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 ⌀ |
|---|---|---|---|---|---|---|
HasSum.matrix_diag {f : X → Matrix n n R} {a : Matrix n n R} (hf : HasSum f a) :
HasSum (fun x => diag (f x)) (diag a) :=
hf.map (diagAddMonoidHom n R) continuous_matrix_diag | 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_diag | null |
Summable.matrix_diag {f : X → Matrix n n R} (hf : Summable f) :
Summable fun x => diag (f x) :=
hf.hasSum.matrix_diag.summable | 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_diag | null |
HasSum.matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R}
{a : p → Matrix m n R} (hf : HasSum f a) :
HasSum (fun x => blockDiagonal (f x)) (blockDiagonal a) :=
hf.map (blockDiagonalAddMonoidHom m n p R) continuous_id.matrix_blockDiagonal | 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_blockDiagonal | null |
Summable.matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R} (hf : Summable f) :
Summable fun x => blockDiagonal (f x) :=
hf.hasSum.matrix_blockDiagonal.summable | 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_blockDiagonal | null |
summable_matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R} :
(Summable fun x => blockDiagonal (f x)) ↔ Summable f :=
Summable.map_iff_of_leftInverse (blockDiagonalAddMonoidHom m n p R)
(blockDiagAddMonoidHom m n p R) continuous_id.matrix_blockDiagonal
continuous_id.matrix_blockDiag fun A => blockDiag_blockDiagonal 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_blockDiagonal | null |
Matrix.blockDiagonal_tsum [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R} :
blockDiagonal (∑' x, f x) = ∑' x, blockDiagonal (f x) := by
by_cases hf : Summable f
· exact hf.hasSum.matrix_blockDiagonal.tsum_eq.symm
· have hft := summable_matrix_blockDiagonal.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]
exact blockDiagonal_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.blockDiagonal_tsum | null |
HasSum.matrix_blockDiag {f : X → Matrix (m × p) (n × p) R} {a : Matrix (m × p) (n × p) R}
(hf : HasSum f a) : HasSum (fun x => blockDiag (f x)) (blockDiag a) :=
(hf.map (blockDiagAddMonoidHom m n p R) <| Continuous.matrix_blockDiag continuous_id :) | 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_blockDiag | null |
Summable.matrix_blockDiag {f : X → Matrix (m × p) (n × p) R} (hf : Summable f) :
Summable fun x => blockDiag (f x) :=
hf.hasSum.matrix_blockDiag.summable | 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_blockDiag | null |
HasSum.matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R}
{a : ∀ i, Matrix (m' i) (n' i) R} (hf : HasSum f a) :
HasSum (fun x => blockDiagonal' (f x)) (blockDiagonal' a) :=
hf.map (blockDiagonal'AddMonoidHom m' n' R) continuous_id.matrix_blockDiagonal' | 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_blockDiagonal' | null |
Summable.matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R}
(hf : Summable f) : Summable fun x => blockDiagonal' (f x) :=
hf.hasSum.matrix_blockDiagonal'.summable | 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_blockDiagonal' | null |
summable_matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R} :
(Summable fun x => blockDiagonal' (f x)) ↔ Summable f :=
Summable.map_iff_of_leftInverse (blockDiagonal'AddMonoidHom m' n' R)
(blockDiag'AddMonoidHom m' n' R) continuous_id.matrix_blockDiagonal'
continuous_id.matrix_blockDiag' fun A => blockDiag'_blockDiagonal' 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_blockDiagonal' | null |
Matrix.blockDiagonal'_tsum [DecidableEq l] [T2Space R]
{f : X → ∀ i, Matrix (m' i) (n' i) R} :
blockDiagonal' (∑' x, f x) = ∑' x, blockDiagonal' (f x) := by
by_cases hf : Summable f
· exact hf.hasSum.matrix_blockDiagonal'.tsum_eq.symm
· have hft := summable_matrix_blockDiagonal'.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]
exact blockDiagonal'_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.blockDiagonal'_tsum | null |
HasSum.matrix_blockDiag' {f : X → Matrix (Σ i, m' i) (Σ i, n' i) R}
{a : Matrix (Σ i, m' i) (Σ i, n' i) R} (hf : HasSum f a) :
HasSum (fun x => blockDiag' (f x)) (blockDiag' a) :=
hf.map (blockDiag'AddMonoidHom m' n' R) continuous_id.matrix_blockDiag' | 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_blockDiag' | null |
Summable.matrix_blockDiag' {f : X → Matrix (Σ i, m' i) (Σ i, n' i) R} (hf : Summable f) :
Summable fun x => blockDiag' (f x) :=
hf.hasSum.matrix_blockDiag'.summable | 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_blockDiag' | null |
@[continuity, fun_prop] protected continuous_det :
Continuous (det : GL n R → Rˣ) := by
simp_rw [Units.continuous_iff, ← map_inv]
constructor <;> fun_prop | 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 | continuous_det | The determinant is continuous as a map from the general linear group to the units. |
topologicalGroup : IsTopologicalGroup (SL n R) where
continuous_inv := by simpa [continuous_induced_rng] using continuous_induced_dom.matrix_adjugate
continuous_mul := by simpa only [continuous_induced_rng] using
(continuous_induced_dom.comp continuous_fst).mul (continuous_induced_dom.comp continuous_snd) | 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 | topologicalGroup | If `R` is a commutative ring with the discrete topology, then `SL(n, R)` has the discrete
topology. -/
instance [DiscreteTopology R] : DiscreteTopology (SL n R) :=
instDiscreteTopologySubtype
/-- The special linear group over a topological ring is a topological group. |
continuous_toGL : Continuous (toGL : SL n R → GL n R) := by
simp_rw [Units.continuous_iff, ← map_inv]
constructor <;> fun_prop | 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 | continuous_toGL | The natural map from `SL n A` to `GL n A` is continuous. |
isInducing_toGL : Topology.IsInducing (toGL : SL n R → GL n R) :=
.of_comp continuous_toGL Units.continuous_val (Topology.IsInducing.induced _) | 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_toGL | The natural map from `SL n A` to `GL n A` is inducing, i.e. the topology on
`SL n A` is the pullback of the topology from `GL n A`. |
isEmbedding_toGL : Topology.IsEmbedding (toGL : SL n R → GL n R) :=
⟨isInducing_toGL, toGL_injective⟩ | 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_toGL | The natural map from `SL n A` in `GL n A` is an embedding, i.e. it is an injection and
the topology on `SL n A` coincides with the subspace topology from `GL n A`. |
range_toGL {A : Type*} [CommRing A] :
Set.range (toGL : SL n A → GL n A) = GeneralLinearGroup.det ⁻¹' {1} := by
ext x
simpa [Units.ext_iff] using ⟨fun ⟨y, hy⟩ ↦ by simp [← hy], fun hx ↦ ⟨⟨x, hx⟩, rfl⟩⟩ | 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 | range_toGL | null |
isClosedEmbedding_toGL [T0Space R] : Topology.IsClosedEmbedding (toGL : SL n R → GL n R) :=
⟨isEmbedding_toGL, by simpa [range_toGL] using isClosed_singleton.preimage <| by fun_prop⟩ | 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_toGL | The natural inclusion of `SL n A` in `GL n A` is a closed embedding. |
isInducing_mapGL (h : Topology.IsInducing (algebraMap A B)) :
Topology.IsInducing (mapGL B : SL n A → GL n B) := by
refine isInducing_toGL.comp ?_
refine .of_comp ?_ continuous_induced_dom (h.matrix_map.comp (Topology.IsInducing.induced _))
rw [continuous_induced_rng]
exact continuous_subtype_val.matrix_map h.continuous | 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_mapGL | null |
isEmbedding_mapGL (h : Topology.IsEmbedding (algebraMap A B)) :
Topology.IsEmbedding (mapGL B : SL n A → GL n B) :=
haveI : FaithfulSMul A B := (faithfulSMul_iff_algebraMap_injective _ _).mpr h.2
⟨isInducing_mapGL h.isInducing, mapGL_injective⟩ | 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_mapGL | null |
dist_eq (x y : ℕ) : dist x y = |(x : ℝ) - y| := rfl | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | dist_eq | null |
dist_coe_int (x y : ℕ) : dist (x : ℤ) (y : ℤ) = dist x y := rfl
@[norm_cast, simp] | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | dist_coe_int | null |
dist_cast_real (x y : ℕ) : dist (x : ℝ) y = dist x y := rfl | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | dist_cast_real | null |
pairwise_one_le_dist : Pairwise fun m n : ℕ => 1 ≤ dist m n := fun _ _ hne =>
Int.pairwise_one_le_dist <| mod_cast hne | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | pairwise_one_le_dist | null |
isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℕ → ℝ) :=
isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | isUniformEmbedding_coe_real | null |
isClosedEmbedding_coe_real : IsClosedEmbedding ((↑) : ℕ → ℝ) :=
isClosedEmbedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | isClosedEmbedding_coe_real | null |
preimage_ball (x : ℕ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r := rfl | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | preimage_ball | null |
preimage_closedBall (x : ℕ) (r : ℝ) : (↑) ⁻¹' closedBall (x : ℝ) r = closedBall x r := rfl | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | preimage_closedBall | null |
closedBall_eq_Icc (x : ℕ) (r : ℝ) : closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊ := by
rcases le_or_gt 0 r with (hr | hr)
· rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]
positivity
· rw [closedBall_eq_empty.2 hr, Icc_eq_empty_of_lt]
calc ⌊(x : ℝ) + r⌋₊ ≤ ⌊(x : ℝ)⌋₊ := floor_mono <| by linarith
_ < ⌈↑x - r⌉₊ := by
rw [floor_natCast, Nat.lt_ceil]
linarith | theorem | Topology | [
"Mathlib.Data.Nat.Lattice",
"Mathlib.Topology.Instances.Int"
] | Mathlib/Topology/Instances/Nat.lean | closedBall_eq_Icc | null |
dist_eq (x y : ℕ+) : dist x y = |(↑x : ℝ) - ↑y| := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/PNat.lean | dist_eq | null |
dist_coe (x y : ℕ+) : dist (↑x : ℕ) (↑y : ℕ) = dist x y := rfl | theorem | Topology | [
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/PNat.lean | dist_coe | null |
isUniformEmbedding_coe : IsUniformEmbedding ((↑) : ℕ+ → ℕ) := isUniformEmbedding_subtype_val | theorem | Topology | [
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/PNat.lean | isUniformEmbedding_coe | null |
dist_eq (x y : ℚ) : dist x y = |(x : ℝ) - y| := rfl
@[norm_cast, simp] | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | dist_eq | null |
dist_cast (x y : ℚ) : dist (x : ℝ) y = dist x y :=
rfl | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | dist_cast | null |
uniformContinuous_coe_real : UniformContinuous ((↑) : ℚ → ℝ) :=
uniformContinuous_comap | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | uniformContinuous_coe_real | null |
isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℚ → ℝ) :=
isUniformEmbedding_comap Rat.cast_injective | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | isUniformEmbedding_coe_real | null |
isDenseEmbedding_coe_real : IsDenseEmbedding ((↑) : ℚ → ℝ) :=
isUniformEmbedding_coe_real.isDenseEmbedding Rat.denseRange_cast | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | isDenseEmbedding_coe_real | null |
isEmbedding_coe_real : IsEmbedding ((↑) : ℚ → ℝ) :=
isDenseEmbedding_coe_real.isEmbedding | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | isEmbedding_coe_real | null |
continuous_coe_real : Continuous ((↑) : ℚ → ℝ) :=
uniformContinuous_coe_real.continuous | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | continuous_coe_real | null |
@[norm_cast, simp]
Nat.dist_cast_rat (x y : ℕ) : dist (x : ℚ) y = dist x y := by
rw [← Nat.dist_cast_real, ← Rat.dist_cast]; congr | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Nat.dist_cast_rat | null |
Nat.isUniformEmbedding_coe_rat : IsUniformEmbedding ((↑) : ℕ → ℚ) :=
isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Nat.pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Nat.isUniformEmbedding_coe_rat | null |
Nat.isClosedEmbedding_coe_rat : IsClosedEmbedding ((↑) : ℕ → ℚ) :=
isClosedEmbedding_of_pairwise_le_dist zero_lt_one <| by simpa using Nat.pairwise_one_le_dist
@[norm_cast, simp] | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Nat.isClosedEmbedding_coe_rat | null |
Int.dist_cast_rat (x y : ℤ) : dist (x : ℚ) y = dist x y := by
rw [← Int.dist_cast_real, ← Rat.dist_cast]; congr | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Int.dist_cast_rat | null |
Int.isUniformEmbedding_coe_rat : IsUniformEmbedding ((↑) : ℤ → ℚ) :=
isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Int.isUniformEmbedding_coe_rat | null |
Int.isClosedEmbedding_coe_rat : IsClosedEmbedding ((↑) : ℤ → ℚ) :=
isClosedEmbedding_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | Int.isClosedEmbedding_coe_rat | null |
uniformContinuous_add : UniformContinuous fun p : ℚ × ℚ => p.1 + p.2 :=
Rat.isUniformEmbedding_coe_real.isUniformInducing.uniformContinuous_iff.2 <| by
simp only [Function.comp_def, Rat.cast_add]
exact Real.uniformContinuous_add.comp
(Rat.uniformContinuous_coe_real.prodMap Rat.uniformContinuous_coe_real) | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | uniformContinuous_add | null |
uniformContinuous_neg : UniformContinuous (@Neg.neg ℚ _) :=
Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨_, ε0, fun _ _ h => by simpa only [abs_sub_comm, dist_eq, cast_neg, neg_sub_neg] using h⟩ | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | uniformContinuous_neg | null |
uniformContinuous_abs : UniformContinuous (abs : ℚ → ℚ) :=
Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨ε, ε0, fun _ _ h =>
lt_of_le_of_lt (by simpa [Rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _) h⟩ | theorem | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | uniformContinuous_abs | null |
@[simp ←, push_cast]
dist_eq (p q : ℚ≥0) : dist p q = dist (p : ℚ) (q : ℚ) := rfl
set_option linter.style.commandStart false in
@[simp ←, push_cast] | lemma | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | dist_eq | null |
nndist_eq (p q : ℚ≥0) : nndist p q = nndist (p : ℚ) (q : ℚ) := rfl | lemma | Topology | [
"Mathlib.Data.NNRat.Order",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Instances.Nat"
] | Mathlib/Topology/Instances/Rat.lean | nndist_eq | null |
interior_compact_eq_empty (hs : IsCompact s) : interior s = ∅ :=
isDenseEmbedding_coe_real.isDenseInducing.interior_compact_eq_empty dense_irrational hs | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | interior_compact_eq_empty | null |
dense_compl_compact (hs : IsCompact s) : Dense sᶜ :=
interior_eq_empty_iff_dense_compl.1 (interior_compact_eq_empty hs) | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | dense_compl_compact | null |
cocompact_inf_nhds_neBot : NeBot (cocompact ℚ ⊓ 𝓝 p) := by
refine (hasBasis_cocompact.inf (nhds_basis_opens _)).neBot_iff.2 ?_
rintro ⟨s, o⟩ ⟨hs, hpo, ho⟩; rw [inter_comm]
exact (dense_compl_compact hs).inter_open_nonempty _ ho ⟨p, hpo⟩ | instance | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | cocompact_inf_nhds_neBot | null |
not_countably_generated_cocompact : ¬IsCountablyGenerated (cocompact ℚ) := by
intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩) | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | not_countably_generated_cocompact | null |
not_countably_generated_nhds_infty_opc : ¬IsCountablyGenerated (𝓝 (∞ : ℚ∞)) := by
intro
have : IsCountablyGenerated (comap (OnePoint.some : ℚ → ℚ∞) (𝓝 ∞)) := by infer_instance
rw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this
exact not_countably_generated_cocompact this | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | not_countably_generated_nhds_infty_opc | null |
not_firstCountableTopology_opc : ¬FirstCountableTopology ℚ∞ := by
intro
exact not_countably_generated_nhds_infty_opc inferInstance | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | not_firstCountableTopology_opc | null |
not_secondCountableTopology_opc : ¬SecondCountableTopology ℚ∞ := by
intro
exact not_firstCountableTopology_opc inferInstance | theorem | Topology | [
"Mathlib.Topology.Instances.Irrational",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Metrizable.Uniformity"
] | Mathlib/Topology/Instances/RatLemmas.lean | not_secondCountableTopology_opc | null |
map_real_smul {G} [FunLike G E F] [AddMonoidHomClass G E F] (f : G) (hf : Continuous f)
(c : ℝ) (x : E) :
f (c • x) = c • f x :=
suffices (fun c : ℝ => f (c • x)) = fun c : ℝ => c • f x from congr_fun this c
Rat.isDenseEmbedding_coe_real.dense.equalizer (hf.comp <| continuous_id.smul continuous_const)
(continuous_id.smul continuous_const) (funext fun r => map_ratCast_smul f ℝ ℝ r x) | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Algebra.Module.Rat"
] | Mathlib/Topology/Instances/RealVectorSpace.lean | map_real_smul | A continuous additive map between two vector spaces over `ℝ` is `ℝ`-linear. |
toRealLinearMap (f : E →+ F) (hf : Continuous f) : E →L[ℝ] F :=
⟨{ toFun := f
map_add' := f.map_add
map_smul' := map_real_smul f hf }, hf⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Algebra.Module.Rat"
] | Mathlib/Topology/Instances/RealVectorSpace.lean | toRealLinearMap | Reinterpret a continuous additive homomorphism between two real vector spaces
as a continuous real-linear map. |
coe_toRealLinearMap (f : E →+ F) (hf : Continuous f) : ⇑(f.toRealLinearMap hf) = f :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Algebra.Module.Rat"
] | Mathlib/Topology/Instances/RealVectorSpace.lean | coe_toRealLinearMap | null |
AddEquiv.toRealLinearEquiv (e : E ≃+ F) (h₁ : Continuous e) (h₂ : Continuous e.symm) :
E ≃L[ℝ] F :=
{ e, e.toAddMonoidHom.toRealLinearMap h₁ with } | def | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Algebra.Module.Rat"
] | Mathlib/Topology/Instances/RealVectorSpace.lean | AddEquiv.toRealLinearEquiv | Reinterpret a continuous additive equivalence between two real vector spaces
as a continuous real-linear map. |
@[simps toEquiv]
noncomputable homeomorph (X : Type u) [TopologicalSpace X] [Small.{v} X] :
X ≃ₜ Shrink.{v} X where
__ := equivShrink X
continuous_toFun := continuous_coinduced_rng
continuous_invFun := by
convert continuous_induced_dom
simp only [Equiv.invFun_as_coe, Equiv.induced_symm]
rfl | def | Topology | [
"Mathlib.Logic.Small.Defs",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Homeomorph.Defs"
] | Mathlib/Topology/Instances/Shrink.lean | homeomorph | `equivShrink` as a homeomorphism. |
continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a := by
refine (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr ?_
rw [Filter.EventuallyEq, eventually_nhds_iff]
exact ⟨{ x | 0 < x }, fun x hx => (sign_pos hx).symm, isOpen_lt' 0, h⟩ | theorem | Topology | [
"Mathlib.Data.Sign.Defs",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Instances/Sign.lean | continuousAt_sign_of_pos | null |
continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by
refine (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr ?_
rw [Filter.EventuallyEq, eventually_nhds_iff]
exact ⟨{ x | x < 0 }, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩ | theorem | Topology | [
"Mathlib.Data.Sign.Defs",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Instances/Sign.lean | continuousAt_sign_of_neg | null |
continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a := by
rcases h.lt_or_gt with (h_neg | h_pos)
· exact continuousAt_sign_of_neg h_neg
· exact continuousAt_sign_of_pos h_pos | theorem | Topology | [
"Mathlib.Data.Sign.Defs",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Instances/Sign.lean | continuousAt_sign_of_ne_zero | null |
instTopologicalSpace : TopologicalSpace (tsze R M) :=
TopologicalSpace.induced fst ‹_› ⊓ TopologicalSpace.induced snd ‹_› | instance | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | instTopologicalSpace | null |
nhds_def (x : tsze R M) : 𝓝 x = 𝓝 x.fst ×ˢ 𝓝 x.snd := nhds_prod_eq | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | nhds_def | null |
nhds_inl [Zero M] (x : R) : 𝓝 (inl x : tsze R M) = 𝓝 x ×ˢ 𝓝 0 :=
nhds_def _ | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | nhds_inl | null |
nhds_inr [Zero R] (m : M) : 𝓝 (inr m : tsze R M) = 𝓝 0 ×ˢ 𝓝 m :=
nhds_def _
nonrec theorem continuous_fst : Continuous (fst : tsze R M → R) :=
continuous_fst
nonrec theorem continuous_snd : Continuous (snd : tsze R M → M) :=
continuous_snd | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | nhds_inr | null |
continuous_inl [Zero M] : Continuous (inl : R → tsze R M) :=
continuous_id.prodMk continuous_const | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | continuous_inl | null |
continuous_inr [Zero R] : Continuous (inr : M → tsze R M) :=
continuous_const.prodMk continuous_id | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | continuous_inr | null |
IsEmbedding.inl [Zero M] : IsEmbedding (inl : R → tsze R M) :=
.of_comp continuous_inl continuous_fst .id | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | IsEmbedding.inl | null |
IsEmbedding.inr [Zero R] : IsEmbedding (inr : M → tsze R M) :=
.of_comp continuous_inr continuous_snd .id
variable (R M) | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | IsEmbedding.inr | null |
@[simps]
fstCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : StrongDual R (tsze R M) :=
{ ContinuousLinearMap.fst R R M with toFun := fst } | def | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | fstCLM | `TrivSqZeroExt.fst` as a continuous linear map. |
@[simps]
sndCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : tsze R M →L[R] M :=
{ ContinuousLinearMap.snd R R M with
toFun := snd
cont := continuous_snd } | def | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | sndCLM | `TrivSqZeroExt.snd` as a continuous linear map. |
@[simps]
inlCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : R →L[R] tsze R M :=
{ ContinuousLinearMap.inl R R M with toFun := inl } | def | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | inlCLM | `TrivSqZeroExt.inl` as a continuous linear map. |
@[simps]
inrCLM [CommSemiring R] [AddCommMonoid M] [Module R M] : M →L[R] tsze R M :=
{ ContinuousLinearMap.inr R R M with toFun := inr }
variable {R M} | def | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | inrCLM | `TrivSqZeroExt.inr` as a continuous linear map. |
topologicalSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
[IsTopologicalSemiring R] [ContinuousAdd M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] :
IsTopologicalSemiring (tsze R M) := { } | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | topologicalSemiring | This is not an instance due to complaints by the `fails_quickly` linter. At any rate, we only
really care about the `IsTopologicalRing` instance below. |
hasSum_inl [AddCommMonoid R] [AddCommMonoid M] {f : α → R} {a : R} (h : HasSum f a) :
HasSum (fun x ↦ inl (f x)) (inl a : tsze R M) :=
h.map (⟨⟨inl, inl_zero _⟩, inl_add _⟩ : R →+ tsze R M) continuous_inl | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | hasSum_inl | null |
hasSum_inr [AddCommMonoid R] [AddCommMonoid M] {f : α → M} {a : M} (h : HasSum f a) :
HasSum (fun x ↦ inr (f x)) (inr a : tsze R M) :=
h.map (⟨⟨inr, inr_zero _⟩, inr_add _⟩ : M →+ tsze R M) continuous_inr | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | hasSum_inr | null |
hasSum_fst [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
(h : HasSum f a) : HasSum (fun x ↦ fst (f x)) (fst a) :=
h.map (⟨⟨fst, fst_zero⟩, fst_add⟩ : tsze R M →+ R) continuous_fst | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | hasSum_fst | null |
hasSum_snd [AddCommMonoid R] [AddCommMonoid M] {f : α → tsze R M} {a : tsze R M}
(h : HasSum f a) : HasSum (fun x ↦ snd (f x)) (snd a) :=
h.map (⟨⟨snd, snd_zero⟩, snd_add⟩ : tsze R M →+ M) continuous_snd | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | hasSum_snd | null |
instUniformSpace : UniformSpace (tsze R M) where
toTopologicalSpace := instTopologicalSpace
__ := instUniformSpaceProd | instance | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | instUniformSpace | null |
uniformity_def :
𝓤 (tsze R M) =
((𝓤 R).comap fun p => (p.1.fst, p.2.fst)) ⊓ ((𝓤 M).comap fun p => (p.1.snd, p.2.snd)) :=
rfl
nonrec theorem uniformContinuous_fst : UniformContinuous (fst : tsze R M → R) :=
uniformContinuous_fst
nonrec theorem uniformContinuous_snd : UniformContinuous (snd : tsze R M → M) :=
uniformContinuous_snd | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | uniformity_def | null |
uniformContinuous_inl [Zero M] : UniformContinuous (inl : R → tsze R M) :=
uniformContinuous_id.prodMk uniformContinuous_const | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | uniformContinuous_inl | null |
uniformContinuous_inr [Zero R] : UniformContinuous (inr : M → tsze R M) :=
uniformContinuous_const.prodMk uniformContinuous_id | theorem | Topology | [
"Mathlib.Algebra.TrivSqZeroExt",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Instances/TrivSqZeroExt.lean | uniformContinuous_inr | null |
tendsto_coe_cofinite : Tendsto ((↑) : ℤ → ℝ) cofinite (cocompact ℝ) := by
apply (castAddHom ℝ).tendsto_coe_cofinite_of_discrete cast_injective
rw [range_castAddHom]
infer_instance | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Metrizable.Basic"
] | Mathlib/Topology/Instances/ZMultiples.lean | tendsto_coe_cofinite | This is a special case of `NormedSpace.discreteTopology_zmultiples`. It exists only to simplify
dependencies. -/
instance {a : ℝ} : DiscreteTopology (AddSubgroup.zmultiples a) := by
rcases eq_or_ne a 0 with (rfl | ha)
· rw [AddSubgroup.zmultiples_zero_eq_bot]
exact Subsingleton.discreteTopology (α := (⊥ : Submodule ℤ ℝ))
rw [discreteTopology_iff_isOpen_singleton_zero, isOpen_induced_iff]
refine ⟨ball 0 |a|, isOpen_ball, ?_⟩
ext ⟨x, hx⟩
obtain ⟨k, rfl⟩ := AddSubgroup.mem_zmultiples_iff.mp hx
simp [ha, Real.dist_eq, abs_mul, (by norm_cast : |(k : ℝ)| < 1 ↔ |k| < 1)]
/-- Under the coercion from `ℤ` to `ℝ`, inverse images of compact sets are finite. |
tendsto_zmultiplesHom_cofinite {a : ℝ} (ha : a ≠ 0) :
Tendsto (zmultiplesHom ℝ a) cofinite (cocompact ℝ) := by
apply (zmultiplesHom ℝ a).tendsto_coe_cofinite_of_discrete <| smul_left_injective ℤ ha
rw [AddSubgroup.range_zmultiplesHom]
infer_instance | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Metrizable.Basic"
] | Mathlib/Topology/Instances/ZMultiples.lean | tendsto_zmultiplesHom_cofinite | For nonzero `a`, the "multiples of `a`" map `zmultiplesHom` from `ℤ` to `ℝ` is discrete, i.e.
inverse images of compact sets are finite. |
tendsto_zmultiples_subtype_cofinite (a : ℝ) :
Tendsto (zmultiples a).subtype cofinite (cocompact ℝ) :=
(zmultiples a).tendsto_coe_cofinite_of_discrete | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.Metrizable.Basic"
] | Mathlib/Topology/Instances/ZMultiples.lean | tendsto_zmultiples_subtype_cofinite | The subgroup "multiples of `a`" (`zmultiples a`) is a discrete subgroup of `ℝ`, i.e. its
intersection with compact sets is finite. |
@[to_additive (attr := simp)]
coe_one [One Y] : ⇑(1 : LocallyConstant X Y) = (1 : X → Y) :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | coe_one | null |
one_apply [One Y] (x : X) : (1 : LocallyConstant X Y) x = 1 :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | one_apply | null |
@[to_additive (attr := simp)]
coe_inv [Inv Y] (f : LocallyConstant X Y) : ⇑(f⁻¹ : LocallyConstant X Y) = (f : X → Y)⁻¹ :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | coe_inv | null |
inv_apply [Inv Y] (f : LocallyConstant X Y) (x : X) : f⁻¹ x = (f x)⁻¹ :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | inv_apply | null |
@[to_additive (attr := simp)]
coe_mul [Mul Y] (f g : LocallyConstant X Y) : ⇑(f * g) = f * g :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | coe_mul | null |
mul_apply [Mul Y] (f g : LocallyConstant X Y) (x : X) : (f * g) x = f x * g x :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | mul_apply | null |
@[to_additive (attr := simps) /-- `DFunLike.coe` as an `AddMonoidHom`. -/]
coeFnMonoidHom [MulOneClass Y] : LocallyConstant X Y →* X → Y where
toFun := DFunLike.coe
map_one' := rfl
map_mul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | coeFnMonoidHom | `DFunLike.coe` as a `MonoidHom`. |
@[to_additive (attr := simps) /-- The constant-function embedding, as an additive monoid hom. -/]
constMonoidHom [MulOneClass Y] : Y →* LocallyConstant X Y where
toFun := const X
map_one' := rfl
map_mul' _ _ := rfl | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.GroupWithZero.Indicator",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.Topology.LocallyConstant.Basic"
] | Mathlib/Topology/LocallyConstant/Algebra.lean | constMonoidHom | The constant-function embedding, as a multiplicative monoid hom. |
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