsInf_of_continuousAt (F := F.map f) (f :=... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_add_const | `liminf (xᵢ + c) = (liminf xᵢ) + c`. |
limsup_const_sub (F : Filter ι) [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R]
[AddLeftMono R] (f : ι → R) (c : R)
(cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F := by
rcases F.eq_or_neBot with rfl ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_const_sub | `limsup (c - xᵢ) = c - liminf xᵢ`. |
limsup_sub_const (F : Filter ι) [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R]
(f : ι → R) (c : R)
(bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c := by
rcases F.eq_or_neBot with rfl | _
· have {a ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | limsup_sub_const | `limsup (xᵢ - c) = (limsup xᵢ) - c`. |
liminf_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
[OrderedSub R] [AddLeftMono R] (f : ι → R) (c : R)
(bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
Filter.liminf (fun i ↦ c - f i) F = c - Filter.limsup f F :=
(Antitone.map_limsSup_o... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_const_sub | `liminf (c - xᵢ) = c - limsup xᵢ`. |
liminf_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
[OrderedSub R] (f : ι → R) (c : R)
(cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
Filter.liminf (fun i ↦ f i - c) F = Filter.liminf f F - c :=
(Monotone.map_limsInf_of_continuousAt (... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.DenselyOrdered",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.Order.LiminfLimsup"
] | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | liminf_sub_const | `liminf (xᵢ - c) = (liminf xᵢ) - c`. |
@[to_additive]
HasCompactMulSupport.sup {f g : X → M} (hf : HasCompactMulSupport f)
(hg : HasCompactMulSupport g) : HasCompactMulSupport (f ⊔ g) := by
apply IsCompact.of_isClosed_subset (IsCompact.union hf hg) (isClosed_mulTSupport _)
rw [mulTSupport, mulTSupport, mulTSupport, ← closure_union]
apply closure_m... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Indicator",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/Order/Support.lean | HasCompactMulSupport.sup | null |
@[to_additive]
HasCompactMulSupport.inf {f g : X → M} (hf : HasCompactMulSupport f)
(hg : HasCompactMulSupport g) : HasCompactMulSupport (f ⊓ g) := by
apply IsCompact.of_isClosed_subset (IsCompact.union hf hg) (isClosed_mulTSupport _)
rw [mulTSupport, mulTSupport, mulTSupport, ← closure_union]
apply closure_m... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Indicator",
"Mathlib.Topology.Algebra.Support"
] | Mathlib/Topology/Algebra/Order/Support.lean | HasCompactMulSupport.inf | null |
HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
isOpen_low... | class | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | HasUpperLowerClosure | Ad hoc class stating that the closure of an upper set is an upper set. This is used to state
lemmas that do not mention algebraic operations for both the additive and multiplicative versions
simultaneously. If you find a satisfying replacement for this typeclass, please remove it! |
protected IsUpperSet.closure : IsUpperSet s → IsUpperSet (closure s) :=
HasUpperLowerClosure.isUpperSet_closure _ | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsUpperSet.closure | null |
protected IsLowerSet.closure : IsLowerSet s → IsLowerSet (closure s) :=
HasUpperLowerClosure.isLowerSet_closure _ | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsLowerSet.closure | null |
protected IsOpen.upperClosure : IsOpen s → IsOpen (upperClosure s : Set α) :=
HasUpperLowerClosure.isOpen_upperClosure _ | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsOpen.upperClosure | null |
protected IsOpen.lowerClosure : IsOpen s → IsOpen (lowerClosure s : Set α) :=
HasUpperLowerClosure.isOpen_lowerClosure _ | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsOpen.lowerClosure | null |
protected IsUpperSet.interior (h : IsUpperSet s) : IsUpperSet (interior s) := by
rw [← isLowerSet_compl, ← closure_compl]
exact h.compl.closure | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsUpperSet.interior | null |
protected IsLowerSet.interior (h : IsLowerSet s) : IsLowerSet (interior s) :=
h.toDual.interior | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | IsLowerSet.interior | null |
protected Set.OrdConnected.interior (h : s.OrdConnected) : (interior s).OrdConnected := by
rw [← h.upperClosure_inter_lowerClosure, interior_inter]
exact
(upperClosure s).upper.interior.ordConnected.inter (lowerClosure s).lower.interior.ordConnected | theorem | Topology | [
"Mathlib.Algebra.Order.UpperLower",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Order/UpperLower.lean | Set.OrdConnected.interior | null |
ProperVAdd (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [AddGroup G]
[AddAction G X] : Prop where
/-- Proper group action in the sense of Bourbaki:
the map `G × X → X × X` is a proper map (see `IsProperMap`). -/
isProperMap_vadd_pair : IsProperMap (fun gx ↦ (gx.1 +ᵥ gx.2, gx.2) : G × X → X × X) | class | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | ProperVAdd | Proper group action in the sense of Bourbaki:
the map `G × X → X × X` is a proper map (see `IsProperMap`). |
@[to_additive existing (attr := mk_iff)]
ProperSMul (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [Group G]
[MulAction G X] : Prop where
/-- Proper group action in the sense of Bourbaki:
the map `G × X → X × X` is a proper map (see `IsProperMap`). -/
isProperMap_smul_pair : IsProperMap (fun gx ↦ (gx... | class | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | ProperSMul | Proper group action in the sense of Bourbaki:
the map `G × X → X × X` is a proper map (see `IsProperMap`). |
@[to_additive /-- An additive group `G` acts properly on a topological space `X` if and only if
for all ultrafilters `𝒰` on `X`, if `𝒰` converges to `(x₁, x₂)`
along the map `(g, x) ↦ (g • x, x)`, then there exists `g : G` such that `g • x₂ = x₁`
and `𝒰.fst` converges to `g`. -/]
properSMul_iff_continuousSMul_ultraf... | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | properSMul_iff_continuousSMul_ultrafilter_tendsto | If a group acts properly then in particular it acts continuously. -/
@[to_additive /-- If a group acts properly then in particular it acts continuously. -/]
-- See note [lower instance property]
instance (priority := 100) ProperSMul.toContinuousSMul [ProperSMul G X] : ContinuousSMul G X where
continuous_smul := isPro... |
properSMul_iff_continuousSMul_ultrafilter_tendsto_t2 [T2Space X] :
ProperSMul G X ↔ ContinuousSMul G X ∧
(∀ 𝒰 : Ultrafilter (G × X), ∀ x₁ x₂ : X,
Tendsto (fun gx : G × X ↦ (gx.1 • gx.2, gx.2)) 𝒰 (𝓝 (x₁, x₂)) →
∃ g : G, Tendsto (Prod.fst : G × X → G) 𝒰 (𝓝 g)) := by
rw [properSMul_iff_con... | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | properSMul_iff_continuousSMul_ultrafilter_tendsto_t2 | A group `G` acts properly on a T2 topological space `X` if and only if for all ultrafilters
`𝒰` on `X × G`, if `𝒰` converges to `(x₁, x₂)` along the map `(g, x) ↦ (g • x, x)`,
then there exists `g : G` such that `𝒰.fst` converges to `g`. |
@[to_additive /-- If `G` acts properly on `X`, then the quotient space is Hausdorff (T2). -/]
t2Space_quotient_mulAction_of_properSMul [ProperSMul G X] :
T2Space (Quotient (MulAction.orbitRel G X)) := by
rw [t2_iff_isClosed_diagonal]
set R := MulAction.orbitRel G X
let π : X → Quotient R := Quotient.mk'
hav... | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | t2Space_quotient_mulAction_of_properSMul | If `G` acts properly on `X`, then the quotient space is Hausdorff (T2). |
@[to_additive /-- If a T1 group acts properly on a topological space,
then this topological space is T2. -/]
t2Space_of_properSMul_of_t1Group [h_proper : ProperSMul G X] [T1Space G] : T2Space X := by
let f := fun x : X ↦ ((1 : G), x)
have proper_f : IsProperMap f := by
refine IsClosedEmbedding.isProperMap ⟨?_, ... | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | t2Space_of_properSMul_of_t1Group | If a T1 group acts properly on a topological space, then this topological space is T2. |
@[to_additive /-- If two groups `H` and `G` act on a topological space `X` such that `G` acts
properly and there exists a group homomorphism `H → G` which is a closed embedding compatible with
the actions, then `H` also acts properly on `X`. -/]
properSMul_of_isClosedEmbedding {H : Type*} [Group H] [MulAction H X] [Top... | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/ProperAction/Basic.lean | properSMul_of_isClosedEmbedding | If two groups `H` and `G` act on a topological space `X` such that `G` acts properly and
there exists a group homomorphism `H → G` which is a closed embedding compatible with the actions,
then `H` also acts properly on `X`. |
properlyDiscontinuousSMul_iff_properSMul [T2Space X] [DiscreteTopology G]
[ContinuousConstSMul G X] [CompactlyGeneratedSpace (X × X)] :
ProperlyDiscontinuousSMul G X ↔ ProperSMul G X := by
constructor
· intro h
rw [properSMul_iff]
refine isProperMap_iff_isCompact_preimage.2
⟨(continuous_prod_o... | theorem | Topology | [
"Mathlib.Topology.Algebra.ProperAction.Basic",
"Mathlib.Topology.Maps.Proper.CompactlyGenerated"
] | Mathlib/Topology/Algebra/ProperAction/ProperlyDiscontinuous.lean | properlyDiscontinuousSMul_iff_properSMul | If a discrete group acts on a T2 space `X` such that `X × X` is compactly
generated, and if the action is continuous in the second variable, then the action is properly
discontinuous if and only if it is proper. This is in particular true if `X` is first-countable or
weakly locally compact.
There was an older version ... |
RestrictedProduct (𝓕 : Filter ι) : Type _ := {x : Π i, R i // ∀ᶠ i in 𝓕, x i ∈ A i}
open Batteries.ExtendedBinder | def | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | RestrictedProduct | The **restricted product** of a family `R : ι → Type*` of types, relative to subsets
`A : (i : ι) → Set (R i)` and the filter `𝓕 : Filter ι`, is the set of all `x : Π i, R i`
such that the set `{j | x j ∈ A j}` belongs to `𝓕`. We denote it by `Πʳ i, [R i, A i]_[𝓕]`.
The most common use case is with `𝓕 = cofinite`,... |
mk (x : Π i, R i) (hx : ∀ᶠ i in 𝓕, x i ∈ A i) : Πʳ i, [R i, A i]_[𝓕] :=
⟨x, hx⟩
@[simp] | abbrev | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | mk | `Πʳ i, [R i, A i]_[𝓕]` is `RestrictedProduct R A 𝓕`. -/
scoped[RestrictedProduct]
notation3 "Πʳ "(...)", ""["r:(scoped R => R)", "a:(scoped A => A)"]_[" f "]" =>
RestrictedProduct r a f
/-- `Πʳ i, [R i, A i]` is `RestrictedProduct R A cofinite`. -/
scoped[RestrictedProduct]
notation3"Πʳ "(...)", ""["r:(scoped R =>... |
mk_apply (x : Π i, R i) (hx : ∀ᶠ i in 𝓕, x i ∈ A i) (i : ι) :
(mk x hx) i = x i := rfl
@[ext] | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | mk_apply | null |
ext {x y : Πʳ i, [R i, A i]_[𝓕]} (h : ∀ i, x i = y i) : x = y :=
Subtype.ext <| funext h | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | ext | null |
range_coe :
range ((↑) : Πʳ i, [R i, A i]_[𝓕] → Π i, R i) = {x | ∀ᶠ i in 𝓕, x i ∈ A i} :=
Subtype.range_val_subtype | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | range_coe | null |
range_coe_principal {S : Set ι} :
range ((↑) : Πʳ i, [R i, A i]_[𝓟 S] → Π i, R i) = S.pi A :=
range_coe R A
@[simp] lemma eventually (x : Πʳ i, [R i, A i]_[𝓕]) : ∀ᶠ i in 𝓕, x i ∈ A i := x.2
variable (𝓕) in | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | range_coe_principal | null |
structureMap (x : Π i, A i) : Πʳ i, [R i, A i]_[𝓕] :=
⟨fun i ↦ x i, .of_forall fun i ↦ (x i).2⟩ | def | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | structureMap | The *structure map* of the restricted product is the obvious inclusion from `Π i, A i`
into `Πʳ i, [R i, A i]_[𝓕]`. |
inclusion (h : 𝓕 ≤ 𝓖) (x : Πʳ i, [R i, A i]_[𝓖]) :
Πʳ i, [R i, A i]_[𝓕] :=
⟨x, x.2.filter_mono h⟩
variable (𝓕) in | def | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | inclusion | If `𝓕 ≤ 𝓖`, the restricted product `Πʳ i, [R i, A i]_[𝓖]` is naturally included in
`Πʳ i, [R i, A i]_[𝓕]`. This is the corresponding map. |
inclusion_eq_id : inclusion R A (le_refl 𝓕) = id := rfl | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | inclusion_eq_id | null |
exists_inclusion_eq_of_eventually (h : 𝓕 ≤ 𝓖) {x : Πʳ i, [R i, A i]_[𝓕]}
(hx𝓖 : ∀ᶠ i in 𝓖, x i ∈ A i) :
∃ x' : Πʳ i, [R i, A i]_[𝓖], inclusion R A h x' = x :=
⟨⟨x.1, hx𝓖⟩, rfl⟩ | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | exists_inclusion_eq_of_eventually | null |
exists_structureMap_eq_of_forall {x : Πʳ i, [R i, A i]_[𝓕]}
(hx : ∀ i, x.1 i ∈ A i) :
∃ x' : Π i, A i, structureMap R A 𝓕 x' = x :=
⟨fun i ↦ ⟨x i, hx i⟩, rfl⟩ | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | exists_structureMap_eq_of_forall | null |
range_inclusion (h : 𝓕 ≤ 𝓖) :
Set.range (inclusion R A h) = {x | ∀ᶠ i in 𝓖, x i ∈ A i} :=
subset_antisymm (range_subset_iff.mpr fun x ↦ x.2)
(fun _ hx ↦ mem_range.mpr <| exists_inclusion_eq_of_eventually R A h hx) | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | range_inclusion | null |
range_structureMap :
Set.range (structureMap R A 𝓕) = {f | ∀ i, f.1 i ∈ A i} :=
subset_antisymm (range_subset_iff.mpr fun x i ↦ (x i).2)
(fun _ hx ↦ mem_range.mpr <| exists_structureMap_eq_of_forall R A hx) | lemma | Topology | [
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] | Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean | range_structureMap | null |
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