fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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protected skewSwap : (M × N) ≃ₗ[R] (N × M) where
toFun x := (-x.2, x.1)
invFun x := (x.2, -x.1)
map_add' _ _ := by
simp [add_comm]
map_smul' _ _ := by
simp
left_inv _ := by
simp
right_inv _ := by
simp
variable {R M N}
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewSwap | The map `(x, y) ↦ (-y, x)` as a linear equivalence. |
skewSwap_apply (x : M × N) : LinearEquiv.skewSwap R M N x = (-x.2, x.1) := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewSwap_apply | null |
skewSwap_symm_apply (x : N × M) : (LinearEquiv.skewSwap R M N).symm x = (x.2, -x.1) := rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewSwap_symm_apply | null |
@[simps apply]
prodProdProdComm : ((M × M₂) × M₃ × M₄) ≃ₗ[R] (M × M₃) × M₂ × M₄ :=
{ AddEquiv.prodProdProdComm M M₂ M₃ M₄ with
toFun := fun mnmn => ((mnmn.1.1, mnmn.2.1), (mnmn.1.2, mnmn.2.2))
invFun := fun mmnn => ((mmnn.1.1, mmnn.2.1), (mmnn.1.2, mmnn.2.2))
map_smul' := fun _c _mnmn => rfl }
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodProdProdComm | Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. |
prodProdProdComm_symm :
(prodProdProdComm R M M₂ M₃ M₄).symm = prodProdProdComm R M M₃ M₂ M₄ :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodProdProdComm_symm | null |
prodProdProdComm_toAddEquiv :
(prodProdProdComm R M M₂ M₃ M₄ : _ ≃+ _) = AddEquiv.prodProdProdComm M M₂ M₃ M₄ :=
rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodProdProdComm_toAddEquiv | null |
protected prodCongr : (M × M₃) ≃ₗ[R] M₂ × M₄ :=
{ e₁.toAddEquiv.prodCongr e₂.toAddEquiv with
map_smul' := fun c _x => Prod.ext (e₁.map_smulₛₗ c _) (e₂.map_smulₛₗ c _) }
@[deprecated (since := "2025-04-17")] alias prod := LinearEquiv.prodCongr | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodCongr | Product of linear equivalences; the maps come from `Equiv.prodCongr`. |
prodCongr_symm : (e₁.prodCongr e₂).symm = e₁.symm.prodCongr e₂.symm :=
rfl
@[deprecated (since := "2025-04-17")] alias prod_symm := prodCongr_symm
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodCongr_symm | null |
prodCongr_apply (p) : e₁.prodCongr e₂ p = (e₁ p.1, e₂ p.2) :=
rfl
@[deprecated (since := "2025-04-17")] alias prod_apply := prodCongr_apply
@[simp, norm_cast] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodCongr_apply | null |
coe_prodCongr :
(e₁.prodCongr e₂ : M × M₃ →ₗ[R] M₂ × M₄) = (e₁ : M →ₗ[R] M₂).prodMap (e₂ : M₃ →ₗ[R] M₄) :=
rfl
@[deprecated (since := "2025-04-17")] alias coe_prod := coe_prodCongr | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | coe_prodCongr | null |
protected skewProd (f : M →ₗ[R] M₄) : (M × M₃) ≃ₗ[R] M₂ × M₄ :=
{ ((e₁ : M →ₗ[R] M₂).comp (LinearMap.fst R M M₃)).prod
((e₂ : M₃ →ₗ[R] M₄).comp (LinearMap.snd R M M₃) +
f.comp (LinearMap.fst R M M₃)) with
invFun := fun p : M₂ × M₄ => (e₁.symm p.1, e₂.symm (p.2 - f (e₁.symm p.1)))
left_inv := fun p => by simp
right_inv := fun p => by simp }
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewProd | Equivalence given by a block lower diagonal matrix. `e₁` and `e₂` are diagonal square blocks,
and `f` is a rectangular block below the diagonal. |
skewProd_apply (f : M →ₗ[R] M₄) (x) : e₁.skewProd e₂ f x = (e₁ x.1, e₂ x.2 + f x.1) :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewProd_apply | null |
skewProd_symm_apply (f : M →ₗ[R] M₄) (x) :
(e₁.skewProd e₂ f).symm x = (e₁.symm x.1, e₂.symm (x.2 - f (e₁.symm x.1))) :=
rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | skewProd_symm_apply | null |
@[simps!]
uniqueProd : (M₂ × M) ≃ₗ[R] M :=
AddEquiv.uniqueProd.toLinearEquiv (by simp [AddEquiv.uniqueProd]) | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | uniqueProd | Multiplying by the trivial module from the left does not change the structure.
This is the `LinearEquiv` version of `AddEquiv.uniqueProd`. |
coe_uniqueProd :
(uniqueProd (R := R) (M := M) (M₂ := M₂) : (M₂ × M) ≃ M) = Equiv.uniqueProd M M₂ := rfl | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | coe_uniqueProd | null |
@[simps!]
prodUnique : (M × M₂) ≃ₗ[R] M :=
AddEquiv.prodUnique.toLinearEquiv (by simp [AddEquiv.prodUnique]) | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | prodUnique | Multiplying by the trivial module from the right does not change the structure.
This is the `LinearEquiv` version of `AddEquiv.prodUnique`. |
coe_prodUnique :
(prodUnique (R := R) (M := M) (M₂ := M₂) : (M × M₂) ≃ M) = Equiv.prodUnique M M₂ := rfl | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | coe_prodUnique | null |
range_prod_eq {f : M →ₗ[R] M₂} {g : M →ₗ[R] M₃} (h : ker f ⊔ ker g = ⊤) :
range (prod f g) = (range f).prod (range g) := by
refine le_antisymm (f.range_prod_le g) ?_
simp only [SetLike.le_def, prod_apply, mem_range, mem_prod, exists_imp, and_imp,
Prod.forall, Pi.prod]
rintro _ _ x rfl y rfl
simp only [Prod.mk_inj, ← sub_mem_ker_iff (f := f)]
have : y - x ∈ ker f ⊔ ker g := by simp only [h, mem_top]
rcases mem_sup.1 this with ⟨x', hx', y', hy', H⟩
refine ⟨x' + x, ?_, ?_⟩
· rwa [add_sub_cancel_right]
· simp [← eq_sub_iff_add_eq.1 H, map_add, mem_ker.mp hy'] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | range_prod_eq | If the union of the kernels `ker f` and `ker g` spans the domain, then the range of
`Prod f g` is equal to the product of `range f` and `range g`. |
graph : Submodule R (M × M₂) where
carrier := { p | p.2 = f p.1 }
add_mem' (ha : _ = _) (hb : _ = _) := by
change _ + _ = f (_ + _)
rw [map_add, ha, hb]
zero_mem' := Eq.symm (map_zero f)
smul_mem' c x (hx : _ = _) := by
change _ • _ = f (_ • _)
rw [map_smul, hx]
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | graph | Graph of a linear map. |
mem_graph_iff (x : M × M₂) : x ∈ f.graph ↔ x.2 = f x.1 :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | mem_graph_iff | null |
graph_eq_ker_coprod : g.graph = ker ((-g).coprod LinearMap.id) := by
ext x
change _ = _ ↔ -g x.1 + x.2 = _
rw [add_comm, add_neg_eq_zero] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | graph_eq_ker_coprod | null |
graph_eq_range_prod : f.graph = range (LinearMap.id.prod f) := by
ext x
exact ⟨fun hx => ⟨x.1, Prod.ext rfl hx.symm⟩, fun ⟨u, hu⟩ => hu ▸ rfl⟩ | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | graph_eq_range_prod | null |
LinearMap.exists_range_eq_graph {f : G →ₛₗ[σ] H × I} (hf₁ : Surjective (Prod.fst ∘ f))
(hf : ∀ g₁ g₂, (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2) :
∃ f' : H →ₗ[S] I, LinearMap.range f = LinearMap.graph f' := by
obtain ⟨f', hf'⟩ :=
AddMonoidHom.exists_mrange_eq_mgraph (G := G) (H := H) (I := I) (f := f) hf₁ hf
simp only [SetLike.ext_iff, AddMonoidHom.mem_mrange, AddMonoidHom.coe_coe,
AddMonoidHom.mem_mgraph] at hf'
use
{ toFun := f'.toFun
map_add' := f'.map_add'
map_smul' := by
intro s h
simp only [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe, RingHom.id_apply]
refine (hf' (s • h, _)).mp ?_
rw [← Prod.smul_mk, ← LinearMap.mem_range]
apply Submodule.smul_mem
rw [LinearMap.mem_range, hf'] }
ext x
simpa only [mem_range, Eq.comm, ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe, mem_graph_iff,
coe_mk, AddHom.coe_mk, AddMonoidHom.coe_coe, Set.mem_range] using hf' x | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | LinearMap.exists_range_eq_graph | **Vertical line test** for linear maps.
Let `f : G → H × I` be a linear (or semilinear) map to a product. Assume that `f` is surjective on
the first factor and that the image of `f` intersects every "vertical line" `{(h, i) | i : I}` at
most once. Then the image of `f` is the graph of some linear map `f' : H → I`. |
Submodule.exists_eq_graph {G : Submodule S (H × I)} (hf₁ : Bijective (Prod.fst ∘ G.subtype)) :
∃ f : H →ₗ[S] I, G = LinearMap.graph f := by
simpa only [range_subtype] using LinearMap.exists_range_eq_graph hf₁.surjective
(fun a b h ↦ congr_arg (Prod.snd ∘ G.subtype) (hf₁.injective h)) | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | Submodule.exists_eq_graph | **Vertical line test** for linear maps.
Let `G ≤ H × I` be a submodule of a product of modules. Assume that `G` maps bijectively to the
first factor. Then `G` is the graph of some linear map `f : H →ₗ[R] I`. |
LinearMap.exists_linearEquiv_eq_graph {f : G →ₛₗ[σ] H × I} (hf₁ : Surjective (Prod.fst ∘ f))
(hf₂ : Surjective (Prod.snd ∘ f)) (hf : ∀ g₁ g₂, (f g₁).1 = (f g₂).1 ↔ (f g₁).2 = (f g₂).2) :
∃ e : H ≃ₗ[S] I, range f = e.toLinearMap.graph := by
obtain ⟨e₁, he₁⟩ := f.exists_range_eq_graph hf₁ fun _ _ ↦ (hf _ _).1
obtain ⟨e₂, he₂⟩ := ((LinearEquiv.prodComm _ _ _).toLinearMap.comp f).exists_range_eq_graph
(by simpa) <| by simp [hf]
have he₁₂ h i : e₁ h = i ↔ e₂ i = h := by
simp only [SetLike.ext_iff, LinearMap.mem_graph_iff] at he₁ he₂
rw [Eq.comm, ← he₁ (h, i), Eq.comm, ← he₂ (i, h)]
simp only [mem_range, coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
LinearEquiv.prodComm_apply, Prod.swap_eq_iff_eq_swap, Prod.swap_prod_mk]
exact ⟨
{ toFun := e₁
map_smul' := e₁.map_smul'
map_add' := e₁.map_add'
invFun := e₂
left_inv := fun h ↦ by rw [← he₁₂]
right_inv := fun i ↦ by rw [he₁₂] }, he₁⟩ | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | LinearMap.exists_linearEquiv_eq_graph | **Line test** for module isomorphisms.
Let `f : G → H × I` be a linear (or semilinear) map to a product of modules. Assume that `f` is
surjective onto both factors and that the image of `f` intersects every "vertical line"
`{(h, i) | i : I}` and every "horizontal line" `{(h, i) | h : H}` at most once. Then the image of
`f` is the graph of some module isomorphism `f' : H ≃ I`. |
Submodule.exists_equiv_eq_graph {G : Submodule S (H × I)}
(hG₁ : Bijective (Prod.fst ∘ G.subtype)) (hG₂ : Bijective (Prod.snd ∘ G.subtype)) :
∃ e : H ≃ₗ[S] I, G = e.toLinearMap.graph := by
simpa only [range_subtype] using LinearMap.exists_linearEquiv_eq_graph
hG₁.surjective hG₂.surjective fun _ _ ↦ hG₁.injective.eq_iff.trans hG₂.injective.eq_iff.symm | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Algebra.Group.Graph",
"Mathlib.LinearAlgebra.Span.Basic"
] | Mathlib/LinearAlgebra/Prod.lean | Submodule.exists_equiv_eq_graph | **Goursat's lemma** for module isomorphisms.
Let `G ≤ H × I` be a submodule of a product of modules. Assume that the natural maps from `G` to
both factors are bijective. Then `G` is the graph of some module isomorphism `f : H ≃ I`. |
ker_id_sub_eq_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) :
ker (id - p.subtype.comp f) = p := by
ext x
simp only [comp_apply, mem_ker, subtype_apply, sub_apply, id_apply, sub_eq_zero]
exact ⟨fun h => h.symm ▸ Submodule.coe_mem _, fun hx => by rw [hf ⟨x, hx⟩, Subtype.coe_mk]⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ker_id_sub_eq_of_proj | null |
range_eq_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) : range f = ⊤ :=
range_eq_top.2 fun x => ⟨x, hf x⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | range_eq_of_proj | null |
isCompl_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) : IsCompl p (ker f) := by
constructor
· rw [disjoint_iff_inf_le]
rintro x ⟨hpx, hfx⟩
rw [SetLike.mem_coe, mem_ker, hf ⟨x, hpx⟩, mk_eq_zero] at hfx
simp only [hfx, zero_mem]
· rw [codisjoint_iff_le_sup]
intro x _
rw [mem_sup']
refine ⟨f x, ⟨x - f x, ?_⟩, add_sub_cancel _ _⟩
rw [mem_ker, LinearMap.map_sub, hf, sub_self] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isCompl_of_proj | null |
quotientEquivOfIsCompl (h : IsCompl p q) : (E ⧸ p) ≃ₗ[R] q :=
LinearEquiv.symm <|
LinearEquiv.ofBijective (p.mkQ.comp q.subtype)
⟨by rw [← ker_eq_bot, ker_comp, ker_mkQ, disjoint_iff_comap_eq_bot.1 h.symm.disjoint], by
rw [← range_eq_top, range_comp, range_subtype, map_mkQ_eq_top, h.sup_eq_top]⟩
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | quotientEquivOfIsCompl | If `q` is a complement of `p`, then `M/p ≃ q`. |
quotientEquivOfIsCompl_symm_apply (h : IsCompl p q) (x : q) :
(quotientEquivOfIsCompl p q h).symm x = (Quotient.mk x : E ⧸ p) := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | quotientEquivOfIsCompl_symm_apply | null |
quotientEquivOfIsCompl_apply_mk_coe (h : IsCompl p q) (x : q) :
quotientEquivOfIsCompl p q h (Quotient.mk x) = x :=
(quotientEquivOfIsCompl p q h).apply_symm_apply x
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | quotientEquivOfIsCompl_apply_mk_coe | null |
mk_quotientEquivOfIsCompl_apply (h : IsCompl p q) (x : E ⧸ p) :
(Quotient.mk (quotientEquivOfIsCompl p q h x) : E ⧸ p) = x :=
(quotientEquivOfIsCompl p q h).symm_apply_apply x | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | mk_quotientEquivOfIsCompl_apply | null |
prodEquivOfIsCompl (h : IsCompl p q) : (p × q) ≃ₗ[R] E := by
apply LinearEquiv.ofBijective (p.subtype.coprod q.subtype)
constructor
· rw [← ker_eq_bot, ker_coprod_of_disjoint_range, ker_subtype, ker_subtype, prod_bot]
rw [range_subtype, range_subtype]
exact h.1
· rw [← range_eq_top, ← sup_eq_range, h.sup_eq_top]
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodEquivOfIsCompl | If `q` is a complement of `p`, then `p × q` is isomorphic to `E`. |
coe_prodEquivOfIsCompl (h : IsCompl p q) :
(prodEquivOfIsCompl p q h : p × q →ₗ[R] E) = p.subtype.coprod q.subtype := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_prodEquivOfIsCompl | null |
coe_prodEquivOfIsCompl' (h : IsCompl p q) (x : p × q) :
prodEquivOfIsCompl p q h x = x.1 + x.2 := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_prodEquivOfIsCompl' | null |
prodEquivOfIsCompl_symm_apply_left (h : IsCompl p q) (x : p) :
(prodEquivOfIsCompl p q h).symm x = (x, 0) :=
(prodEquivOfIsCompl p q h).symm_apply_eq.2 <| by simp
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodEquivOfIsCompl_symm_apply_left | null |
prodEquivOfIsCompl_symm_apply_right (h : IsCompl p q) (x : q) :
(prodEquivOfIsCompl p q h).symm x = (0, x) :=
(prodEquivOfIsCompl p q h).symm_apply_eq.2 <| by simp
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodEquivOfIsCompl_symm_apply_right | null |
prodEquivOfIsCompl_symm_apply_fst_eq_zero (h : IsCompl p q) {x : E} :
((prodEquivOfIsCompl p q h).symm x).1 = 0 ↔ x ∈ q := by
conv_rhs => rw [← (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_left _ (Submodule.coe_mem _),
mem_right_iff_eq_zero_of_disjoint h.disjoint]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodEquivOfIsCompl_symm_apply_fst_eq_zero | null |
prodEquivOfIsCompl_symm_apply_snd_eq_zero (h : IsCompl p q) {x : E} :
((prodEquivOfIsCompl p q h).symm x).2 = 0 ↔ x ∈ p := by
conv_rhs => rw [← (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_right _ (Submodule.coe_mem _),
mem_left_iff_eq_zero_of_disjoint h.disjoint]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodEquivOfIsCompl_symm_apply_snd_eq_zero | null |
prodComm_trans_prodEquivOfIsCompl (h : IsCompl p q) :
LinearEquiv.prodComm R q p ≪≫ₗ prodEquivOfIsCompl p q h = prodEquivOfIsCompl q p h.symm :=
LinearEquiv.ext fun _ => add_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | prodComm_trans_prodEquivOfIsCompl | null |
linearProjOfIsCompl (h : IsCompl p q) : E →ₗ[R] p :=
LinearMap.fst R p q ∘ₗ ↑(prodEquivOfIsCompl p q h).symm
variable {p q} | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl | Projection to a submodule along a complement. It is the unique
linear map `f : E → p` such that `f x = x` for `x ∈ p` and `f x = 0` for `x ∈ q`.
See also `LinearMap.linearProjOfIsCompl`. |
noncomputable IsCompl.projection (hpq : IsCompl p q) :=
p.subtype ∘ₗ p.linearProjOfIsCompl q hpq
open Submodule | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection | The linear projection onto a subspace along its complement
as a map from the full space to itself, as opposed to `Submodule.linearProjOfIsCompl`,
which maps into the subtype.
This version is important as it satisfies `IsIdempotentElem`. |
IsCompl.projection_apply (hpq : IsCompl p q) (x : E) :
hpq.projection x = p.linearProjOfIsCompl q hpq x :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_apply | null |
coe_linearProjOfIsCompl_apply (hpq : IsCompl p q) (x : E) :
(p.linearProjOfIsCompl q hpq x : E) = hpq.projection x :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_linearProjOfIsCompl_apply | null |
IsCompl.projection_apply_mem (hpq : IsCompl p q) (x : E) :
hpq.projection x ∈ p :=
SetLike.coe_mem _
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_apply_mem | null |
linearProjOfIsCompl_apply_left (h : IsCompl p q) (x : p) :
linearProjOfIsCompl p q h x = x := by simp [linearProjOfIsCompl]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_apply_left | null |
IsCompl.projection_apply_left (hpq : IsCompl p q) (x : p) :
hpq.projection x = x := by simp [projection]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_apply_left | null |
linearProjOfIsCompl_range (h : IsCompl p q) : range (linearProjOfIsCompl p q h) = ⊤ :=
range_eq_of_proj (linearProjOfIsCompl_apply_left h)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_range | null |
IsCompl.projection_range (hpq : IsCompl p q) : range hpq.projection = p := by
simp [projection, range_comp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_range | null |
linearProjOfIsCompl_surjective (h : IsCompl p q) :
Function.Surjective (linearProjOfIsCompl p q h) :=
range_eq_top.mp (linearProjOfIsCompl_range h)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_surjective | null |
linearProjOfIsCompl_apply_eq_zero_iff (h : IsCompl p q) {x : E} :
linearProjOfIsCompl p q h x = 0 ↔ x ∈ q := by simp [linearProjOfIsCompl]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_apply_eq_zero_iff | null |
IsCompl.projection_apply_eq_zero_iff (hpq : IsCompl p q) {x : E} :
hpq.projection x = 0 ↔ x ∈ q := by
simp [projection, -coe_linearProjOfIsCompl_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_apply_eq_zero_iff | null |
linearProjOfIsCompl_apply_right' (h : IsCompl p q) (x : E) (hx : x ∈ q) :
linearProjOfIsCompl p q h x = 0 :=
(linearProjOfIsCompl_apply_eq_zero_iff h).2 hx
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_apply_right' | null |
linearProjOfIsCompl_apply_right (h : IsCompl p q) (x : q) :
linearProjOfIsCompl p q h x = 0 :=
linearProjOfIsCompl_apply_right' h x x.2
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_apply_right | null |
linearProjOfIsCompl_ker (h : IsCompl p q) : ker (linearProjOfIsCompl p q h) = q :=
ext fun _ => mem_ker.trans (linearProjOfIsCompl_apply_eq_zero_iff h)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_ker | null |
IsCompl.projection_ker (hpq : IsCompl p q) :
ker hpq.projection = q := by
simp [projection, ker_comp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_ker | null |
linearProjOfIsCompl_comp_subtype (h : IsCompl p q) :
(linearProjOfIsCompl p q h).comp p.subtype = LinearMap.id :=
LinearMap.ext <| linearProjOfIsCompl_apply_left h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_comp_subtype | null |
linearProjOfIsCompl_isCompl_projection (h : IsCompl p q) (x : E) :
linearProjOfIsCompl p q h (h.projection x) = linearProjOfIsCompl p q h x :=
linearProjOfIsCompl_apply_left h _
@[deprecated (since := "2025-07-29")] alias linearProjOfIsCompl_idempotent :=
linearProjOfIsCompl_isCompl_projection | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_isCompl_projection | null |
@[simp]
IsCompl.projection_isIdempotentElem (hpq : IsCompl p q) :
IsIdempotentElem hpq.projection :=
LinearMap.ext fun _ ↦ congr($(linearProjOfIsCompl_isCompl_projection hpq _)) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_isIdempotentElem | The linear projection onto a subspace along its complement is an idempotent. |
existsUnique_add_of_isCompl_prod (hc : IsCompl p q) (x : E) :
∃! u : p × q, (u.fst : E) + u.snd = x :=
(prodEquivOfIsCompl _ _ hc).toEquiv.bijective.existsUnique _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | existsUnique_add_of_isCompl_prod | null |
existsUnique_add_of_isCompl (hc : IsCompl p q) (x : E) :
∃ (u : p) (v : q), (u : E) + v = x ∧ ∀ (r : p) (s : q), (r : E) + s = x → r = u ∧ s = v :=
let ⟨u, hu₁, hu₂⟩ := existsUnique_add_of_isCompl_prod hc x
⟨u.1, u.2, hu₁, fun r s hrs => Prod.eq_iff_fst_eq_snd_eq.1 (hu₂ ⟨r, s⟩ hrs)⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | existsUnique_add_of_isCompl | null |
IsCompl.projection_add_projection_eq_self (hpq : IsCompl p q) (x : E) :
hpq.projection x + hpq.symm.projection x = x := by
dsimp only [IsCompl.projection, linearProjOfIsCompl]
rw [← prodComm_trans_prodEquivOfIsCompl _ _ hpq]
exact (prodEquivOfIsCompl _ _ hpq).apply_symm_apply x
@[deprecated (since := "2025-07-29")] alias linearProjOfIsCompl_add_linearProjOfIsCompl_eq_self :=
IsCompl.projection_add_projection_eq_self
@[deprecated (since := "2025-07-11")] alias linear_proj_add_linearProjOfIsCompl_eq_self :=
linearProjOfIsCompl_add_linearProjOfIsCompl_eq_self | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_add_projection_eq_self | null |
IsCompl.projection_eq_self_sub_projection (hpq : IsCompl p q) (x : E) :
hpq.symm.projection x = x - hpq.projection x := by
rw [eq_sub_iff_add_eq, projection_add_projection_eq_self]
@[deprecated (since := "2025-07-29")] alias linearProjOfIsCompl_eq_self_sub_linearProjOfIsCompl :=
IsCompl.projection_eq_self_sub_projection | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_eq_self_sub_projection | null |
@[simp] IsCompl.projection_eq_self_iff (hpq : IsCompl p q) (x : E) :
hpq.projection x = x ↔ x ∈ p := by
rw [eq_comm, ← sub_eq_zero, ← projection_eq_self_sub_projection, projection_apply_eq_zero_iff]
@[deprecated (since := "2025-07-29")] alias linearProjOfIsCompl_eq_self_iff :=
IsCompl.projection_eq_self_iff | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsCompl.projection_eq_self_iff | The projection to `p` along `q` of `x` equals `x` if and only if `x ∈ p`. |
linearProjOfIsCompl {F : Type*} [AddCommGroup F] [Module R F]
(i : F →ₗ[R] E) (hi : Function.Injective i)
(h : IsCompl (LinearMap.range i) q) : E →ₗ[R] F :=
(LinearEquiv.ofInjective i hi).symm ∘ₗ (LinearMap.range i).linearProjOfIsCompl q h
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl | Projection to the image of an injection along a complement.
This has an advantage over `Submodule.linearProjOfIsCompl` in that it allows the user better
definitional control over the type. |
linearProjOfIsCompl_apply_left {F : Type*} [AddCommGroup F] [Module R F]
(i : F →ₗ[R] E) (hi : Function.Injective i)
(h : IsCompl (LinearMap.range i) q) (x : F) :
linearProjOfIsCompl q i hi h (i x) = x := by
let ix : LinearMap.range i := ⟨i x, mem_range_self i x⟩
change linearProjOfIsCompl q i hi h ix = x
rw [linearProjOfIsCompl, coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
LinearEquiv.symm_apply_eq, Submodule.linearProjOfIsCompl_apply_left, Subtype.ext_iff,
LinearEquiv.ofInjective_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_apply_left | null |
ofIsCompl {p q : Submodule R E} (h : IsCompl p q) (φ : p →ₗ[R] F) (ψ : q →ₗ[R] F) : E →ₗ[R] F :=
LinearMap.coprod φ ψ ∘ₗ ↑(Submodule.prodEquivOfIsCompl _ _ h).symm
variable {p q}
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl | Given linear maps `φ` and `ψ` from complement submodules, `LinearMap.ofIsCompl` is
the induced linear map over the entire module. |
ofIsCompl_left_apply (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (u : p) :
ofIsCompl h φ ψ (u : E) = φ u := by simp [ofIsCompl]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_left_apply | null |
ofIsCompl_right_apply (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (v : q) :
ofIsCompl h φ ψ (v : E) = ψ v := by simp [ofIsCompl] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_right_apply | null |
ofIsCompl_eq (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} {χ : E →ₗ[R] F}
(hφ : ∀ u, φ u = χ u) (hψ : ∀ u, ψ u = χ u) : ofIsCompl h φ ψ = χ := by
ext x
obtain ⟨_, _, rfl, _⟩ := existsUnique_add_of_isCompl h x
simp [ofIsCompl, hφ, hψ] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_eq | null |
ofIsCompl_eq' (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} {χ : E →ₗ[R] F}
(hφ : φ = χ.comp p.subtype) (hψ : ψ = χ.comp q.subtype) : ofIsCompl h φ ψ = χ :=
ofIsCompl_eq h (fun _ => hφ.symm ▸ rfl) fun _ => hψ.symm ▸ rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_eq' | null |
ofIsCompl_eq_add (hpq : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} :
ofIsCompl hpq φ ψ = (φ ∘ₗ p.linearProjOfIsCompl q hpq)
+ (ψ ∘ₗ q.linearProjOfIsCompl p hpq.symm) := by
ext x
obtain ⟨a, b, rfl, _⟩ := existsUnique_add_of_isCompl hpq x
simp
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_eq_add | null |
ofIsCompl_zero (h : IsCompl p q) : (ofIsCompl h 0 0 : E →ₗ[R] F) = 0 :=
ofIsCompl_eq _ (fun _ => rfl) fun _ => rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_zero | null |
ofIsCompl_add (h : IsCompl p q) {φ₁ φ₂ : p →ₗ[R] F} {ψ₁ ψ₂ : q →ₗ[R] F} :
ofIsCompl h (φ₁ + φ₂) (ψ₁ + ψ₂) = ofIsCompl h φ₁ ψ₁ + ofIsCompl h φ₂ ψ₂ :=
ofIsCompl_eq _ (by simp) (by simp)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_add | null |
ofIsCompl_smul {R : Type*} [CommRing R] {E : Type*} [AddCommGroup E] [Module R E]
{F : Type*} [AddCommGroup F] [Module R F] {p q : Submodule R E} (h : IsCompl p q)
{φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (c : R) : ofIsCompl h (c • φ) (c • ψ) = c • ofIsCompl h φ ψ :=
ofIsCompl_eq _ (by simp) (by simp) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_smul | null |
surjective_comp_linearProjOfIsCompl (h : IsCompl p q) [Module R M] :
Function.Surjective (comp (p.linearProjOfIsCompl q h) : (M →ₗ[R] E) → _) :=
fun f ↦ ⟨p.subtype ∘ₗ f, by ext; simp⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | surjective_comp_linearProjOfIsCompl | null |
surjective_comp_subtype_of_isComplemented (h : IsComplemented p) [Module R M] :
Function.Surjective fun f : E →ₗ[R] M ↦ f ∘ₗ p.subtype :=
have ⟨q, h⟩ := h; fun f ↦ ⟨f ∘ₗ p.linearProjOfIsCompl q h, by ext; simp⟩
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | surjective_comp_subtype_of_isComplemented | null |
range_ofIsCompl (hpq : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} :
range (ofIsCompl hpq φ ψ) = range φ ⊔ range ψ := by
rw [ofIsCompl_eq_add]
apply le_antisymm
· apply range_add_le _ _ |>.trans
gcongr
all_goals exact range_comp_le_range ..
· apply sup_le
all_goals rintro - ⟨x, rfl⟩; exact ⟨x, by simp⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | range_ofIsCompl | null |
ofIsCompl_subtype_zero_eq (hpq : IsCompl p q) :
ofIsCompl hpq p.subtype 0 = hpq.projection := by
simp [ofIsCompl_eq_add, IsCompl.projection] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_subtype_zero_eq | null |
ofIsCompl_symm (hpq : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} :
ofIsCompl hpq.symm ψ φ = ofIsCompl hpq φ ψ := by
simp [ofIsCompl_eq_add, add_comm] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsCompl_symm | null |
ofIsComplProd {p q : Submodule R₁ E} (h : IsCompl p q) :
(p →ₗ[R₁] F) × (q →ₗ[R₁] F) →ₗ[R₁] E →ₗ[R₁] F where
toFun φ := ofIsCompl h φ.1 φ.2
map_add' := by intro φ ψ; rw [Prod.snd_add, Prod.fst_add, ofIsCompl_add]
map_smul' := by intro c φ; simp [Prod.smul_snd, Prod.smul_fst, ofIsCompl_smul]
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsComplProd | The linear map from `(p →ₗ[R₁] F) × (q →ₗ[R₁] F)` to `E →ₗ[R₁] F`. |
ofIsComplProd_apply {p q : Submodule R₁ E} (h : IsCompl p q)
(φ : (p →ₗ[R₁] F) × (q →ₗ[R₁] F)) : ofIsComplProd h φ = ofIsCompl h φ.1 φ.2 :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsComplProd_apply | null |
ofIsComplProdEquiv {p q : Submodule R₁ E} (h : IsCompl p q) :
((p →ₗ[R₁] F) × (q →ₗ[R₁] F)) ≃ₗ[R₁] E →ₗ[R₁] F :=
{ ofIsComplProd h with
invFun := fun φ => ⟨φ.domRestrict p, φ.domRestrict q⟩
left_inv := fun φ ↦ by
ext x
· exact ofIsCompl_left_apply h x
· exact ofIsCompl_right_apply h x
right_inv := fun φ ↦ by
ext x
obtain ⟨a, b, hab, _⟩ := existsUnique_add_of_isCompl h x
rw [← hab]; simp } | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | ofIsComplProdEquiv | The natural linear equivalence between `(p →ₗ[R₁] F) × (q →ₗ[R₁] F)` and `E →ₗ[R₁] F`. |
@[simp]
linearProjOfIsCompl_of_proj (f : E →ₗ[R] p) (hf : ∀ x : p, f x = x) :
p.linearProjOfIsCompl (ker f) (isCompl_of_proj hf) = f := by
ext x
have : x ∈ p ⊔ (ker f) := by simp only [(isCompl_of_proj hf).sup_eq_top, mem_top]
rcases mem_sup'.1 this with ⟨x, y, rfl⟩
simp [hf] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | linearProjOfIsCompl_of_proj | null |
equivProdOfSurjectiveOfIsCompl (f : E →ₗ[R] F) (g : E →ₗ[R] G) (hf : range f = ⊤)
(hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) : E ≃ₗ[R] F × G :=
LinearEquiv.ofBijective (f.prod g)
⟨by simp [← ker_eq_bot, hfg.inf_eq_bot], by
rw [← range_eq_top]
simp [range_prod_eq hfg.sup_eq_top, *]⟩
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | equivProdOfSurjectiveOfIsCompl | If `f : E →ₗ[R] F` and `g : E →ₗ[R] G` are two surjective linear maps and
their kernels are complement of each other, then `x ↦ (f x, g x)` defines
a linear equivalence `E ≃ₗ[R] F × G`. |
coe_equivProdOfSurjectiveOfIsCompl {f : E →ₗ[R] F} {g : E →ₗ[R] G} (hf : range f = ⊤)
(hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) :
(equivProdOfSurjectiveOfIsCompl f g hf hg hfg : E →ₗ[R] F × G) = f.prod g := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_equivProdOfSurjectiveOfIsCompl | null |
equivProdOfSurjectiveOfIsCompl_apply {f : E →ₗ[R] F} {g : E →ₗ[R] G} (hf : range f = ⊤)
(hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) (x : E) :
equivProdOfSurjectiveOfIsCompl f g hf hg hfg x = (f x, g x) := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | equivProdOfSurjectiveOfIsCompl_apply | null |
isComplEquivProj : { q // IsCompl p q } ≃ { f : E →ₗ[R] p // ∀ x : p, f x = x } where
toFun q := ⟨linearProjOfIsCompl p q q.2, linearProjOfIsCompl_apply_left q.2⟩
invFun f := ⟨ker (f : E →ₗ[R] p), isCompl_of_proj f.2⟩
left_inv := fun ⟨q, hq⟩ => by simp only [linearProjOfIsCompl_ker]
right_inv := fun ⟨f, hf⟩ => Subtype.eq <| f.linearProjOfIsCompl_of_proj hf
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isComplEquivProj | Equivalence between submodules `q` such that `IsCompl p q` and linear maps `f : E →ₗ[R] p`
such that `∀ x : p, f x = x`. |
coe_isComplEquivProj_apply (q : { q // IsCompl p q }) :
(p.isComplEquivProj q : E →ₗ[R] p) = linearProjOfIsCompl p q q.2 := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_isComplEquivProj_apply | null |
coe_isComplEquivProj_symm_apply (f : { f : E →ₗ[R] p // ∀ x : p, f x = x }) :
(p.isComplEquivProj.symm f : Submodule R E) = ker (f : E →ₗ[R] p) := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | coe_isComplEquivProj_symm_apply | null |
@[simps] isIdempotentElemEquiv :
{ f : Module.End R E // IsIdempotentElem f ∧ range f = p } ≃
{ f : E →ₗ[R] p // ∀ x : p, f x = x } where
toFun f := ⟨f.1.codRestrict _ fun x ↦ by simp_rw [← f.2.2]; exact mem_range_self f.1 x,
fun ⟨x, hx⟩ ↦ Subtype.ext <| by
obtain ⟨x, rfl⟩ := f.2.2.symm ▸ hx
exact DFunLike.congr_fun f.2.1 x⟩
invFun f := ⟨p.subtype ∘ₗ f.1, LinearMap.ext fun x ↦ by simp [f.2], le_antisymm
((range_comp_le_range _ _).trans_eq p.range_subtype)
fun x hx ↦ ⟨x, Subtype.ext_iff.1 <| f.2 ⟨x, hx⟩⟩⟩ | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isIdempotentElemEquiv | The idempotent endomorphisms of a module with range equal to a submodule are in 1-1
correspondence with linear maps to the submodule that restrict to the identity on the submodule. |
IsProj {F : Type*} [FunLike F M M] (f : F) : Prop where
map_mem : ∀ x, f x ∈ m
map_id : ∀ x ∈ m, f x = x | structure | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | IsProj | A linear endomorphism of a module `E` is a projection onto a submodule `p` if it sends every element
of `E` to `p` and fixes every element of `p`.
The definition allow more generally any `FunLike` type and not just linear maps, so that it can be
used for example with `ContinuousLinearMap` or `Matrix`. |
isProj_range_iff_isIdempotentElem (f : M →ₗ[S] M) :
IsProj (range f) f ↔ IsIdempotentElem f := by
refine ⟨fun ⟨h1, h2⟩ => ?_, fun hf =>
⟨fun x => mem_range_self f x, fun x ⟨y, hy⟩ => by rw [← hy, ← Module.End.mul_apply, hf.eq]⟩⟩
ext x
exact h2 (f x) (h1 x)
alias ⟨_, IsIdempotentElem.isProj_range⟩ := isProj_range_iff_isIdempotentElem | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isProj_range_iff_isIdempotentElem | null |
isProj_iff_isIdempotentElem (f : M →ₗ[S] M) :
(∃ p : Submodule S M, IsProj p f) ↔ IsIdempotentElem f := by
refine ⟨fun ⟨p, hp⟩ => ?_, fun h => ⟨_, IsIdempotentElem.isProj_range _ h⟩⟩
ext x
exact hp.map_id (f x) (hp.map_mem x) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isProj_iff_isIdempotentElem | null |
isIdempotentElem {f : M →ₗ[S] M} (h : IsProj m f) : IsIdempotentElem f :=
f.isProj_iff_isIdempotentElem.mp ⟨m, h⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | isIdempotentElem | null |
mem_iff_map_id {f : M →ₗ[S] M} (hf : IsProj m f) {x : M} :
x ∈ m ↔ f x = x :=
⟨hf.map_id x, fun h ↦ h ▸ hf.map_mem x⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | mem_iff_map_id | null |
codRestrict {f : M →ₗ[S] M} (h : IsProj m f) : M →ₗ[S] m :=
f.codRestrict m h.map_mem
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | codRestrict | Restriction of the codomain of a projection of onto a subspace `p` to `p` instead of the whole
space. |
codRestrict_apply {f : M →ₗ[S] M} (h : IsProj m f) (x : M) : ↑(h.codRestrict x) = f x :=
f.codRestrict_apply m x
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | codRestrict_apply | null |
codRestrict_apply_cod {f : M →ₗ[S] M} (h : IsProj m f) (x : m) : h.codRestrict x = x := by
ext
rw [codRestrict_apply]
exact h.map_id x x.2 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.Algebra.Ring.Idempotent"
] | Mathlib/LinearAlgebra/Projection.lean | codRestrict_apply_cod | null |
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