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 ⌀ |
|---|---|---|---|---|---|---|
smulCommClass [SMulCommClass S₂ T₂ M₂] : SMulCommClass S₂ T₂ (M₁ →SL[σ₁₂] M₂) :=
⟨fun a b f => ext fun x => smul_comm a b (f x)⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulCommClass | null |
zero : Zero (M₁ →SL[σ₁₂] M₂) :=
⟨⟨0, continuous_zero⟩⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | zero | The continuous map that is constantly zero. |
inhabited : Inhabited (M₁ →SL[σ₁₂] M₂) :=
⟨0⟩
@[simp] | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | inhabited | null |
default_def : (default : M₁ →SL[σ₁₂] M₂) = 0 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | default_def | null |
zero_apply (x : M₁) : (0 : M₁ →SL[σ₁₂] M₂) x = 0 :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | zero_apply | null |
coe_zero : ((0 : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂) = 0 :=
rfl
/- no simp attribute on the next line as simp does not always simplify `0 x` to `0`
when `0` is the zero function, while it does for the zero continuous linear map,
and this is the most important property we care about. -/
@[norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_zero | null |
coe_zero' : ⇑(0 : M₁ →SL[σ₁₂] M₂) = 0 :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_zero' | null |
toContinuousAddMonoidHom_zero :
((0 : M₁ →SL[σ₁₂] M₂) : ContinuousAddMonoidHom M₁ M₂) = 0 := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_zero | null |
uniqueOfLeft [Subsingleton M₁] : Unique (M₁ →SL[σ₁₂] M₂) :=
coe_injective.unique | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | uniqueOfLeft | null |
uniqueOfRight [Subsingleton M₂] : Unique (M₁ →SL[σ₁₂] M₂) :=
coe_injective.unique | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | uniqueOfRight | null |
exists_ne_zero {f : M₁ →SL[σ₁₂] M₂} (hf : f ≠ 0) : ∃ x, f x ≠ 0 := by
by_contra! h
exact hf (ContinuousLinearMap.ext h) | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | exists_ne_zero | null |
id : M₁ →L[R₁] M₁ :=
⟨LinearMap.id, continuous_id⟩ | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | id | the identity map as a continuous linear map. |
one : One (M₁ →L[R₁] M₁) :=
⟨id R₁ M₁⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | one | null |
one_def : (1 : M₁ →L[R₁] M₁) = id R₁ M₁ :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | one_def | null |
id_apply (x : M₁) : id R₁ M₁ x = x :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | id_apply | null |
coe_id : (id R₁ M₁ : M₁ →ₗ[R₁] M₁) = LinearMap.id :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_id | null |
coe_id' : ⇑(id R₁ M₁) = _root_.id :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_id' | null |
toContinuousAddMonoidHom_id :
(id R₁ M₁ : ContinuousAddMonoidHom M₁ M₁) = .id _ := rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_id | null |
coe_eq_id {f : M₁ →L[R₁] M₁} : (f : M₁ →ₗ[R₁] M₁) = LinearMap.id ↔ f = id _ _ := by
rw [← coe_id, coe_inj]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_eq_id | null |
one_apply (x : M₁) : (1 : M₁ →L[R₁] M₁) x = x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | one_apply | null |
add : Add (M₁ →SL[σ₁₂] M₂) :=
⟨fun f g => ⟨f + g, f.2.add g.2⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | add | null |
add_apply (f g : M₁ →SL[σ₁₂] M₂) (x : M₁) : (f + g) x = f x + g x :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | add_apply | null |
coe_add (f g : M₁ →SL[σ₁₂] M₂) : (↑(f + g) : M₁ →ₛₗ[σ₁₂] M₂) = f + g :=
rfl
@[norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_add | null |
coe_add' (f g : M₁ →SL[σ₁₂] M₂) : ⇑(f + g) = f + g :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_add' | null |
toContinuousAddMonoidHom_add (f g : M₁ →SL[σ₁₂] M₂) :
↑(f + g) = (f + g : ContinuousAddMonoidHom M₁ M₂) := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_add | null |
addCommMonoid : AddCommMonoid (M₁ →SL[σ₁₂] M₂) where
zero_add := by
intros
ext
apply_rules [zero_add, add_assoc, add_zero, neg_add_cancel, add_comm]
add_zero := by
intros
ext
apply_rules [zero_add, add_assoc, add_zero, neg_add_cancel, add_comm]
add_comm := by
intros
ext
apply_rules [zero_add, add_assoc, add_zero, neg_add_cancel, add_comm]
add_assoc := by
intros
ext
apply_rules [zero_add, add_assoc, add_zero, neg_add_cancel, add_comm]
nsmul := (· • ·)
nsmul_zero f := by
ext
simp
nsmul_succ n f := by
ext
simp [add_smul]
@[simp, norm_cast] | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | addCommMonoid | null |
coe_sum {ι : Type*} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :
↑(∑ d ∈ t, f d) = (∑ d ∈ t, f d : M₁ →ₛₗ[σ₁₂] M₂) :=
map_sum (AddMonoidHom.mk ⟨((↑) : (M₁ →SL[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₂] M₂), rfl⟩ fun _ _ => rfl) _ _
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_sum | null |
coe_sum' {ι : Type*} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :
⇑(∑ d ∈ t, f d) = ∑ d ∈ t, ⇑(f d) := by simp only [← coe_coe, coe_sum, LinearMap.coeFn_sum] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_sum' | null |
sum_apply {ι : Type*} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) (b : M₁) :
(∑ d ∈ t, f d) b = ∑ d ∈ t, f d b := by simp only [coe_sum', Finset.sum_apply] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | sum_apply | null |
comp (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : M₁ →SL[σ₁₃] M₃ :=
⟨(g : M₂ →ₛₗ[σ₂₃] M₃).comp (f : M₁ →ₛₗ[σ₁₂] M₂), g.2.comp f.2⟩
@[inherit_doc comp]
infixr:80 " ∘L " =>
@ContinuousLinearMap.comp _ _ _ _ _ _ (RingHom.id _) (RingHom.id _) (RingHom.id _) _ _ _ _ _ _ _ _
_ _ _ _ RingHomCompTriple.ids
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp | Composition of bounded linear maps. |
coe_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(h.comp f : M₁ →ₛₗ[σ₁₃] M₃) = (h : M₂ →ₛₗ[σ₂₃] M₃).comp (f : M₁ →ₛₗ[σ₁₂] M₂) :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_comp | null |
coe_comp' (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ⇑(h.comp f) = h ∘ f :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_comp' | null |
toContinuousAddMonoidHom_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(↑(h.comp f) : ContinuousAddMonoidHom M₁ M₃) = (h : ContinuousAddMonoidHom M₂ M₃).comp f := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_comp | null |
comp_apply (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (g.comp f) x = g (f x) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_apply | null |
comp_id (f : M₁ →SL[σ₁₂] M₂) : f.comp (id R₁ M₁) = f :=
ext fun _x => rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_id | null |
id_comp (f : M₁ →SL[σ₁₂] M₂) : (id R₂ M₂).comp f = f :=
ext fun _x => rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | id_comp | null |
leftInverse_of_comp {f : E →L[R] F} {g : F →L[R] E}
(hinv : g.comp f = ContinuousLinearMap.id R E) : Function.LeftInverse g f := by
simpa [← Function.rightInverse_iff_comp] using congr(⇑$hinv) | lemma | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | leftInverse_of_comp | `g ∘ f = id` as `ContinuousLinearMap`s implies `g ∘ f = id` as functions. |
rightInverse_of_comp {f : E →L[R] F} {g : F →L[R] E}
(hinv : f.comp g = ContinuousLinearMap.id R F) : Function.RightInverse g f :=
leftInverse_of_comp hinv | lemma | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | rightInverse_of_comp | `f ∘ g = id` as `ContinuousLinearMap`s implies `f ∘ g = id` as functions. |
@[simp]
comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁₂] M₂) = 0 := by
ext
simp
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_zero | null |
zero_comp (f : M₁ →SL[σ₁₂] M₂) : (0 : M₂ →SL[σ₂₃] M₃).comp f = 0 := by
ext
simp
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | zero_comp | null |
comp_add [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃)
(f₁ f₂ : M₁ →SL[σ₁₂] M₂) : g.comp (f₁ + f₂) = g.comp f₁ + g.comp f₂ := by
ext
simp
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_add | null |
add_comp [ContinuousAdd M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(g₁ + g₂).comp f = g₁.comp f + g₂.comp f := by
ext
simp | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | add_comp | null |
comp_finset_sum {ι : Type*} {s : Finset ι}
[ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃)
(f : ι → M₁ →SL[σ₁₂] M₂) : g.comp (∑ i ∈ s, f i) = ∑ i ∈ s, g.comp (f i) := by
ext
simp | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_finset_sum | null |
finset_sum_comp {ι : Type*} {s : Finset ι}
[ContinuousAdd M₃] (g : ι → M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂) : (∑ i ∈ s, g i).comp f = ∑ i ∈ s, (g i).comp f := by
ext
simp only [coe_comp', coe_sum', Function.comp_apply, Finset.sum_apply] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | finset_sum_comp | null |
comp_assoc {R₄ : Type*} [Semiring R₄] [Module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄}
{σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]
[RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(h.comp g).comp f = h.comp (g.comp f) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_assoc | null |
instMul : Mul (M₁ →L[R₁] M₁) :=
⟨comp⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | instMul | null |
mul_def (f g : M₁ →L[R₁] M₁) : f * g = f.comp g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | mul_def | null |
coe_mul (f g : M₁ →L[R₁] M₁) : ⇑(f * g) = f ∘ g :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_mul | null |
mul_apply (f g : M₁ →L[R₁] M₁) (x : M₁) : (f * g) x = f (g x) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | mul_apply | null |
monoidWithZero : MonoidWithZero (M₁ →L[R₁] M₁) where
mul_zero f := ext fun _ => map_zero f
zero_mul _ := ext fun _ => rfl
mul_one _ := ext fun _ => rfl
one_mul _ := ext fun _ => rfl
mul_assoc _ _ _ := ext fun _ => rfl | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | monoidWithZero | null |
coe_pow (f : M₁ →L[R₁] M₁) (n : ℕ) : ⇑(f ^ n) = f^[n] :=
hom_coe_pow _ rfl (fun _ _ ↦ rfl) _ _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_pow | null |
instNatCast [ContinuousAdd M₁] : NatCast (M₁ →L[R₁] M₁) where
natCast n := n • (1 : M₁ →L[R₁] M₁) | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | instNatCast | null |
semiring [ContinuousAdd M₁] : Semiring (M₁ →L[R₁] M₁) where
__ := ContinuousLinearMap.monoidWithZero
__ := ContinuousLinearMap.addCommMonoid
left_distrib f g h := ext fun x => map_add f (g x) (h x)
right_distrib _ _ _ := ext fun _ => LinearMap.add_apply _ _ _
toNatCast := instNatCast
natCast_zero := zero_smul ℕ (1 : M₁ →L[R₁] M₁)
natCast_succ n := AddMonoid.nsmul_succ n (1 : M₁ →L[R₁] M₁) | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | semiring | null |
@[simps]
toLinearMapRingHom [ContinuousAdd M₁] : (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁ where
toFun := toLinearMap
map_zero' := rfl
map_one' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toLinearMapRingHom | `ContinuousLinearMap.toLinearMap` as a `RingHom`. |
natCast_apply [ContinuousAdd M₁] (n : ℕ) (m : M₁) : (↑n : M₁ →L[R₁] M₁) m = n • m :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | natCast_apply | null |
ofNat_apply [ContinuousAdd M₁] (n : ℕ) [n.AtLeastTwo] (m : M₁) :
(ofNat(n) : M₁ →L[R₁] M₁) m = OfNat.ofNat n • m :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ofNat_apply | null |
applyModule : Module (M₁ →L[R₁] M₁) M₁ :=
Module.compHom _ toLinearMapRingHom
@[simp] | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | applyModule | The tautological action by `M₁ →L[R₁] M₁` on `M`.
This generalizes `Function.End.applyMulAction`. |
protected smul_def (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smul_def | null |
applyFaithfulSMul : FaithfulSMul (M₁ →L[R₁] M₁) M₁ :=
⟨fun {_ _} => ContinuousLinearMap.ext⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | applyFaithfulSMul | `ContinuousLinearMap.applyModule` is faithful. |
applySMulCommClass : SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁ where
smul_comm r e m := (e.map_smul r m).symm | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | applySMulCommClass | null |
applySMulCommClass' : SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁ where
smul_comm := ContinuousLinearMap.map_smul | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | applySMulCommClass' | null |
continuousConstSMul_apply : ContinuousConstSMul (M₁ →L[R₁] M₁) M₁ :=
⟨ContinuousLinearMap.continuous⟩ | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | continuousConstSMul_apply | null |
isClosed_ker [T1Space M₂] [FunLike F M₁ M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂]
(f : F) :
IsClosed (ker f : Set M₁) :=
continuous_iff_isClosed.1 (map_continuous f) _ isClosed_singleton | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | isClosed_ker | null |
isComplete_ker {M' : Type*} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M']
[Module R₁ M'] [T1Space M₂] [FunLike F M' M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂]
(f : F) :
IsComplete (ker f : Set M') :=
(isClosed_ker f).isComplete | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | isComplete_ker | null |
completeSpace_ker {M' : Type*} [UniformSpace M'] [CompleteSpace M']
[AddCommMonoid M'] [Module R₁ M'] [T1Space M₂]
[FunLike F M' M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂]
(f : F) : CompleteSpace (ker f) :=
(isComplete_ker f).completeSpace_coe | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | completeSpace_ker | null |
completeSpace_eqLocus {M' : Type*} [UniformSpace M'] [CompleteSpace M']
[AddCommMonoid M'] [Module R₁ M'] [T2Space M₂]
[FunLike F M' M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂]
(f g : F) : CompleteSpace (LinearMap.eqLocus f g) :=
IsClosed.completeSpace_coe (hs := isClosed_eq (map_continuous f) (map_continuous g)) | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | completeSpace_eqLocus | null |
codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
M₁ →SL[σ₁₂] p where
cont := f.continuous.subtype_mk _
toLinearMap := (f : M₁ →ₛₗ[σ₁₂] M₂).codRestrict p h
@[norm_cast] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | codRestrict | Restrict codomain of a continuous linear map. |
coe_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
(f.codRestrict p h : M₁ →ₛₗ[σ₁₂] p) = (f : M₁ →ₛₗ[σ₁₂] M₂).codRestrict p h :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_codRestrict | null |
coe_codRestrict_apply (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) (x) :
(f.codRestrict p h x : M₂) = f x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_codRestrict_apply | null |
ker_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
ker (f.codRestrict p h) = ker f :=
(f : M₁ →ₛₗ[σ₁₂] M₂).ker_codRestrict p h | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | ker_codRestrict | null |
rangeRestrict [RingHomSurjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) :=
f.codRestrict (LinearMap.range f) (LinearMap.mem_range_self f)
@[simp] | abbrev | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | rangeRestrict | Restrict the codomain of a continuous linear map `f` to `f.range`. |
coe_rangeRestrict [RingHomSurjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) :
(f.rangeRestrict : M₁ →ₛₗ[σ₁₂] LinearMap.range f) = (f : M₁ →ₛₗ[σ₁₂] M₂).rangeRestrict :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_rangeRestrict | null |
_root_.Submodule.subtypeL (p : Submodule R₁ M₁) : p →L[R₁] M₁ where
cont := continuous_subtype_val
toLinearMap := p.subtype
@[simp, norm_cast] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.subtypeL | `Submodule.subtype` as a `ContinuousLinearMap`. |
_root_.Submodule.coe_subtypeL (p : Submodule R₁ M₁) :
(p.subtypeL : p →ₗ[R₁] M₁) = p.subtype :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.coe_subtypeL | null |
_root_.Submodule.coe_subtypeL' (p : Submodule R₁ M₁) : ⇑p.subtypeL = p.subtype :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.coe_subtypeL' | null |
_root_.Submodule.subtypeL_apply (p : Submodule R₁ M₁) (x : p) : p.subtypeL x = x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.subtypeL_apply | null |
_root_.Submodule.range_subtypeL (p : Submodule R₁ M₁) : range p.subtypeL = p :=
Submodule.range_subtype _
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.range_subtypeL | null |
_root_.Submodule.ker_subtypeL (p : Submodule R₁ M₁) : ker p.subtypeL = ⊥ :=
Submodule.ker_subtype _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | _root_.Submodule.ker_subtypeL | null |
@[simps coe]
smulRight (c : M₁ →L[R] S) (f : M₂) : M₁ →L[R] M₂ :=
{ c.toLinearMap.smulRight f with cont := c.2.smul continuous_const }
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulRight | The linear map `fun x => c x • f`. Associates to a scalar-valued linear map and an element of
`M₂` the `M₂`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M₂`).
See also `ContinuousLinearMap.smulRightₗ` and `ContinuousLinearMap.smulRightL`. |
smulRight_apply {c : M₁ →L[R] S} {f : M₂} {x : M₁} :
(smulRight c f : M₁ → M₂) x = c x • f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulRight_apply | null |
@[simp]
smulRight_one_one (c : R₁ →L[R₁] M₂) : smulRight (1 : R₁ →L[R₁] R₁) (c 1) = c := by
ext
simp [← ContinuousLinearMap.map_smul_of_tower]
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulRight_one_one | null |
smulRight_one_eq_iff {f f' : M₂} :
smulRight (1 : R₁ →L[R₁] R₁) f = smulRight (1 : R₁ →L[R₁] R₁) f' ↔ f = f' := by
simp only [ContinuousLinearMap.ext_ring_iff, smulRight_apply, one_apply, one_smul] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulRight_one_eq_iff | null |
smulRight_comp [ContinuousMul R₁] {x : M₂} {c : R₁} :
(smulRight (1 : R₁ →L[R₁] R₁) x).comp (smulRight (1 : R₁ →L[R₁] R₁) c) =
smulRight (1 : R₁ →L[R₁] R₁) (c • x) := by
ext
simp | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | smulRight_comp | null |
range_smulRight_apply {R : Type*} [DivisionSemiring R] [Module R M₁] [Module R M₂]
[TopologicalSpace R] [ContinuousSMul R M₂] {f : M₁ →L[R] R} (hf : f ≠ 0) (x : M₂) :
range (f.smulRight x) = Submodule.span R {x} :=
LinearMap.range_smulRight_apply (by simpa [coe_inj, ← coe_zero] using hf) x | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | range_smulRight_apply | null |
toSpanSingleton (x : M₁) : R₁ →L[R₁] M₁ where
toLinearMap := LinearMap.toSpanSingleton R₁ M₁ x
cont := continuous_id.smul continuous_const
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toSpanSingleton | Given an element `x` of a topological space `M` over a semiring `R`, the natural continuous
linear map from `R` to `M` by taking multiples of `x`. |
toSpanSingleton_apply (x : M₁) (r : R₁) : toSpanSingleton R₁ x r = r • x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toSpanSingleton_apply | null |
toSpanSingleton_one (x : M₁) : toSpanSingleton R₁ x 1 = x :=
one_smul _ _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toSpanSingleton_one | null |
toSpanSingleton_add [ContinuousAdd M₁] (x y : M₁) :
toSpanSingleton R₁ (x + y) = toSpanSingleton R₁ x + toSpanSingleton R₁ y :=
coe_inj.mp <| LinearMap.toSpanSingleton_add _ _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toSpanSingleton_add | null |
toSpanSingleton_smul {α} [Monoid α] [DistribMulAction α M₁] [ContinuousConstSMul α M₁]
[SMulCommClass R₁ α M₁] (c : α) (x : M₁) :
toSpanSingleton R₁ (c • x) = c • toSpanSingleton R₁ x :=
coe_inj.mp <| LinearMap.toSpanSingleton_smul _ _
@[deprecated (since := "2025-08-28")] alias toSpanSingleton_smul' := toSpanSingleton_smul | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toSpanSingleton_smul | null |
one_smulRight_eq_toSpanSingleton (x : M₁) :
(1 : R₁ →L[R₁] R₁).smulRight x = toSpanSingleton R₁ x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | one_smulRight_eq_toSpanSingleton | null |
toLinearMap_toSpanSingleton (x : M₁) :
(toSpanSingleton R₁ x).toLinearMap = LinearMap.toSpanSingleton R₁ M₁ x := rfl
variable {R₁} in | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toLinearMap_toSpanSingleton | null |
comp_toSpanSingleton (f : M₁ →L[R₁] M₂) (x : M₁) :
f ∘L toSpanSingleton R₁ x = toSpanSingleton R₁ (f x) :=
coe_inj.mp <| LinearMap.comp_toSpanSingleton _ _ | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | comp_toSpanSingleton | null |
protected map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = -f x := by
exact map_neg f x | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_neg | null |
protected map_sub (f : M →SL[σ₁₂] M₂) (x y : M) : f (x - y) = f x - f y := by
exact map_sub f x y
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | map_sub | null |
sub_apply' (f g : M →SL[σ₁₂] M₂) (x : M) : ((f : M →ₛₗ[σ₁₂] M₂) - g) x = f x - g x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | sub_apply' | null |
neg : Neg (M →SL[σ₁₂] M₂) :=
⟨fun f => ⟨-f, f.2.neg⟩⟩
@[simp] | instance | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | neg | null |
neg_apply (f : M →SL[σ₁₂] M₂) (x : M) : (-f) x = -f x :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | neg_apply | null |
coe_neg (f : M →SL[σ₁₂] M₂) : (↑(-f) : M →ₛₗ[σ₁₂] M₂) = -f :=
rfl
@[norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_neg | null |
coe_neg' (f : M →SL[σ₁₂] M₂) : ⇑(-f) = -f :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | coe_neg' | null |
toContinuousAddMonoidHom_neg (f : M →SL[σ₁₂] M₂) :
↑(-f) = -(f : ContinuousAddMonoidHom M M₂) := rfl | theorem | Topology | [
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/LinearMap.lean | toContinuousAddMonoidHom_neg | null |
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