Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 99 | 101 | theorem csInf_mem {s : Set ℤ} (h1 : s.Nonempty) (h2 : BddBelow s) : sInf s ∈ s := by |
convert (leastOfBdd _ (Classical.choose_spec h2) h1).2.1
exact dif_pos ⟨h1, h2⟩
| 1,104 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set α → Filter β) : (⊤ : Filter α).lift g = g univ := by | simp [Filter.lift]
| 1,105 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 45 | 53 | theorem HasBasis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(hf : f.HasBasis p s) {β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ}
{g : Set α → Filter γ} (hg : ∀ i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set γ} : s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg... |
refine (mem_biInf_of_directed ?_ ⟨univ, univ_sets _⟩).trans ?_
· intro t₁ ht₁ t₂ ht₂
exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm inter_subset_left, gm inter_subset_right⟩
· simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H
| 1,105 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 65 | 70 | theorem HasBasis.lift {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α} (hf : f.HasBasis p s)
{β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ} {g : Set α → Filter γ}
(hg : ∀ i, (g (s i)).HasBasis (pg i) (sg i)) (gm : Monotone g) :
(f.lift g).HasBasis (fun i : Σi, β i => p i.1 ∧ pg i.1 i.2) fun... |
refine ⟨fun t => (hf.mem_lift_iff hg gm).trans ?_⟩
simp [Sigma.exists, and_assoc, exists_and_left]
| 1,105 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 78 | 81 | theorem sInter_lift_sets (hg : Monotone g) :
⋂₀ { s | s ∈ f.lift g } = ⋂ s ∈ f, ⋂₀ { t | t ∈ g s } := by |
simp only [sInter_eq_biInter, mem_setOf_eq, Filter.mem_sets, mem_lift_sets hg, iInter_exists,
iInter_and, @iInter_comm _ (Set β)]
| 1,105 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 106 | 108 | theorem tendsto_lift {m : γ → β} {l : Filter γ} :
Tendsto m l (f.lift g) ↔ ∀ s ∈ f, Tendsto m l (g s) := by |
simp only [Filter.lift, tendsto_iInf]
| 1,105 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 117 | 118 | theorem comap_lift_eq {m : γ → β} : comap m (f.lift g) = f.lift (comap m ∘ g) := by |
simp only [Filter.lift, comap_iInf]; rfl
| 1,105 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 55 | 56 | theorem isOpen_setOf_mem {s : Set α} : IsOpen { l : Filter α | s ∈ l } := by |
simpa only [Iic_principal] using isOpen_Iic_principal
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 59 | 66 | theorem isTopologicalBasis_Iic_principal :
IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) :=
{ exists_subset_inter := by |
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl
exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩
sUnion_eq := sUnion_eq_univ_iff.2 fun l => ⟨Iic ⊤, ⟨univ, congr_arg Iic principal_univ⟩,
mem_Iic.2 le_top⟩
eq_generateFrom := rfl }
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 105 | 106 | theorem HasBasis.nhds' {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) :
HasBasis (𝓝 l) p fun i => { l' | s i ∈ l' } := by | simpa only [Iic_principal] using h.nhds
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 125 | 125 | theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by | simp [nhds_eq]
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 134 | 135 | theorem nhds_pure (x : α) : 𝓝 (pure x : Filter α) = 𝓟 {⊥, pure x} := by |
rw [← principal_singleton, nhds_principal, principal_singleton, Iic_pure]
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 139 | 141 | theorem nhds_iInf (f : ι → Filter α) : 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i) := by |
simp only [nhds_eq]
apply lift'_iInf_of_map_univ <;> simp
| 1,106 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 115 | 116 | theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by |
rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
| 1,107 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 153 | 153 | theorem isOpen_const {p : Prop} : IsOpen { _x : X | p } := by | by_cases p <;> simp [*]
| 1,107 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 164 | 167 | theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s := by |
rw [TopologicalSpace.ext_iff, compl_surjective.forall]
simp only [@isOpen_compl_iff _ _ t₁, @isOpen_compl_iff _ _ t₂]
| 1,107 |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 30 | 34 | theorem rtendsto'_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
RTendsto' r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.preimage s ∈ l := by |
rw [rtendsto'_def]
apply all_mem_nhds_filter
apply Rel.preimage_mono
| 1,108 |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 57 | 58 | theorem open_dom_of_pcontinuous {f : X →. Y} (h : PContinuous f) : IsOpen f.Dom := by |
rw [← PFun.preimage_univ]; exact h _ isOpen_univ
| 1,108 |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 61 | 83 | theorem pcontinuous_iff' {f : X →. Y} :
PContinuous f ↔ ∀ {x y} (h : y ∈ f x), PTendsto' f (𝓝 x) (𝓝 y) := by |
constructor
· intro h x y h'
simp only [ptendsto'_def, mem_nhds_iff]
rintro s ⟨t, tsubs, opent, yt⟩
exact ⟨f.preimage t, PFun.preimage_mono _ tsubs, h _ opent, ⟨y, yt, h'⟩⟩
intro hf s os
rw [isOpen_iff_nhds]
rintro x ⟨y, ys, fxy⟩ t
rw [mem_principal]
intro (h : f.preimage s ⊆ t)
change t ∈ ... | 1,108 |
import Mathlib.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Set OmegaCompletePartialOrder
open scoped Classical
universe ... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 41 | 43 | theorem isωSup_iff_isLUB {α : Type u} [Preorder α] {c : Chain α} {x : α} :
IsωSup c x ↔ IsLUB (range c) x := by |
simp [IsωSup, IsLUB, IsLeast, upperBounds, lowerBounds]
| 1,109 |
import Mathlib.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Set OmegaCompletePartialOrder
open scoped Classical
universe ... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 62 | 66 | theorem isOpen_sUnion (s : Set (Set α)) (hs : ∀ t ∈ s, IsOpen α t) : IsOpen α (⋃₀ s) := by |
simp only [IsOpen] at hs ⊢
convert CompleteLattice.sSup_continuous' (setOf ⁻¹' s) hs
simp only [sSup_apply, setOf_bijective.surjective.exists, exists_prop, mem_preimage,
SetCoe.exists, iSup_Prop_eq, mem_setOf_eq, mem_sUnion]
| 1,109 |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 78 | 90 | theorem nhds_generateFrom {g : Set (Set α)} {a : α} :
@nhds α (generateFrom g) a = ⨅ s ∈ { s | a ∈ s ∧ s ∈ g }, 𝓟 s := by |
letI := generateFrom g
rw [nhds_def]
refine le_antisymm (biInf_mono fun s ⟨as, sg⟩ => ⟨as, .basic _ sg⟩) <| le_iInf₂ ?_
rintro s ⟨ha, hs⟩
induction hs with
| basic _ hs => exact iInf₂_le _ ⟨ha, hs⟩
| univ => exact le_top.trans_eq principal_univ.symm
| inter _ _ _ _ hs ht => exact (le_inf (hs ha.1) (ht ... | 1,110 |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 110 | 121 | theorem nhds_mkOfNhds_of_hasBasis {n : α → Filter α} {ι : α → Sort*} {p : ∀ a, ι a → Prop}
{s : ∀ a, ι a → Set α} (hb : ∀ a, (n a).HasBasis (p a) (s a))
(hpure : ∀ a i, p a i → a ∈ s a i) (hopen : ∀ a i, p a i → ∀ᶠ x in n a, s a i ∈ n x) (a : α) :
@nhds α (.mkOfNhds n) a = n a := by |
let t : TopologicalSpace α := .mkOfNhds n
apply le_antisymm
· intro U hU
replace hpure : pure ≤ n := fun x ↦ (hb x).ge_iff.2 (hpure x)
refine mem_nhds_iff.2 ⟨{x | U ∈ n x}, fun x hx ↦ hpure x hx, fun x hx ↦ ?_, hU⟩
rcases (hb x).mem_iff.1 hx with ⟨i, hpi, hi⟩
exact (hopen x i hpi).mono fun y hy ↦... | 1,110 |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 129 | 138 | theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b =
(update pure a₀ l : α → Filter α) b := by |
refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_
rcases eq_or_ne a a₀ with (rfl | ha)
· filter_upwards [hs] with b hb
rcases eq_or_ne b a with (rfl | hb)
· exact hs
· rwa [update_noteq hb]
· simpa only [update_noteq ha, mem_pure, eventually_pure] using hs
| 1,110 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 69 | 72 | theorem Inducing.of_comp_iff (hg : Inducing g) :
Inducing (g ∘ f) ↔ Inducing f := by |
refine ⟨fun h ↦ ?_, hg.comp⟩
rw [inducing_iff, hg.induced, induced_compose, h.induced]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 97 | 99 | theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by |
simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 112 | 115 | theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
MapClusterPt (f x) l f ↔ ClusterPt x l := by |
delta MapClusterPt ClusterPt
rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 122 | 124 | theorem tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Inducing g) :
Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by |
rw [hg.nhds_eq_comap, tendsto_comap_iff]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 132 | 134 | theorem continuous_iff (hg : Inducing g) :
Continuous f ↔ Continuous (g ∘ f) := by |
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 137 | 139 | theorem continuousAt_iff' (hf : Inducing f) {x : X} (h : range f ∈ 𝓝 (f x)) :
ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by |
simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 146 | 149 | theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set X) :
closure s = f ⁻¹' closure (f '' s) := by |
ext x
rw [Set.mem_preimage, ← closure_induced, hf.induced]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 152 | 153 | theorem isClosed_iff (hf : Inducing f) {s : Set X} :
IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by | rw [hf.induced, isClosed_induced_iff]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 156 | 157 | theorem isClosed_iff' (hf : Inducing f) {s : Set X} :
IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by | rw [hf.induced, isClosed_induced_iff']
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 338 | 340 | theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by |
rw [← image_univ]
exact hf _ isOpen_univ
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 371 | 378 | theorem of_sections
(h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
of_nhds_le fun x =>
let ⟨g, hgc, hgx, hgf⟩ := h x
calc
𝓝 (f x) = map f (map g (𝓝 (f x))) := by | rw [map_map, hgf.comp_eq_id, map_id]
_ ≤ map f (𝓝 (g (f x))) := map_mono hgc
_ = map f (𝓝 x) := by rw [hgx]
| 1,111 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 478 | 482 | theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
IsClosedMap f := by |
intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
· simp_rw [h2s, image_empty, isClosed_empty]
· exact h s hs h2s
| 1,111 |
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.t0_sierpinski from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
namespace TopologicalSpace
| Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean | 28 | 37 | theorem eq_induced_by_maps_to_sierpinski (X : Type*) [t : TopologicalSpace X] :
t = ⨅ u : Opens X, sierpinskiSpace.induced (· ∈ u) := by |
apply le_antisymm
· rw [le_iInf_iff]
exact fun u => Continuous.le_induced (isOpen_iff_continuous_mem.mp u.2)
· intro u h
rw [← generateFrom_iUnion_isOpen]
apply isOpen_generateFrom_of_mem
simp only [Set.mem_iUnion, Set.mem_setOf_eq, isOpen_induced_iff]
exact ⟨⟨u, h⟩, {True}, isOpen_singleton_... | 1,112 |
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.t0_sierpinski from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
namespace TopologicalSpace
theorem eq_in... | Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean | 50 | 52 | theorem productOfMemOpens_inducing : Inducing (productOfMemOpens X) := by |
convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x ∈ u
apply eq_induced_by_maps_to_sierpinski
| 1,112 |
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.t0_sierpinski from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
namespace TopologicalSpace
theorem eq_in... | Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean | 55 | 58 | theorem productOfMemOpens_injective [T0Space X] : Function.Injective (productOfMemOpens X) := by |
intro x1 x2 h
apply Inseparable.eq
rw [← Inducing.inseparable_iff (productOfMemOpens_inducing X), h]
| 1,112 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
| Mathlib/Topology/NhdsSet.lean | 35 | 38 | theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] :
𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by |
rw [nhdsSet, ← range_diag, ← range_comp]
rfl
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 41 | 42 | theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by |
simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 52 | 53 | theorem subset_interior_iff_mem_nhdsSet : s ⊆ interior t ↔ t ∈ 𝓝ˢ s := by |
simp_rw [mem_nhdsSet_iff_forall, subset_interior_iff_nhds]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 56 | 58 | theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by |
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 60 | 61 | theorem disjoint_nhdsSet_principal : Disjoint (𝓝ˢ s) (𝓟 t) ↔ Disjoint s (closure t) := by |
rw [disjoint_comm, disjoint_principal_nhdsSet, disjoint_comm]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 63 | 64 | theorem mem_nhdsSet_iff_exists : s ∈ 𝓝ˢ t ↔ ∃ U : Set X, IsOpen U ∧ t ⊆ U ∧ U ⊆ s := by |
rw [← subset_interior_iff_mem_nhdsSet, subset_interior_iff]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 90 | 91 | theorem IsOpen.mem_nhdsSet (hU : IsOpen s) : s ∈ 𝓝ˢ t ↔ t ⊆ s := by |
rw [← subset_interior_iff_mem_nhdsSet, hU.interior_eq]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 110 | 112 | theorem nhdsSet_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ IsOpen s := by |
rw [← principal_le_nhdsSet.le_iff_eq, le_principal_iff, mem_nhdsSet_iff_forall,
isOpen_iff_mem_nhds]
| 1,113 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 124 | 124 | theorem nhdsSet_singleton : 𝓝ˢ {x} = 𝓝 x := by | simp [nhdsSet]
| 1,113 |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 94 | 102 | theorem Filter.Eventually.germ_congr_set
{P : ∀ x : X, Germ (𝓝 x) Y → Prop} (hf : ∀ᶠ x in 𝓝ˢ A, P x f)
(h : ∀ᶠ z in 𝓝ˢ A, g z = f z) : ∀ᶠ x in 𝓝ˢ A, P x g := by |
rw [eventually_nhdsSet_iff_forall] at *
intro x hx
apply ((hf x hx).and (h x hx).eventually_nhds).mono
intro y hy
convert hy.1 using 1
exact Germ.coe_eq.mpr hy.2
| 1,114 |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 104 | 110 | theorem restrictGermPredicate_congr {P : ∀ x : X, Germ (𝓝 x) Y → Prop}
(hf : RestrictGermPredicate P A x f) (h : ∀ᶠ z in 𝓝ˢ A, g z = f z) :
RestrictGermPredicate P A x g := by |
intro hx
apply ((hf hx).and <| (eventually_nhdsSet_iff_forall.mp h x hx).eventually_nhds).mono
rintro y ⟨hy, h'y⟩
rwa [Germ.coe_eq.mpr h'y]
| 1,114 |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 112 | 115 | theorem forall_restrictGermPredicate_iff {P : ∀ x : X, Germ (𝓝 x) Y → Prop} :
(∀ x, RestrictGermPredicate P A x f) ↔ ∀ᶠ x in 𝓝ˢ A, P x f := by |
rw [eventually_nhdsSet_iff_forall]
rfl
| 1,114 |
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import algebra.bounds from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29... | Mathlib/Algebra/Bounds.lean | 175 | 176 | theorem ciSup_div (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by |
simp only [div_eq_mul_inv, ciSup_mul hf]
| 1,115 |
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import algebra.bounds from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29... | Mathlib/Algebra/Bounds.lean | 185 | 186 | theorem ciInf_div (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by |
simp only [div_eq_mul_inv, ciInf_mul hf]
| 1,115 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
open scoped Classical
open Pointwise CauSeq
namespace Real
... | Mathlib/Data/Real/Archimedean.lean | 58 | 106 | theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by |
rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
cases' exists_int_gt U with k hk
refine fun d => ⟨k * d, fun z h => ?_⟩
rcases h with ⟨y, yS, hy⟩
refine Int.cast_le.1 (hy.trans ?_)
push_cast
exact mul_le_mul_of_nonneg_right ((hU y... | 1,116 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 37 | 46 | theorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRight ha').map_csInf' hs h).symm
· rw [Real.sInf_of_not_bddBelow (mt (b... | 1,117 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 53 | 62 | theorem Real.sSup_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sSup (a • s) = a • sSup s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddAbove s
· exact ((OrderIso.smulRight ha').map_csSup' hs h).symm
· rw [Real.sSup_of_not_bddAbove (mt (b... | 1,117 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 75 | 84 | theorem Real.sInf_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sInf (a • s) = a • sSup s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddAbove s
· exact ((OrderIso.smulRightDual ℝ ha').map_csSup' hs h).symm
· rw [Real.sInf... | 1,117 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 91 | 100 | theorem Real.sSup_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sSup (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sSup... | 1,117 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Algebra.Order.Nonneg.Floor
import Mathlib.Data.Real.Pointwise
import Mathlib.Order.ConditionallyCompleteLattice.Group
imp... | Mathlib/Data/Real/NNReal.lean | 125 | 126 | theorem _root_.Real.toNNReal_of_nonneg {r : ℝ} (hr : 0 ≤ r) : r.toNNReal = ⟨r, hr⟩ := by |
simp_rw [Real.toNNReal, max_eq_left hr]
| 1,118 |
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
namespace Valuation
class RankOne (v : Valuation R Γ₀) where
hom : Γ₀ →*₀ ℝ≥0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 51 | 55 | theorem zero_of_hom_zero {x : Γ₀} (hx : hom v x = 0) : x = 0 := by |
refine (eq_of_le_of_not_lt (zero_le' (a := x)) fun h_lt ↦ ?_).symm
have hs := strictMono v h_lt
rw [_root_.map_zero, hx] at hs
exact hs.false
| 1,119 |
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
namespace Valuation
class RankOne (v : Valuation R Γ₀) where
hom : Γ₀ →*₀ ℝ≥0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 67 | 69 | theorem unit_ne_one : unit v ≠ 1 := by |
rw [Ne, ← Units.eq_iff, Units.val_one]
exact ((nontrivial v).choose_spec ).2
| 1,119 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K → ℝ))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 68 | 68 | theorem norm_nonneg (x : L) : 0 ≤ norm x := by | simp only [norm, NNReal.zero_le_coe]
| 1,120 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K → ℝ))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 70 | 72 | theorem norm_add_le (x y : L) : norm (x + y) ≤ max (norm x) (norm y) := by |
simp only [norm, NNReal.coe_le_coe, le_max_iff, StrictMono.le_iff_le hv.strictMono]
exact le_max_iff.mp (Valuation.map_add_le_max' val.v _ _)
| 1,120 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K → ℝ))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 74 | 75 | theorem norm_eq_zero {x : L} (hx : norm x = 0) : x = 0 := by |
simpa [norm, NNReal.coe_eq_zero, RankOne.hom_eq_zero_iff, zero_iff] using hx
| 1,120 |
import Mathlib.Data.Real.NNReal
import Mathlib.Tactic.GCongr.Core
#align_import analysis.normed.group.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
open Set
open NNReal
variable {ι R R' E F G : Type*}
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
-- Port... | Mathlib/Analysis/Normed/Group/Seminorm.lean | 148 | 150 | theorem map_sub_le_max : f (x - y) ≤ max (f x) (f y) := by |
rw [sub_eq_add_neg, ← NonarchAddGroupSeminormClass.map_neg_eq_map' f y]
exact map_add_le_max _ _ _
| 1,121 |
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.WithBotTop
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Function Set NNReal
variable {α : Typ... | Mathlib/Data/ENNReal/Basic.lean | 212 | 213 | theorem ofReal_toReal {a : ℝ≥0∞} (h : a ≠ ∞) : ENNReal.ofReal a.toReal = a := by |
simp [ENNReal.toReal, ENNReal.ofReal, h]
| 1,122 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 33 | 41 | theorem mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d := by |
rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩
lift a to ℝ≥0 using ne_top_of_lt aa'
rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩
lift b to ℝ≥0 using ne_top_of_lt bb'
norm_cast at *
calc
↑(a * b) < ↑(a' * b') := coe_lt_coe.2 (mul_lt_mul₀ aa' bb')
_ ≤ c * d := mul_le_mul' a'c.... | 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 71 | 77 | theorem mul_left_strictMono (h0 : a ≠ 0) (hinf : a ≠ ∞) : StrictMono (a * ·) := by |
lift a to ℝ≥0 using hinf
rw [coe_ne_zero] at h0
intro x y h
contrapose! h
simpa only [← mul_assoc, ← coe_mul, inv_mul_cancel h0, coe_one, one_mul]
using mul_le_mul_left' h (↑a⁻¹)
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 130 | 130 | theorem not_lt_zero : ¬a < 0 := by | simp
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 177 | 178 | theorem lt_add_right (ha : a ≠ ∞) (hb : b ≠ 0) : a < a + b := by |
rwa [← pos_iff_ne_zero, ← ENNReal.add_lt_add_iff_left ha, add_zero] at hb
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 193 | 197 | theorem toNNReal_add {r₁ r₂ : ℝ≥0∞} (h₁ : r₁ ≠ ∞) (h₂ : r₂ ≠ ∞) :
(r₁ + r₂).toNNReal = r₁.toNNReal + r₂.toNNReal := by |
lift r₁ to ℝ≥0 using h₁
lift r₂ to ℝ≥0 using h₂
rfl
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 200 | 200 | theorem not_lt_top {x : ℝ≥0∞} : ¬x < ∞ ↔ x = ∞ := by | rw [lt_top_iff_ne_top, Classical.not_not]
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 203 | 203 | theorem add_ne_top : a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞ := by | simpa only [lt_top_iff_ne_top] using add_lt_top
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 206 | 206 | theorem mul_top' : a * ∞ = if a = 0 then 0 else ∞ := by | convert WithTop.mul_top' a
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 212 | 212 | theorem top_mul' : ∞ * a = if a = 0 then 0 else ∞ := by | convert WithTop.top_mul' a
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 235 | 235 | theorem mul_ne_top : a ≠ ∞ → b ≠ ∞ → a * b ≠ ∞ := by | simpa only [lt_top_iff_ne_top] using mul_lt_top
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 246 | 252 | theorem mul_lt_top_iff {a b : ℝ≥0∞} : a * b < ∞ ↔ a < ∞ ∧ b < ∞ ∨ a = 0 ∨ b = 0 := by |
constructor
· intro h
rw [← or_assoc, or_iff_not_imp_right, or_iff_not_imp_right]
intro hb ha
exact ⟨lt_top_of_mul_ne_top_left h.ne hb, lt_top_of_mul_ne_top_right h.ne ha⟩
· rintro (⟨ha, hb⟩ | rfl | rfl) <;> [exact mul_lt_top ha.ne hb.ne; simp; simp]
| 1,123 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 255 | 258 | theorem mul_self_lt_top_iff {a : ℝ≥0∞} : a * a < ⊤ ↔ a < ⊤ := by |
rw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp]
rintro rfl
exact zero_lt_top
| 1,123 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 68 | 68 | theorem coe_inv_two : ((2⁻¹ : ℝ≥0) : ℝ≥0∞) = 2⁻¹ := by | rw [coe_inv _root_.two_ne_zero, coe_two]
| 1,124 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 72 | 73 | theorem coe_div (hr : r ≠ 0) : (↑(p / r) : ℝ≥0∞) = p / r := by |
rw [div_eq_mul_inv, div_eq_mul_inv, coe_mul, coe_inv hr]
| 1,124 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 79 | 79 | theorem div_zero (h : a ≠ 0) : a / 0 = ∞ := by | simp [div_eq_mul_inv, h]
| 1,124 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 133 | 133 | theorem inv_ne_top : a⁻¹ ≠ ∞ ↔ a ≠ 0 := by | simp
| 1,124 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 137 | 138 | theorem inv_lt_top {x : ℝ≥0∞} : x⁻¹ < ∞ ↔ 0 < x := by |
simp only [lt_top_iff_ne_top, inv_ne_top, pos_iff_ne_zero]
| 1,124 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
| Mathlib/Data/ENNReal/Real.lean | 37 | 40 | theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal + b.toReal := by |
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
rfl
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 43 | 47 | theorem toReal_sub_of_le {a b : ℝ≥0∞} (h : b ≤ a) (ha : a ≠ ∞) :
(a - b).toReal = a.toReal - b.toReal := by |
lift b to ℝ≥0 using ne_top_of_le_ne_top ha h
lift a to ℝ≥0 using ha
simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 50 | 55 | theorem le_toReal_sub {a b : ℝ≥0∞} (hb : b ≠ ∞) : a.toReal - b.toReal ≤ (a - b).toReal := by |
lift b to ℝ≥0 using hb
induction a
· simp
· simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]
exact le_max_left _ _
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 65 | 68 | theorem ofReal_add {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) :
ENNReal.ofReal (p + q) = ENNReal.ofReal p + ENNReal.ofReal q := by |
rw [ENNReal.ofReal, ENNReal.ofReal, ENNReal.ofReal, ← coe_add, coe_inj,
Real.toNNReal_add hp hq]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 76 | 79 | theorem toReal_le_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal ≤ b.toReal ↔ a ≤ b := by |
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
norm_cast
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 88 | 91 | theorem toReal_mono' (h : a ≤ b) (ht : b = ∞ → a = ∞) : a.toReal ≤ b.toReal := by |
rcases eq_or_ne a ∞ with rfl | ha
· exact toReal_nonneg
· exact toReal_mono (mt ht ha) h
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 94 | 97 | theorem toReal_lt_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal < b.toReal ↔ a < b := by |
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
norm_cast
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 114 | 117 | theorem toReal_le_add' (hle : a ≤ b + c) (hb : b = ∞ → a = ∞) (hc : c = ∞ → a = ∞) :
a.toReal ≤ b.toReal + c.toReal := by |
refine le_trans (toReal_mono' hle ?_) toReal_add_le
simpa only [add_eq_top, or_imp] using And.intro hb hc
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 132 | 133 | theorem toNNReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toNNReal < b.toNNReal := by |
simpa [← ENNReal.coe_lt_coe, hb, h.ne_top]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
| Mathlib/Data/ENNReal/Real.lean | 541 | 545 | theorem toNNReal_iInf (hf : ∀ i, f i ≠ ∞) : (iInf f).toNNReal = ⨅ i, (f i).toNNReal := by |
cases isEmpty_or_nonempty ι
· rw [iInf_of_empty, top_toNNReal, NNReal.iInf_empty]
· lift f to ι → ℝ≥0 using hf
simp_rw [← coe_iInf, toNNReal_coe]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 548 | 553 | theorem toNNReal_sInf (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toNNReal = sInf (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sInf_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 556 | 561 | theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal := by |
lift f to ι → ℝ≥0 using hf
simp_rw [toNNReal_coe]
by_cases h : BddAbove (range f)
· rw [← coe_iSup h, toNNReal_coe]
· rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 564 | 569 | theorem toNNReal_sSup (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sSup s).toNNReal = sSup (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sSup_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sSup_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iSup hf)
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 572 | 573 | theorem toReal_iInf (hf : ∀ i, f i ≠ ∞) : (iInf f).toReal = ⨅ i, (f i).toReal := by |
simp only [ENNReal.toReal, toNNReal_iInf hf, NNReal.coe_iInf]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 576 | 578 | theorem toReal_sInf (s : Set ℝ≥0∞) (hf : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toReal = sInf (ENNReal.toReal '' s) := by |
simp only [ENNReal.toReal, toNNReal_sInf s hf, NNReal.coe_sInf, Set.image_image]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 581 | 582 | theorem toReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toReal = ⨆ i, (f i).toReal := by |
simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup]
| 1,125 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 585 | 587 | theorem toReal_sSup (s : Set ℝ≥0∞) (hf : ∀ r ∈ s, r ≠ ∞) :
(sSup s).toReal = sSup (ENNReal.toReal '' s) := by |
simp only [ENNReal.toReal, toNNReal_sSup s hf, NNReal.coe_sSup, Set.image_image]
| 1,125 |
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