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 ⌀ |
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projLeft (p : Path.Homotopic.Quotient c₁ c₂) : Path.Homotopic.Quotient c₁.1 c₂.1 :=
p.mapFn ⟨_, continuous_fst⟩ | abbrev | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Homotopy.Path"
] | Mathlib/Topology/Homotopy/Product.lean | projLeft | Abbreviation for projection onto the left coordinate of a path class. |
projRight (p : Path.Homotopic.Quotient c₁ c₂) : Path.Homotopic.Quotient c₁.2 c₂.2 :=
p.mapFn ⟨_, continuous_snd⟩ | abbrev | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Homotopy.Path"
] | Mathlib/Topology/Homotopy/Product.lean | projRight | Abbreviation for projection onto the right coordinate of a path class. |
@[simp]
projLeft_prod : projLeft (prod q₁ q₂) = q₁ := by
induction q₁, q₂ using Quotient.inductionOn₂
rw [projLeft, prod_lift, ← Path.Homotopic.map_lift]
congr
@[simp] | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Homotopy.Path"
] | Mathlib/Topology/Homotopy/Product.lean | projLeft_prod | Lemmas showing projection is the inverse of product. |
projRight_prod : projRight (prod q₁ q₂) = q₂ := by
induction q₁, q₂ using Quotient.inductionOn₂
rw [projRight, prod_lift, ← Path.Homotopic.map_lift]
congr
@[simp] | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Homotopy.Path"
] | Mathlib/Topology/Homotopy/Product.lean | projRight_prod | null |
prod_projLeft_projRight (p : Path.Homotopic.Quotient (a₁, b₁) (a₂, b₂)) :
prod (projLeft p) (projRight p) = p := by
induction p using Quotient.inductionOn
simp only [projLeft, projRight, ← Path.Homotopic.map_lift]
congr | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Homotopy.Path"
] | Mathlib/Topology/Homotopy/Product.lean | prod_projLeft_projRight | null |
preCantorSet : ℕ → Set ℝ
| 0 => Set.Icc 0 1
| n + 1 => (· / 3) '' preCantorSet n ∪ (fun x ↦ (2 + x) / 3) '' preCantorSet n
@[simp] lemma preCantorSet_zero : preCantorSet 0 = Set.Icc 0 1 := rfl
@[simp] lemma preCantorSet_succ (n : ℕ) :
preCantorSet (n + 1) = (· / 3) '' preCantorSet n ∪ (fun x ↦ (2 + x) / 3) '' preCantorSet n :=
rfl | def | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | preCantorSet | The order `n` pre-Cantor set, defined starting from `[0, 1]` and successively removing the
middle third of each interval. Formally, the order `n + 1` pre-Cantor set is the
union of the images under the functions `(· / 3)` and `((2 + ·) / 3)` of `preCantorSet n`. |
cantorSet : Set ℝ := ⋂ n, preCantorSet n
/-! | def | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | cantorSet | The Cantor set is the subset of the unit interval obtained as the intersection of all
pre-Cantor sets. This means that the Cantor set is obtained by iteratively removing the
open middle third of each subinterval, starting from the unit interval `[0, 1]`. |
quarters_mem_preCantorSet (n : ℕ) : 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n := by
induction n with
| zero =>
simp only [preCantorSet_zero]
refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩ <;> norm_num
| succ n ih =>
apply And.intro
· -- goal: 1 / 4 ∈ preCantorSet (n + 1)
exact Or.inl ⟨3 / 4, ih.2, by norm_num⟩
· -- goal: 3 / 4 ∈ preCantorSet (n + 1)
exact Or.inr ⟨1 / 4, ih.1, by norm_num⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | quarters_mem_preCantorSet | null |
quarter_mem_preCantorSet (n : ℕ) : 1 / 4 ∈ preCantorSet n := (quarters_mem_preCantorSet n).1 | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | quarter_mem_preCantorSet | null |
quarter_mem_cantorSet : 1 / 4 ∈ cantorSet :=
Set.mem_iInter.mpr quarter_mem_preCantorSet | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | quarter_mem_cantorSet | null |
zero_mem_preCantorSet (n : ℕ) : 0 ∈ preCantorSet n := by
induction n with
| zero =>
simp [preCantorSet]
| succ n ih =>
exact Or.inl ⟨0, ih, by simp only [zero_div]⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | zero_mem_preCantorSet | null |
zero_mem_cantorSet : 0 ∈ cantorSet := by simp [cantorSet, zero_mem_preCantorSet] | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | zero_mem_cantorSet | null |
preCantorSet_antitone : Antitone preCantorSet := by
apply antitone_nat_of_succ_le
intro m
simp only [Set.le_eq_subset, preCantorSet_succ, Set.union_subset_iff]
induction m with
| zero =>
simp only [preCantorSet_zero]
constructor <;> intro x <;>
simp only [Set.mem_image, Set.mem_Icc, forall_exists_index, and_imp] <;>
intro y _ _ _ <;> constructor <;> linarith
| succ m ih => grind [preCantorSet_succ, Set.image_union] | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | preCantorSet_antitone | null |
preCantorSet_subset_unitInterval {n : ℕ} : preCantorSet n ⊆ Set.Icc 0 1 := by
rw [← preCantorSet_zero]
exact preCantorSet_antitone (by simp) | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | preCantorSet_subset_unitInterval | null |
cantorSet_subset_unitInterval : cantorSet ⊆ Set.Icc 0 1 :=
Set.iInter_subset _ 0 | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | cantorSet_subset_unitInterval | The ternary Cantor set is a subset of [0,1]. |
cantorSet_eq_union_halves :
cantorSet = (· / 3) '' cantorSet ∪ (fun x ↦ (2 + x) / 3) '' cantorSet := by
simp only [cantorSet]
rw [Set.image_iInter, Set.image_iInter]
rotate_left
· exact (mulRight_bijective₀ 3⁻¹ (by simp)).comp (AddGroup.addLeft_bijective 2)
· exact mulRight_bijective₀ 3⁻¹ (by simp)
simp_rw [← Function.comp_def,
← Set.iInter_union_of_antitone
(Set.monotone_image.comp_antitone preCantorSet_antitone)
(Set.monotone_image.comp_antitone preCantorSet_antitone),
Function.comp_def, ← preCantorSet_succ]
exact (preCantorSet_antitone.iInter_nat_add _).symm | theorem | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | cantorSet_eq_union_halves | The ternary Cantor set satisfies the equation `C = C / 3 ∪ (2 / 3 + C / 3)`. |
isClosed_preCantorSet (n : ℕ) : IsClosed (preCantorSet n) := by
let f := Homeomorph.mulLeft₀ (1 / 3 : ℝ) (by simp)
let g := (Homeomorph.addLeft (2 : ℝ)).trans f
induction n with
| zero => exact isClosed_Icc
| succ n ih =>
refine IsClosed.union ?_ ?_
· simpa [f, div_eq_inv_mul] using f.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
· simpa [g, f, div_eq_inv_mul] using g.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | isClosed_preCantorSet | The preCantor sets are closed. |
isClosed_cantorSet : IsClosed cantorSet :=
isClosed_iInter isClosed_preCantorSet | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | isClosed_cantorSet | The ternary Cantor set is closed. |
isCompact_cantorSet : IsCompact cantorSet :=
isCompact_Icc.of_isClosed_subset isClosed_cantorSet cantorSet_subset_unitInterval | lemma | Topology | [
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Topology.Algebra.Ring.Real"
] | Mathlib/Topology/Instances/CantorSet.lean | isCompact_cantorSet | The ternary Cantor set is compact. |
Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set ℂ)) :
K = ofRealHom.fieldRange ∨ K = ⊤ := by
suffices range (ofReal : ℝ → ℂ) ⊆ K by
rw [range_subset_iff, ← coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] at this ⊢
exact this
suffices range (ofReal : ℝ → ℂ) ⊆ closure (Set.range ((ofReal : ℝ → ℂ) ∘ ((↑) : ℚ → ℝ))) by
refine subset_trans this ?_
rw [← IsClosed.closure_eq hc]
apply closure_mono
rintro _ ⟨_, rfl⟩
simp only [Function.comp_apply, ofReal_ratCast, SetLike.mem_coe, SubfieldClass.ratCast_mem]
nth_rw 1 [range_comp]
refine subset_trans ?_ (image_closure_subset_closure_image continuous_ofReal)
rw [DenseRange.closure_range Rat.isDenseEmbedding_coe_real.dense]
simp only [image_univ]
rfl | theorem | Topology | [
"Mathlib.Analysis.Complex.Basic",
"Mathlib.FieldTheory.IntermediateField.Basic",
"Mathlib.LinearAlgebra.Complex.FiniteDimensional",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Instances/Complex.lean | Complex.subfield_eq_of_closed | The only closed subfields of `ℂ` are `ℝ` and `ℂ`. |
Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype := by
letI : IsTopologicalDivisionRing ℂ := IsTopologicalDivisionRing.mk
letI : IsTopologicalRing K.topologicalClosure :=
Subring.instIsTopologicalRing K.topologicalClosure.toSubring
set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure)
have ui : IsUniformInducing ι :=
⟨by
rw [uniformity_subtype, uniformity_subtype, Filter.comap_comap]
congr ⟩
let di := ui.isDenseInducing (?_ : DenseRange ι)
· -- extψ : closure(K) →+* ℂ is the extension of ψ : K →+* ℂ
let extψ := IsDenseInducing.extendRingHom ui di.dense hc
haveI hψ := (uniformContinuous_uniformly_extend ui di.dense hc).continuous
rcases Complex.subfield_eq_of_closed (Subfield.isClosed_topologicalClosure K) with h | h
· left
let j := RingEquiv.subfieldCongr h
let ψ₁ := RingHom.comp extψ (RingHom.comp j.symm.toRingHom ofRealHom.rangeRestrict)
have hψ₁ : Continuous ψ₁ := by
simpa only [RingHom.coe_comp] using hψ.comp ((continuous_algebraMap ℝ ℂ).subtype_mk _)
ext1 x
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ofRealHom.rangeRestrict r = j (ι x)
· have :=
RingHom.congr_fun (ringHom_eq_ofReal_of_continuous hψ₁) r
rw [RingHom.comp_apply, RingHom.comp_apply] at this
erw [hr] at this
rw [RingEquiv.toRingHom_eq_coe] at this
convert this using 1
· exact (IsDenseInducing.extend_eq di hc.continuous _).symm
· rw [← ofRealHom.coe_rangeRestrict, hr]
rfl
obtain ⟨r, hr⟩ := SetLike.coe_mem (j (ι x))
exact ⟨r, Subtype.ext hr⟩
· -- ψ₁ is the continuous ring hom `ℂ →+* ℂ` constructed from `closure (K) ≃+* ℂ`
let ψ₁ :=
RingHom.comp extψ
(RingHom.comp (RingEquiv.subfieldCongr h).symm.toRingHom
(@Subfield.topEquiv ℂ _).symm.toRingHom)
have hψ₁ : Continuous ψ₁ := by
simpa only [RingHom.coe_comp] using hψ.comp (continuous_id.subtype_mk _)
rcases ringHom_eq_id_or_conj_of_continuous hψ₁ with h | h
· left
ext1 z
convert RingHom.congr_fun h z using 1
exact (IsDenseInducing.extend_eq di hc.continuous z).symm
· right
ext1 z
convert RingHom.congr_fun h z using 1
exact (IsDenseInducing.extend_eq di hc.continuous z).symm
· let j : { x // x ∈ closure (id '' { x | (K : Set ℂ) x }) } → (K.topologicalClosure : Set ℂ) :=
fun x =>
... | theorem | Topology | [
"Mathlib.Analysis.Complex.Basic",
"Mathlib.FieldTheory.IntermediateField.Basic",
"Mathlib.LinearAlgebra.Complex.FiniteDimensional",
"Mathlib.LinearAlgebra.FiniteDimensional.Lemmas",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.UniformRing"
] | Mathlib/Topology/Instances/Complex.lean | Complex.uniformContinuous_ringHom_eq_id_or_conj | Let `K` a subfield of `ℂ` and let `ψ : K →+* ℂ` a ring homomorphism. Assume that `ψ` is uniform
continuous, then `ψ` is either the inclusion map or the composition of the inclusion map with the
complex conjugation. |
LinearOrder.bot_topologicalSpace_eq_generateFrom {α} [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
let _ := Preorder.topology α
have : OrderTopology α := ⟨rfl⟩
exact DiscreteTopology.of_predOrder_succOrder.eq_bot.symm | theorem | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.SuccPred.LinearLocallyFinite"
] | Mathlib/Topology/Instances/Discrete.lean | LinearOrder.bot_topologicalSpace_eq_generateFrom | null |
discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine ⟨fun h ↦ ⟨?_⟩, fun h ↦ .of_predOrder_succOrder⟩
rw [h.eq_bot, LinearOrder.bot_topologicalSpace_eq_generateFrom] | theorem | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.SuccPred.LinearLocallyFinite"
] | Mathlib/Topology/Instances/Discrete.lean | discreteTopology_iff_orderTopology_of_pred_succ | null |
OrderTopology.of_discreteTopology [LinearOrder α] [PredOrder α] [SuccOrder α]
[DiscreteTopology α] : OrderTopology α :=
discreteTopology_iff_orderTopology_of_pred_succ.mp ‹_› | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.SuccPred.LinearLocallyFinite"
] | Mathlib/Topology/Instances/Discrete.lean | OrderTopology.of_discreteTopology | null |
OrderTopology.of_linearLocallyFinite
[LinearOrder α] [LocallyFiniteOrder α] [DiscreteTopology α] : OrderTopology α :=
haveI := LinearLocallyFiniteOrder.succOrder α
haveI := LinearLocallyFiniteOrder.predOrder α
inferInstance | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.SuccPred.LinearLocallyFinite"
] | Mathlib/Topology/Instances/Discrete.lean | OrderTopology.of_linearLocallyFinite | null |
@[simp] range_natCast : range ((↑) : ℕ → ℕ∞) = Iio ⊤ :=
WithTop.range_coe | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | range_natCast | null |
isEmbedding_natCast : IsEmbedding ((↑) : ℕ → ℕ∞) :=
Nat.strictMono_cast.isEmbedding_of_ordConnected <| range_natCast ▸ ordConnected_Iio | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | isEmbedding_natCast | null |
isOpenEmbedding_natCast : IsOpenEmbedding ((↑) : ℕ → ℕ∞) :=
⟨isEmbedding_natCast, range_natCast ▸ isOpen_Iio⟩ | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | isOpenEmbedding_natCast | null |
nhds_natCast (n : ℕ) : 𝓝 (n : ℕ∞) = pure (n : ℕ∞) := by
simp [← isOpenEmbedding_natCast.map_nhds_eq]
@[simp] | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | nhds_natCast | null |
protected nhds_eq_pure {n : ℕ∞} (h : n ≠ ⊤) : 𝓝 n = pure n := by
lift n to ℕ using h
simp [nhds_natCast] | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | nhds_eq_pure | null |
isOpen_singleton {x : ℕ∞} (hx : x ≠ ⊤) : IsOpen {x} := by
rw [isOpen_singleton_iff_nhds_eq_pure, ENat.nhds_eq_pure hx] | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | isOpen_singleton | null |
mem_nhds_iff {x : ℕ∞} {s : Set ℕ∞} (hx : x ≠ ⊤) : s ∈ 𝓝 x ↔ x ∈ s := by
simp [hx] | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | mem_nhds_iff | null |
mem_nhds_natCast_iff (n : ℕ) {s : Set ℕ∞} : s ∈ 𝓝 (n : ℕ∞) ↔ (n : ℕ∞) ∈ s :=
mem_nhds_iff (coe_ne_top _) | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | mem_nhds_natCast_iff | null |
tendsto_nhds_top_iff_natCast_lt {α : Type*} {l : Filter α} {f : α → ℕ∞} :
Tendsto f l (𝓝 ⊤) ↔ ∀ n : ℕ, ∀ᶠ a in l, n < f a := by
simp_rw [nhds_top_order, lt_top_iff_ne_top, tendsto_iInf, tendsto_principal]
exact Option.forall_ne_none | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | tendsto_nhds_top_iff_natCast_lt | null |
protected continuousAt_sub {a b : ℕ∞} (h : a ≠ ⊤ ∨ b ≠ ⊤) :
ContinuousAt (· - ·).uncurry (a, b) := by
match a, b, h with
| (a : ℕ), (b : ℕ), _ => simp [ContinuousAt, nhds_prod_eq]
| (a : ℕ), ⊤, _ =>
suffices ∀ᶠ b in 𝓝 ⊤, (a - b : ℕ∞) = 0 by
simpa [ContinuousAt, nhds_prod_eq, tsub_eq_zero_of_le]
filter_upwards [le_mem_nhds (WithTop.coe_lt_top a)] with b using tsub_eq_zero_of_le
| ⊤, (b : ℕ), _ =>
suffices ∀ n : ℕ, ∀ᶠ a : ℕ∞ in 𝓝 ⊤, b + n < a by
simpa [ContinuousAt, nhds_prod_eq, (· ∘ ·), lt_tsub_iff_left, tendsto_nhds_top_iff_natCast_lt]
exact fun n ↦ lt_mem_nhds <| WithTop.coe_lt_top (b + n) | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | continuousAt_sub | null |
Filter.Tendsto.enatSub {α : Type*} {l : Filter α} {f g : α → ℕ∞} {a b : ℕ∞}
(hf : Tendsto f l (𝓝 a)) (hg : Tendsto g l (𝓝 b)) (h : a ≠ ⊤ ∨ b ≠ ⊤) :
Tendsto (fun x ↦ f x - g x) l (𝓝 (a - b)) :=
(ENat.continuousAt_sub h).tendsto.comp (hf.prodMk_nhds hg)
variable {X : Type*} [TopologicalSpace X] {f g : X → ℕ∞} {s : Set X} {x : X}
nonrec theorem ContinuousWithinAt.enatSub
(hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : f x ≠ ⊤ ∨ g x ≠ ⊤) :
ContinuousWithinAt (fun x ↦ f x - g x) s x :=
hf.enatSub hg h
nonrec theorem ContinuousAt.enatSub
(hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : f x ≠ ⊤ ∨ g x ≠ ⊤) :
ContinuousAt (fun x ↦ f x - g x) x :=
hf.enatSub hg h
nonrec theorem ContinuousOn.enatSub
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (h : ∀ x ∈ s, f x ≠ ⊤ ∨ g x ≠ ⊤) :
ContinuousOn (fun x ↦ f x - g x) s := fun x hx ↦
(hf x hx).enatSub (hg x hx) (h x hx)
nonrec theorem Continuous.enatSub
(hf : Continuous f) (hg : Continuous g) (h : ∀ x, f x ≠ ⊤ ∨ g x ≠ ⊤) :
Continuous (fun x ↦ f x - g x) :=
continuous_iff_continuousAt.2 fun x ↦ hf.continuousAt.enatSub hg.continuousAt (h x) | theorem | Topology | [
"Mathlib.Data.ENat.Basic",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Order.Interval.Set.WithBotTop",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Instances/ENat.lean | Filter.Tendsto.enatSub | null |
dist_eq (x y : ℤ) : dist x y = |(x : ℝ) - y| := rfl | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | dist_eq | null |
dist_eq' (m n : ℤ) : dist m n = |m - n| := by rw [dist_eq]; norm_cast
@[norm_cast, simp] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | dist_eq' | null |
dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y :=
rfl | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | dist_cast_real | null |
pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by
intro m n hne
rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | pairwise_one_le_dist | null |
isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℤ → ℝ) :=
isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | isUniformEmbedding_coe_real | null |
isClosedEmbedding_coe_real : IsClosedEmbedding ((↑) : ℤ → ℝ) :=
isClosedEmbedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | isClosedEmbedding_coe_real | null |
preimage_ball (x : ℤ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r := rfl | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | preimage_ball | null |
preimage_closedBall (x : ℤ) (r : ℝ) : (↑) ⁻¹' closedBall (x : ℝ) r = closedBall x r := rfl | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | preimage_closedBall | null |
ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ := by
rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | ball_eq_Ioo | null |
closedBall_eq_Icc (x : ℤ) (r : ℝ) : closedBall x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋ := by
rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | closedBall_eq_Icc | null |
@[simp]
cobounded_eq : Bornology.cobounded ℤ = atBot ⊔ atTop := by
simp_rw [← comap_dist_right_atTop (0 : ℤ), dist_eq', sub_zero,
← comap_abs_atTop, ← @Int.comap_cast_atTop ℝ, comap_comap]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | cobounded_eq | null |
cofinite_eq : (cofinite : Filter ℤ) = atBot ⊔ atTop := by
rw [← cocompact_eq_cofinite, cocompact_eq_atBot_atTop] | theorem | Topology | [
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Order.Filter.AtTopBot.Archimedean",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Order.Bornology"
] | Mathlib/Topology/Instances/Int.lean | cofinite_eq | null |
protected IsGδ.setOf_irrational : IsGδ { x | Irrational x } :=
(countable_range _).isGδ_compl | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | IsGδ.setOf_irrational | null |
dense_irrational : Dense { x : ℝ | Irrational x } := by
refine Real.isTopologicalBasis_Ioo_rat.dense_iff.2 ?_
simp only [mem_iUnion, mem_singleton_iff, exists_prop, forall_exists_index, and_imp]
rintro _ a b hlt rfl _
rw [inter_comm]
exact exists_irrational_btwn (Rat.cast_lt.2 hlt) | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | dense_irrational | null |
eventually_residual_irrational : ∀ᶠ x in residual ℝ, Irrational x :=
residual_of_dense_Gδ .setOf_irrational dense_irrational | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | eventually_residual_irrational | null |
eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) := by
have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.isClosedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ⟨ε, ε0, hε⟩
refine (ge_mem_nhds ε0).mono fun δ hδ m => not_lt.1 fun hlt => ?_
rw [dist_comm] at hlt
refine hε (ball_subset_ball hδ hlt) ⟨m, ?_⟩
simp [div_eq_inv_mul] | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | eventually_forall_le_dist_cast_div | null |
eventually_forall_le_dist_cast_div_of_denom_le (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (m / k) :=
(finite_le_nat n).eventually_all.2 fun k _ => hx.eventually_forall_le_dist_cast_div k | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | eventually_forall_le_dist_cast_div_of_denom_le | null |
eventually_forall_le_dist_cast_rat_of_den_le (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ r : ℚ, r.den ≤ n → ε ≤ dist x r :=
(hx.eventually_forall_le_dist_cast_div_of_denom_le n).mono fun ε H r hr => by
simpa only [Rat.cast_def] using H r.den hr r.num | theorem | Topology | [
"Mathlib.Data.Real.Irrational",
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.Separation.GDelta",
"Mathlib.Topology.Instances.Real.Lemmas"
] | Mathlib/Topology/Instances/Irrational.lean | eventually_forall_le_dist_cast_rat_of_den_le | null |
@[continuity]
continuous_matrix [TopologicalSpace α] {f : α → Matrix m n R}
(h : ∀ i j, Continuous fun a => f a i j) : Continuous f :=
continuous_pi fun _ => continuous_pi fun _ => h _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | continuous_matrix | The topology on finite matrices over a discrete space is discrete. -/
instance [TopologicalSpace R] [Finite m] [Finite n] [DiscreteTopology R] :
DiscreteTopology (Matrix m n R) :=
Pi.discreteTopology
section Set
theorem IsOpen.matrix [Finite m] [Finite n]
[TopologicalSpace R] {S : Set R} (hS : IsOpen S) :
IsOpen (S.matrix : Set (Matrix m n R)) :=
Set.matrix_eq_pi ▸
(isOpen_set_pi Set.finite_univ fun _ _ =>
isOpen_set_pi Set.finite_univ fun _ _ => hS).preimage continuous_id
theorem IsCompact.matrix [TopologicalSpace R] {S : Set R} (hS : IsCompact S) :
IsCompact (S.matrix : Set (Matrix m n R)) :=
isCompact_pi_infinite fun _ => isCompact_pi_infinite fun _ => hS
end Set
/-! ### Lemmas about continuity of operations -/
section Continuity
variable [TopologicalSpace X] [TopologicalSpace R]
instance [SMul α R] [ContinuousConstSMul α R] : ContinuousConstSMul α (Matrix m n R) :=
inferInstanceAs (ContinuousConstSMul α (m → n → R))
instance [TopologicalSpace α] [SMul α R] [ContinuousSMul α R] : ContinuousSMul α (Matrix m n R) :=
inferInstanceAs (ContinuousSMul α (m → n → R))
instance [Add R] [ContinuousAdd R] : ContinuousAdd (Matrix m n R) :=
Pi.continuousAdd
instance [Neg R] [ContinuousNeg R] : ContinuousNeg (Matrix m n R) :=
Pi.continuousNeg
instance [AddGroup R] [IsTopologicalAddGroup R] : IsTopologicalAddGroup (Matrix m n R) :=
Pi.topologicalAddGroup
/-- To show a function into matrices is continuous it suffices to show the coefficients of the
resulting matrix are continuous |
Continuous.matrix_elem {A : X → Matrix m n R} (hA : Continuous A) (i : m) (j : n) :
Continuous fun x => A x i j :=
(continuous_apply_apply i j).comp hA
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_elem | null |
Continuous.matrix_map [TopologicalSpace S] {A : X → Matrix m n S} {f : S → R}
(hA : Continuous A) (hf : Continuous f) : Continuous fun x => (A x).map f :=
continuous_matrix fun _ _ => hf.comp <| hA.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_map | null |
Continuous.matrix_transpose {A : X → Matrix m n R} (hA : Continuous A) :
Continuous fun x => (A x)ᵀ :=
continuous_matrix fun i j => hA.matrix_elem j i
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_transpose | null |
Continuous.matrix_conjTranspose [Star R] [ContinuousStar R] {A : X → Matrix m n R}
(hA : Continuous A) : Continuous fun x => (A x)ᴴ :=
hA.matrix_transpose.matrix_map continuous_star | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_conjTranspose | null |
@[continuity, fun_prop]
Continuous.matrix_replicateCol {ι : Type*} {A : X → n → R} (hA : Continuous A) :
Continuous fun x => replicateCol ι (A x) :=
continuous_matrix fun i _ => (continuous_apply i).comp hA
@[deprecated (since := "2025-03-15")] alias Continuous.matrix_col := Continuous.matrix_replicateCol
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_replicateCol | null |
Continuous.matrix_replicateRow {ι : Type*} {A : X → n → R} (hA : Continuous A) :
Continuous fun x => replicateRow ι (A x) :=
continuous_matrix fun _ _ => (continuous_apply _).comp hA
@[deprecated (since := "2025-03-15")] alias Continuous.matrix_row := Continuous.matrix_replicateRow
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_replicateRow | null |
Continuous.matrix_diagonal [Zero R] [DecidableEq n] {A : X → n → R} (hA : Continuous A) :
Continuous fun x => diagonal (A x) :=
continuous_matrix fun i _ => ((continuous_apply i).comp hA).if_const _ continuous_zero
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_diagonal | null |
protected Continuous.dotProduct [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
[ContinuousMul R] {A : X → n → R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
Continuous fun x => A x ⬝ᵥ B x := by
dsimp only [dotProduct]
fun_prop
@[deprecated (since := "2025-05-09")]
alias Continuous.matrix_dotProduct := Continuous.dotProduct | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.dotProduct | null |
@[continuity, fun_prop]
Continuous.matrix_mul [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
[ContinuousMul R] {A : X → Matrix m n R} {B : X → Matrix n p R} (hA : Continuous A)
(hB : Continuous B) : Continuous fun x => A x * B x :=
continuous_matrix fun _ _ =>
continuous_finset_sum _ fun _ _ => (hA.matrix_elem _ _).mul (hB.matrix_elem _ _) | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_mul | For square matrices the usual `continuous_mul` can be used. |
Matrix.topologicalRing [Fintype n] [NonUnitalNonAssocRing R] [IsTopologicalRing R] :
IsTopologicalRing (Matrix n n R) where
@[continuity, fun_prop] | instance | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Matrix.topologicalRing | null |
Continuous.matrix_vecMulVec [Mul R] [ContinuousMul R] {A : X → m → R} {B : X → n → R}
(hA : Continuous A) (hB : Continuous B) : Continuous fun x => vecMulVec (A x) (B x) :=
continuous_matrix fun _ _ => ((continuous_apply _).comp hA).mul ((continuous_apply _).comp hB)
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_vecMulVec | null |
Continuous.matrix_mulVec [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
[Fintype n] {A : X → Matrix m n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
Continuous fun x => A x *ᵥ B x :=
continuous_pi fun i => ((continuous_apply i).comp hA).dotProduct hB
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_mulVec | null |
Continuous.matrix_vecMul [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
[Fintype m] {A : X → m → R} {B : X → Matrix m n R} (hA : Continuous A) (hB : Continuous B) :
Continuous fun x => A x ᵥ* B x :=
continuous_pi fun _i => hA.dotProduct <| continuous_pi fun _j => hB.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_vecMul | null |
Continuous.matrix_submatrix {A : X → Matrix l n R} (hA : Continuous A) (e₁ : m → l)
(e₂ : p → n) : Continuous fun x => (A x).submatrix e₁ e₂ :=
continuous_matrix fun _i _j => hA.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_submatrix | null |
Continuous.matrix_reindex {A : X → Matrix l n R} (hA : Continuous A) (e₁ : l ≃ m)
(e₂ : n ≃ p) : Continuous fun x => reindex e₁ e₂ (A x) :=
hA.matrix_submatrix _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_reindex | null |
Continuous.matrix_diag {A : X → Matrix n n R} (hA : Continuous A) :
Continuous fun x => Matrix.diag (A x) :=
continuous_pi fun _ => hA.matrix_elem _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_diag | null |
continuous_matrix_diag : Continuous (Matrix.diag : Matrix n n R → n → R) :=
show Continuous fun x : Matrix n n R => Matrix.diag x from continuous_id.matrix_diag
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | continuous_matrix_diag | null |
Continuous.matrix_trace [Fintype n] [AddCommMonoid R] [ContinuousAdd R]
{A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => trace (A x) :=
continuous_finset_sum _ fun _ _ => hA.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_trace | null |
Continuous.matrix_det [Fintype n] [DecidableEq n] [CommRing R] [IsTopologicalRing R]
{A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).det := by
simp_rw [Matrix.det_apply]
refine continuous_finset_sum _ fun l _ => Continuous.const_smul ?_ _
exact continuous_finset_prod _ fun l _ => hA.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_det | null |
Continuous.matrix_updateCol [DecidableEq n] (i : n) {A : X → Matrix m n R}
{B : X → m → R} (hA : Continuous A) (hB : Continuous B) :
Continuous fun x => (A x).updateCol i (B x) :=
continuous_matrix fun _j k =>
(continuous_apply k).comp <|
((continuous_apply _).comp hA).update i ((continuous_apply _).comp hB)
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_updateCol | null |
Continuous.matrix_updateRow [DecidableEq m] (i : m) {A : X → Matrix m n R} {B : X → n → R}
(hA : Continuous A) (hB : Continuous B) : Continuous fun x => (A x).updateRow i (B x) :=
hA.update i hB
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_updateRow | null |
Continuous.matrix_cramer [Fintype n] [DecidableEq n] [CommRing R] [IsTopologicalRing R]
{A : X → Matrix n n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
Continuous fun x => cramer (A x) (B x) :=
continuous_pi fun _ => (hA.matrix_updateCol _ hB).matrix_det
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_cramer | null |
Continuous.matrix_adjugate [Fintype n] [DecidableEq n] [CommRing R] [IsTopologicalRing R]
{A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).adjugate :=
continuous_matrix fun _j k =>
(hA.matrix_transpose.matrix_updateCol k continuous_const).matrix_det | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_adjugate | null |
continuousAt_matrix_inv [Fintype n] [DecidableEq n] [CommRing R] [IsTopologicalRing R]
(A : Matrix n n R) (h : ContinuousAt Ring.inverse A.det) : ContinuousAt Inv.inv A :=
(h.comp continuous_id.matrix_det.continuousAt).smul continuous_id.matrix_adjugate.continuousAt | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | continuousAt_matrix_inv | When `Ring.inverse` is continuous at the determinant (such as in a `NormedRing`, or a
topological field), so is `Matrix.inv`. |
IsInducing.matrix_map (hf : IsInducing f) :
IsInducing (map · f : Matrix m n R → Matrix m n S) :=
IsInducing.piMap fun _ : m ↦ (IsInducing.piMap fun _ : n ↦ hf) | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | IsInducing.matrix_map | null |
IsEmbedding.matrix_map (hf : IsEmbedding f) :
IsEmbedding (map · f : Matrix m n R → Matrix m n S) :=
IsEmbedding.piMap fun _ : m ↦ (IsEmbedding.piMap fun _ : n ↦ hf) | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | IsEmbedding.matrix_map | null |
IsClosedEmbedding.matrix_map (hf : IsClosedEmbedding f) :
IsClosedEmbedding (map · f : Matrix m n R → Matrix m n S) :=
IsClosedEmbedding.piMap fun _ : m ↦ (IsClosedEmbedding.piMap fun _ : n ↦ hf) | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | IsClosedEmbedding.matrix_map | null |
IsOpenEmbedding.matrix_map [Finite m] [Finite n] (hf : IsOpenEmbedding f) :
IsOpenEmbedding (map · f : Matrix m n R → Matrix m n S) :=
IsOpenEmbedding.piMap fun _ : m ↦ (IsOpenEmbedding.piMap fun _ : n ↦ hf) | lemma | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | IsOpenEmbedding.matrix_map | null |
@[continuity, fun_prop]
Continuous.matrix_fromBlocks {A : X → Matrix n l R} {B : X → Matrix n m R}
{C : X → Matrix p l R} {D : X → Matrix p m R} (hA : Continuous A) (hB : Continuous B)
(hC : Continuous C) (hD : Continuous D) :
Continuous fun x => Matrix.fromBlocks (A x) (B x) (C x) (D x) :=
continuous_matrix <| by
rintro (i | i) (j | j) <;> refine Continuous.matrix_elem ?_ i j <;> assumption
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_fromBlocks | null |
Continuous.matrix_blockDiagonal [Zero R] [DecidableEq p] {A : X → p → Matrix m n R}
(hA : Continuous A) : Continuous fun x => blockDiagonal (A x) :=
continuous_matrix fun ⟨i₁, i₂⟩ ⟨j₁, _j₂⟩ =>
(((continuous_apply i₂).comp hA).matrix_elem i₁ j₁).if_const _ continuous_zero
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_blockDiagonal | null |
Continuous.matrix_blockDiag {A : X → Matrix (m × p) (n × p) R} (hA : Continuous A) :
Continuous fun x => blockDiag (A x) :=
continuous_pi fun _i => continuous_matrix fun _j _k => hA.matrix_elem _ _
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_blockDiag | null |
Continuous.matrix_blockDiagonal' [Zero R] [DecidableEq l]
{A : X → ∀ i, Matrix (m' i) (n' i) R} (hA : Continuous A) :
Continuous fun x => blockDiagonal' (A x) :=
continuous_matrix fun ⟨i₁, i₂⟩ ⟨j₁, j₂⟩ => by
dsimp only [blockDiagonal'_apply']
split_ifs with h
· subst h
exact ((continuous_apply i₁).comp hA).matrix_elem i₂ j₂
· exact continuous_const
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_blockDiagonal' | null |
Continuous.matrix_blockDiag'
{A : X → Matrix (Σ i, m' i) (Σ i, n' i) R} (hA : Continuous A) :
Continuous fun x => blockDiag' (A x) :=
continuous_pi fun _i => continuous_matrix fun _j _k => hA.matrix_elem _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Continuous.matrix_blockDiag' | null |
HasSum.matrix_transpose {f : X → Matrix m n R} {a : Matrix m n R} (hf : HasSum f a) :
HasSum (fun x => (f x)ᵀ) aᵀ :=
(hf.map (Matrix.transposeAddEquiv m n R) continuous_id.matrix_transpose :) | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | HasSum.matrix_transpose | null |
Summable.matrix_transpose {f : X → Matrix m n R} (hf : Summable f) :
Summable fun x => (f x)ᵀ :=
hf.hasSum.matrix_transpose.summable
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_transpose | null |
summable_matrix_transpose {f : X → Matrix m n R} :
(Summable fun x => (f x)ᵀ) ↔ Summable f :=
Summable.map_iff_of_equiv (Matrix.transposeAddEquiv m n R)
continuous_id.matrix_transpose continuous_id.matrix_transpose | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | summable_matrix_transpose | null |
Matrix.transpose_tsum [T2Space R] {f : X → Matrix m n R} : (∑' x, f x)ᵀ = ∑' x, (f x)ᵀ := by
by_cases hf : Summable f
· exact hf.hasSum.matrix_transpose.tsum_eq.symm
· have hft := summable_matrix_transpose.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, transpose_zero] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Matrix.transpose_tsum | null |
HasSum.matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R}
{a : Matrix m n R} (hf : HasSum f a) : HasSum (fun x => (f x)ᴴ) aᴴ :=
(hf.map (Matrix.conjTransposeAddEquiv m n R) continuous_id.matrix_conjTranspose :) | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | HasSum.matrix_conjTranspose | null |
Summable.matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R}
(hf : Summable f) : Summable fun x => (f x)ᴴ :=
hf.hasSum.matrix_conjTranspose.summable
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_conjTranspose | null |
summable_matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R} :
(Summable fun x => (f x)ᴴ) ↔ Summable f :=
Summable.map_iff_of_equiv (Matrix.conjTransposeAddEquiv m n R)
continuous_id.matrix_conjTranspose continuous_id.matrix_conjTranspose | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | summable_matrix_conjTranspose | null |
Matrix.conjTranspose_tsum [StarAddMonoid R] [ContinuousStar R] [T2Space R]
{f : X → Matrix m n R} : (∑' x, f x)ᴴ = ∑' x, (f x)ᴴ := by
by_cases hf : Summable f
· exact hf.hasSum.matrix_conjTranspose.tsum_eq.symm
· have hft := summable_matrix_conjTranspose.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, conjTranspose_zero] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Matrix.conjTranspose_tsum | null |
HasSum.matrix_diagonal [DecidableEq n] {f : X → n → R} {a : n → R} (hf : HasSum f a) :
HasSum (fun x => diagonal (f x)) (diagonal a) :=
hf.map (diagonalAddMonoidHom n R) continuous_id.matrix_diagonal | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | HasSum.matrix_diagonal | null |
Summable.matrix_diagonal [DecidableEq n] {f : X → n → R} (hf : Summable f) :
Summable fun x => diagonal (f x) :=
hf.hasSum.matrix_diagonal.summable
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_diagonal | null |
summable_matrix_diagonal [DecidableEq n] {f : X → n → R} :
(Summable fun x => diagonal (f x)) ↔ Summable f :=
Summable.map_iff_of_leftInverse (Matrix.diagonalAddMonoidHom n R) (Matrix.diagAddMonoidHom n R)
continuous_id.matrix_diagonal continuous_matrix_diag fun A => diag_diagonal A | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | summable_matrix_diagonal | null |
Matrix.diagonal_tsum [DecidableEq n] [T2Space R] {f : X → n → R} :
diagonal (∑' x, f x) = ∑' x, diagonal (f x) := by
by_cases hf : Summable f
· exact hf.hasSum.matrix_diagonal.tsum_eq.symm
· have hft := summable_matrix_diagonal.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]
exact diagonal_zero | theorem | Topology | [
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Star",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs",
"Mathlib.LinearAlgebra.Matrix.Trace... | Mathlib/Topology/Instances/Matrix.lean | Matrix.diagonal_tsum | null |
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