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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map (H : M.LinearDisjoint N) {T : Type w} [Semiring T] [Algebra R T]
{F : Type*} [FunLike F S T] [AlgHomClass F R S T] (f : F) (hf : Function.Injective f) :
(M.map f).LinearDisjoint (N.map f) := by
rw [linearDisjoint_iff] at H ⊢
have : _ ∘ₗ
(TensorProduct.congr (M.equivMapOfInjective f hf) (N.equivMapOfInjective f hf)).toLinearMap
= _ := M.mulMap_map_comp_eq N f
replace H : Function.Injective ((f : S →ₗ[R] T) ∘ₗ mulMap M N) := hf.comp H
simpa only [← this, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.injective_comp] using H
variable (M N) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | map | Linear disjointness is preserved by injective algebra homomorphisms. |
of_basis_left' {ι : Type*} (m : Basis ι R M)
(H : Function.Injective (mulLeftMap N m)) : M.LinearDisjoint N := by
classical simp_rw [mulLeftMap_eq_mulMap_comp, ← Basis.coe_repr_symm,
← LinearEquiv.coe_rTensor, LinearEquiv.comp_coe, LinearMap.coe_comp,
LinearEquiv.coe_coe, EquivLike.injective_comp] at H
exact ⟨H⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_left' | If `{ m_i }` is an `R`-basis of `M`, which is also `N`-linearly independent
(in this result it is stated as `Submodule.mulLeftMap` is injective),
then `M` and `N` are linearly disjoint. |
of_basis_right' {ι : Type*} (n : Basis ι R N)
(H : Function.Injective (mulRightMap M n)) : M.LinearDisjoint N := by
classical simp_rw [mulRightMap_eq_mulMap_comp, ← Basis.coe_repr_symm,
← LinearEquiv.coe_lTensor, LinearEquiv.comp_coe, LinearMap.coe_comp,
LinearEquiv.coe_coe, EquivLike.injective_comp] at H
exact ⟨H⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_right' | If `{ n_i }` is an `R`-basis of `N`, which is also `M`-linearly independent
(in this result it is stated as `Submodule.mulRightMap` is injective),
then `M` and `N` are linearly disjoint. |
of_basis_mul' {κ ι : Type*} (m : Basis κ R M) (n : Basis ι R N)
(H : Function.Injective (Finsupp.linearCombination R fun i : κ × ι ↦ (m i.1 * n i.2 : S))) :
M.LinearDisjoint N := by
let i0 := (finsuppTensorFinsupp' R κ ι).symm
let i1 := TensorProduct.congr m.repr n.repr
let i := mulMap M N ∘ₗ (i0.trans i1.symm).toLinearMap
have : i = Finsupp.linearCombination R fun i : κ × ι ↦ (m i.1 * n i.2 : S) := by
ext x
simp [i, i0, i1, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
simp_rw [← this, i, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.injective_comp] at H
exact ⟨H⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_mul' | If `{ m_i }` is an `R`-basis of `M`, if `{ n_i }` is an `R`-basis of `N`,
such that the family `{ m_i * n_j }` in `S` is `R`-linearly independent
(in this result it is stated as the relevant `Finsupp.linearCombination` is injective),
then `M` and `N` are linearly disjoint. |
bot_left : (⊥ : Submodule R S).LinearDisjoint N :=
⟨Function.injective_of_subsingleton _⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | bot_left | The zero module is linearly disjoint with any other submodules. |
bot_right : M.LinearDisjoint (⊥ : Submodule R S) :=
⟨Function.injective_of_subsingleton _⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | bot_right | The zero module is linearly disjoint with any other submodules. |
one_left : (1 : Submodule R S).LinearDisjoint N := by
rw [linearDisjoint_iff, ← Algebra.toSubmodule_bot, mulMap_one_left_eq]
exact N.injective_subtype.comp N.lTensorOne.injective | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | one_left | The image of `R` in `S` is linearly disjoint with any other submodules. |
one_right : M.LinearDisjoint (1 : Submodule R S) := by
rw [linearDisjoint_iff, ← Algebra.toSubmodule_bot, mulMap_one_right_eq]
exact M.injective_subtype.comp M.rTensorOne.injective | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | one_right | The image of `R` in `S` is linearly disjoint with any other submodules. |
of_linearDisjoint_fg_left
(H : ∀ M' : Submodule R S, M' ≤ M → M'.FG → M'.LinearDisjoint N) :
M.LinearDisjoint N := (linearDisjoint_iff _ _).2 fun x y hxy ↦ by
obtain ⟨M', hM, hFG, h⟩ :=
TensorProduct.exists_finite_submodule_left_of_finite' {x, y} (Set.toFinite _)
rw [Module.Finite.iff_fg] at hFG
obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp)
obtain ⟨y', hy'⟩ := h (show y ∈ {x, y} by simp)
rw [← hx', ← hy']; congr
exact (H M' hM hFG).injective (by simp [← mulMap_comp_rTensor _ hM, hx', hy', hxy]) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_linearDisjoint_fg_left | If for any finitely generated submodules `M'` of `M`, `M'` and `N` are linearly disjoint,
then `M` and `N` themselves are linearly disjoint. |
of_linearDisjoint_fg_right
(H : ∀ N' : Submodule R S, N' ≤ N → N'.FG → M.LinearDisjoint N') :
M.LinearDisjoint N := (linearDisjoint_iff _ _).2 fun x y hxy ↦ by
obtain ⟨N', hN, hFG, h⟩ :=
TensorProduct.exists_finite_submodule_right_of_finite' {x, y} (Set.toFinite _)
rw [Module.Finite.iff_fg] at hFG
obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp)
obtain ⟨y', hy'⟩ := h (show y ∈ {x, y} by simp)
rw [← hx', ← hy']; congr
exact (H N' hN hFG).injective (by simp [← mulMap_comp_lTensor _ hN, hx', hy', hxy]) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_linearDisjoint_fg_right | If for any finitely generated submodules `N'` of `N`, `M` and `N'` are linearly disjoint,
then `M` and `N` themselves are linearly disjoint. |
of_linearDisjoint_fg
(H : ∀ (M' N' : Submodule R S), M' ≤ M → N' ≤ N → M'.FG → N'.FG → M'.LinearDisjoint N') :
M.LinearDisjoint N :=
of_linearDisjoint_fg_left _ _ fun _ hM hM' ↦
of_linearDisjoint_fg_right _ _ fun _ hN hN' ↦ H _ _ hM hN hM' hN' | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_linearDisjoint_fg | If for any finitely generated submodules `M'` and `N'` of `M` and `N`, respectively,
`M'` and `N'` are linearly disjoint, then `M` and `N` themselves are linearly disjoint. |
LinearDisjoint.symm (H : M.LinearDisjoint N) : N.LinearDisjoint M :=
H.symm_of_commute fun _ _ ↦ mul_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | LinearDisjoint.symm | Linear disjointness is symmetric in a commutative ring. |
linearDisjoint_comm : M.LinearDisjoint N ↔ N.LinearDisjoint M :=
⟨LinearDisjoint.symm, LinearDisjoint.symm⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearDisjoint_comm | Linear disjointness is symmetric in a commutative ring. |
linearIndependent_left_of_flat (H : M.LinearDisjoint N) [Module.Flat R N]
{ι : Type*} {m : ι → M} (hm : LinearIndependent R m) : LinearMap.ker (mulLeftMap N m) = ⊥ := by
refine LinearMap.ker_eq_bot_of_injective ?_
classical simp_rw [mulLeftMap_eq_mulMap_comp, LinearMap.coe_comp, LinearEquiv.coe_coe,
← Function.comp_assoc, EquivLike.injective_comp]
rw [LinearIndependent] at hm
exact H.injective.comp (Module.Flat.rTensor_preserves_injective_linearMap (M := N) _ hm) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearIndependent_left_of_flat | If `M` and `N` are linearly disjoint, if `N` is a flat `R`-module, then for any family of
`R`-linearly independent elements `{ m_i }` of `M`, they are also `N`-linearly independent,
in the sense that the `R`-linear map from `ι →₀ N` to `S` which maps `{ n_i }`
to the sum of `m_i * n_i` (`Submodule.mulLeftMap N m`) has trivial kernel. |
of_basis_left {ι : Type*} (m : Basis ι R M)
(H : LinearMap.ker (mulLeftMap N m) = ⊥) : M.LinearDisjoint N := by
letI : AddCommGroup (ι →₀ N) := Finsupp.instAddCommGroup
exact of_basis_left' M N m (LinearMap.ker_eq_bot.1 H)
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_left | If `{ m_i }` is an `R`-basis of `M`, which is also `N`-linearly independent,
then `M` and `N` are linearly disjoint. |
linearIndependent_right_of_flat (H : M.LinearDisjoint N) [Module.Flat R M]
{ι : Type*} {n : ι → N} (hn : LinearIndependent R n) : LinearMap.ker (mulRightMap M n) = ⊥ := by
refine LinearMap.ker_eq_bot_of_injective ?_
classical simp_rw [mulRightMap_eq_mulMap_comp, LinearMap.coe_comp, LinearEquiv.coe_coe,
← Function.comp_assoc, EquivLike.injective_comp]
rw [LinearIndependent] at hn
exact H.injective.comp (Module.Flat.lTensor_preserves_injective_linearMap (M := M) _ hn) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearIndependent_right_of_flat | If `M` and `N` are linearly disjoint, if `M` is a flat `R`-module, then for any family of
`R`-linearly independent elements `{ n_i }` of `N`, they are also `M`-linearly independent,
in the sense that the `R`-linear map from `ι →₀ M` to `S` which maps `{ m_i }`
to the sum of `m_i * n_i` (`Submodule.mulRightMap M n`) has trivial kernel. |
of_basis_right {ι : Type*} (n : Basis ι R N)
(H : LinearMap.ker (mulRightMap M n) = ⊥) : M.LinearDisjoint N := by
letI : AddCommGroup (ι →₀ M) := Finsupp.instAddCommGroup
exact of_basis_right' M N n (LinearMap.ker_eq_bot.1 H)
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_right | If `{ n_i }` is an `R`-basis of `N`, which is also `M`-linearly independent,
then `M` and `N` are linearly disjoint. |
linearIndependent_mul_of_flat_left (H : M.LinearDisjoint N) [Module.Flat R M]
{κ ι : Type*} {m : κ → M} {n : ι → N} (hm : LinearIndependent R m)
(hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 := by
rw [LinearIndependent] at hm hn ⊢
let i0 := (finsuppTensorFinsupp' R κ ι).symm
let i1 := LinearMap.rTensor (ι →₀ R) (Finsupp.linearCombination R m)
let i2 := LinearMap.lTensor M (Finsupp.linearCombination R n)
let i := mulMap M N ∘ₗ i2 ∘ₗ i1 ∘ₗ i0.toLinearMap
have h1 : Function.Injective i1 := Module.Flat.rTensor_preserves_injective_linearMap _ hm
have h2 : Function.Injective i2 := Module.Flat.lTensor_preserves_injective_linearMap _ hn
have h : Function.Injective i := H.injective.comp h2 |>.comp h1 |>.comp i0.injective
have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
ext x
simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
rwa [this] at h
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearIndependent_mul_of_flat_left | If `M` and `N` are linearly disjoint, if `M` is flat, then for any family of
`R`-linearly independent elements `{ m_i }` of `M`, and any family of
`R`-linearly independent elements `{ n_j }` of `N`, the family `{ m_i * n_j }` in `S` is
also `R`-linearly independent. |
linearIndependent_mul_of_flat_right (H : M.LinearDisjoint N) [Module.Flat R N]
{κ ι : Type*} {m : κ → M} {n : ι → N} (hm : LinearIndependent R m)
(hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 := by
rw [LinearIndependent] at hm hn ⊢
let i0 := (finsuppTensorFinsupp' R κ ι).symm
let i1 := LinearMap.lTensor (κ →₀ R) (Finsupp.linearCombination R n)
let i2 := LinearMap.rTensor N (Finsupp.linearCombination R m)
let i := mulMap M N ∘ₗ i2 ∘ₗ i1 ∘ₗ i0.toLinearMap
have h1 : Function.Injective i1 := Module.Flat.lTensor_preserves_injective_linearMap _ hn
have h2 : Function.Injective i2 := Module.Flat.rTensor_preserves_injective_linearMap _ hm
have h : Function.Injective i := H.injective.comp h2 |>.comp h1 |>.comp i0.injective
have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
ext x
simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
rwa [this] at h
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearIndependent_mul_of_flat_right | If `M` and `N` are linearly disjoint, if `N` is flat, then for any family of
`R`-linearly independent elements `{ m_i }` of `M`, and any family of
`R`-linearly independent elements `{ n_j }` of `N`, the family `{ m_i * n_j }` in `S` is
also `R`-linearly independent. |
linearIndependent_mul_of_flat (H : M.LinearDisjoint N)
(hf : Module.Flat R M ∨ Module.Flat R N)
{κ ι : Type*} {m : κ → M} {n : ι → N} (hm : LinearIndependent R m)
(hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 := by
rcases hf with _ | _
· exact H.linearIndependent_mul_of_flat_left hm hn
· exact H.linearIndependent_mul_of_flat_right hm hn | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | linearIndependent_mul_of_flat | If `M` and `N` are linearly disjoint, if one of `M` and `N` is flat, then for any family of
`R`-linearly independent elements `{ m_i }` of `M`, and any family of
`R`-linearly independent elements `{ n_j }` of `N`, the family `{ m_i * n_j }` in `S` is
also `R`-linearly independent. |
of_basis_mul {κ ι : Type*} (m : Basis κ R M) (n : Basis ι R N)
(H : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1) : M.LinearDisjoint N := by
rw [LinearIndependent] at H
exact of_basis_mul' M N m n H
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_basis_mul | If `{ m_i }` is an `R`-basis of `M`, if `{ n_j }` is an `R`-basis of `N`,
such that the family `{ m_i * n_j }` in `S` is `R`-linearly independent,
then `M` and `N` are linearly disjoint. |
of_le_left_of_flat (H : M.LinearDisjoint N) {M' : Submodule R S}
(h : M' ≤ M) [Module.Flat R N] : M'.LinearDisjoint N := by
let i := mulMap M N ∘ₗ (inclusion h).rTensor N
have hi : Function.Injective i := H.injective.comp <|
Module.Flat.rTensor_preserves_injective_linearMap _ <| inclusion_injective h
have : i = mulMap M' N := by ext; simp [i]
exact ⟨this ▸ hi⟩
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_le_left_of_flat | If `M` and `N` are linearly disjoint, if `N` is flat, then for any submodule `M'` of `M`,
`M'` and `N` are also linearly disjoint. |
of_le_right_of_flat (H : M.LinearDisjoint N) {N' : Submodule R S}
(h : N' ≤ N) [Module.Flat R M] : M.LinearDisjoint N' := by
let i := mulMap M N ∘ₗ (inclusion h).lTensor M
have hi : Function.Injective i := H.injective.comp <|
Module.Flat.lTensor_preserves_injective_linearMap _ <| inclusion_injective h
have : i = mulMap M N' := by ext; simp [i]
exact ⟨this ▸ hi⟩
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_le_right_of_flat | If `M` and `N` are linearly disjoint, if `M` is flat, then for any submodule `N'` of `N`,
`M` and `N'` are also linearly disjoint. |
of_le_of_flat_right (H : M.LinearDisjoint N) {M' N' : Submodule R S}
(hm : M' ≤ M) (hn : N' ≤ N) [Module.Flat R N] [Module.Flat R M'] :
M'.LinearDisjoint N' := (H.of_le_left_of_flat hm).of_le_right_of_flat hn
variable {M N} in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_le_of_flat_right | If `M` and `N` are linearly disjoint, `M'` and `N'` are submodules of `M` and `N`,
respectively, such that `N` and `M'` are flat, then `M'` and `N'` are also linearly disjoint. |
of_le_of_flat_left (H : M.LinearDisjoint N) {M' N' : Submodule R S}
(hm : M' ≤ M) (hn : N' ≤ N) [Module.Flat R M] [Module.Flat R N'] :
M'.LinearDisjoint N' := (H.of_le_right_of_flat hn).of_le_left_of_flat hm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_le_of_flat_left | If `M` and `N` are linearly disjoint, `M'` and `N'` are submodules of `M` and `N`,
respectively, such that `M` and `N'` are flat, then `M'` and `N'` are also linearly disjoint. |
of_left_le_one_of_flat (h : M ≤ 1) [Module.Flat R N] :
M.LinearDisjoint N := (one_left N).of_le_left_of_flat h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_left_le_one_of_flat | If `N` is flat, `M` is contained in `i(R)`, where `i : R → S` is the structure map,
then `M` and `N` are linearly disjoint. |
of_right_le_one_of_flat (h : N ≤ 1) [Module.Flat R M] :
M.LinearDisjoint N := (one_right M).of_le_right_of_flat h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | of_right_le_one_of_flat | If `M` is flat, `N` is contained in `i(R)`, where `i : R → S` is the structure map,
then `M` and `N` are linearly disjoint. |
not_linearIndependent_pair_of_commute_of_flat_left [Module.Flat R M]
(a b : ↥(M ⊓ N)) (hc : Commute a.1 b.1) : ¬LinearIndependent R ![a, b] := fun h ↦ by
let n : Fin 2 → N := (inclusion inf_le_right) ∘ ![a, b]
have hn : LinearIndependent R n := h.map' _ (ker_inclusion _ _ _)
letI : AddCommGroup (Fin 2 →₀ M) := Finsupp.instAddCommGroup
let m : Fin 2 →₀ M := .single 0 ⟨b.1, b.2.1⟩ - .single 1 ⟨a.1, a.2.1⟩
have hm : mulRightMap M n m = 0 := by simp [m, n, show _ * _ = _ * _ from hc]
rw [← LinearMap.mem_ker, H.linearIndependent_right_of_flat hn, mem_bot] at hm
simp only [Fin.isValue, sub_eq_zero, Finsupp.single_eq_single_iff, zero_ne_one, Subtype.mk.injEq,
SetLike.coe_eq_coe, false_and, false_or, m] at hm
repeat rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hm
exact h.ne_zero 0 hm.2 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_commute_of_flat_left | If `M` and `N` are linearly disjoint, if `M` is flat, then any two commutative
elements of `↥(M ⊓ N)` are not `R`-linearly independent (namely, their span is not `R ^ 2`). |
not_linearIndependent_pair_of_commute_of_flat_right [Module.Flat R N]
(a b : ↥(M ⊓ N)) (hc : Commute a.1 b.1) : ¬LinearIndependent R ![a, b] := fun h ↦ by
let m : Fin 2 → M := (inclusion inf_le_left) ∘ ![a, b]
have hm : LinearIndependent R m := h.map' _ (ker_inclusion _ _ _)
letI : AddCommGroup (Fin 2 →₀ N) := Finsupp.instAddCommGroup
let n : Fin 2 →₀ N := .single 0 ⟨b.1, b.2.2⟩ - .single 1 ⟨a.1, a.2.2⟩
have hn : mulLeftMap N m n = 0 := by simp [m, n, show _ * _ = _ * _ from hc]
rw [← LinearMap.mem_ker, H.linearIndependent_left_of_flat hm, mem_bot] at hn
simp only [Fin.isValue, sub_eq_zero, Finsupp.single_eq_single_iff, zero_ne_one, Subtype.mk.injEq,
SetLike.coe_eq_coe, false_and, false_or, n] at hn
repeat rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hn
exact h.ne_zero 0 hn.2 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_commute_of_flat_right | If `M` and `N` are linearly disjoint, if `N` is flat, then any two commutative
elements of `↥(M ⊓ N)` are not `R`-linearly independent (namely, their span is not `R ^ 2`). |
not_linearIndependent_pair_of_commute_of_flat (hf : Module.Flat R M ∨ Module.Flat R N)
(a b : ↥(M ⊓ N)) (hc : Commute a.1 b.1) : ¬LinearIndependent R ![a, b] := by
rcases hf with _ | _
· exact H.not_linearIndependent_pair_of_commute_of_flat_left a b hc
· exact H.not_linearIndependent_pair_of_commute_of_flat_right a b hc | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_commute_of_flat | If `M` and `N` are linearly disjoint, if one of `M` and `N` is flat, then any two commutative
elements of `↥(M ⊓ N)` are not `R`-linearly independent (namely, their span is not `R ^ 2`). |
rank_inf_le_one_of_commute_of_flat (hf : Module.Flat R M ∨ Module.Flat R N)
(hc : ∀ (m n : ↥(M ⊓ N)), Commute m.1 n.1) : Module.rank R ↥(M ⊓ N) ≤ 1 := by
nontriviality R
refine _root_.rank_le fun s h ↦ ?_
by_contra hs
rw [not_le, ← Fintype.card_coe, Fintype.one_lt_card_iff_nontrivial] at hs
obtain ⟨a, b, hab⟩ := hs.exists_pair_ne
refine H.not_linearIndependent_pair_of_commute_of_flat hf a.1 b.1 (hc a.1 b.1) ?_
have := h.comp ![a, b] fun i j hij ↦ by
fin_cases i <;> fin_cases j
· rfl
· simp [hab] at hij
· simp [hab.symm] at hij
· rfl
convert this
ext i
fin_cases i <;> simp | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_commute_of_flat | If `M` and `N` are linearly disjoint, if one of `M` and `N` is flat,
if any two elements of `↥(M ⊓ N)` are commutative, then the rank of `↥(M ⊓ N)` is at most one. |
rank_inf_le_one_of_commute_of_flat_left [Module.Flat R M]
(hc : ∀ (m n : ↥(M ⊓ N)), Commute m.1 n.1) : Module.rank R ↥(M ⊓ N) ≤ 1 :=
H.rank_inf_le_one_of_commute_of_flat (Or.inl ‹_›) hc | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_commute_of_flat_left | If `M` and `N` are linearly disjoint, if `M` is flat,
if any two elements of `↥(M ⊓ N)` are commutative, then the rank of `↥(M ⊓ N)` is at most one. |
rank_inf_le_one_of_commute_of_flat_right [Module.Flat R N]
(hc : ∀ (m n : ↥(M ⊓ N)), Commute m.1 n.1) : Module.rank R ↥(M ⊓ N) ≤ 1 :=
H.rank_inf_le_one_of_commute_of_flat (Or.inr ‹_›) hc | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_commute_of_flat_right | If `M` and `N` are linearly disjoint, if `N` is flat,
if any two elements of `↥(M ⊓ N)` are commutative, then the rank of `↥(M ⊓ N)` is at most one. |
rank_le_one_of_commute_of_flat_of_self (H : M.LinearDisjoint M) [Module.Flat R M]
(hc : ∀ (m n : M), Commute m.1 n.1) : Module.rank R M ≤ 1 := by
rw [← inf_of_le_left (le_refl M)] at hc ⊢
exact H.rank_inf_le_one_of_commute_of_flat_left hc | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_le_one_of_commute_of_flat_of_self | If `M` and itself are linearly disjoint, if `M` is flat,
if any two elements of `M` are commutative, then the rank of `M` is at most one. |
not_linearIndependent_pair_of_flat_left [Module.Flat R M]
(a b : ↥(M ⊓ N)) : ¬LinearIndependent R ![a, b] :=
H.not_linearIndependent_pair_of_commute_of_flat_left a b (mul_comm _ _) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_flat_left | The `Submodule.LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_left`
for commutative rings. |
not_linearIndependent_pair_of_flat_right [Module.Flat R N]
(a b : ↥(M ⊓ N)) : ¬LinearIndependent R ![a, b] :=
H.not_linearIndependent_pair_of_commute_of_flat_right a b (mul_comm _ _) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_flat_right | The `Submodule.LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_right`
for commutative rings. |
not_linearIndependent_pair_of_flat (hf : Module.Flat R M ∨ Module.Flat R N)
(a b : ↥(M ⊓ N)) : ¬LinearIndependent R ![a, b] :=
H.not_linearIndependent_pair_of_commute_of_flat hf a b (mul_comm _ _) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | not_linearIndependent_pair_of_flat | The `Submodule.LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat`
for commutative rings. |
rank_inf_le_one_of_flat (hf : Module.Flat R M ∨ Module.Flat R N) :
Module.rank R ↥(M ⊓ N) ≤ 1 :=
H.rank_inf_le_one_of_commute_of_flat hf fun _ _ ↦ mul_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_flat | The `Submodule.LinearDisjoint.rank_inf_le_one_of_commute_of_flat`
for commutative rings. |
rank_inf_le_one_of_flat_left [Module.Flat R M] : Module.rank R ↥(M ⊓ N) ≤ 1 :=
H.rank_inf_le_one_of_commute_of_flat_left fun _ _ ↦ mul_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_flat_left | The `Submodule.LinearDisjoint.rank_inf_le_one_of_commute_of_flat_left`
for commutative rings. |
rank_inf_le_one_of_flat_right [Module.Flat R N] : Module.rank R ↥(M ⊓ N) ≤ 1 :=
H.rank_inf_le_one_of_commute_of_flat_right fun _ _ ↦ mul_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_inf_le_one_of_flat_right | The `Submodule.LinearDisjoint.rank_inf_le_one_of_commute_of_flat_right`
for commutative rings. |
rank_le_one_of_flat_of_self (H : M.LinearDisjoint M) [Module.Flat R M] :
Module.rank R M ≤ 1 :=
H.rank_le_one_of_commute_of_flat_of_self fun _ _ ↦ mul_comm _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Submodule",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.RingTheory.Flat.Basic"
] | Mathlib/LinearAlgebra/LinearDisjoint.lean | rank_le_one_of_flat_of_self | The `Submodule.LinearDisjoint.rank_le_one_of_commute_of_flat_of_self`
for commutative rings. |
LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domain : Submodule R E
toFun : domain →ₗ[R] F
@[inherit_doc] notation:25 E " →ₗ.[" R:25 "] " F:0 => LinearPMap R E F
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E] {F : Type*}
[AddCommGroup F] [Module R F] {G : Type*} [AddCommGroup G] [Module R G] | structure | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap | A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to `F`. |
@[coe]
toFun' (f : E →ₗ.[R] F) : f.domain → F := f.toFun | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | toFun' | null |
@[simp]
toFun_eq_coe (f : E →ₗ.[R] F) (x : f.domain) : f.toFun x = f x :=
rfl
@[ext (iff := false)] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | toFun_eq_coe | null |
ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : E⦄ ⦃hf : x ∈ f.domain⦄ ⦃hg : x ∈ g.domain⦄, f ⟨x, hf⟩ = g ⟨x, hg⟩) : f = g := by
rcases f with ⟨f_dom, f⟩
rcases g with ⟨g_dom, g⟩
obtain rfl : f_dom = g_dom := h
congr
apply LinearMap.ext
intro x
apply h' | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | ext | null |
dExt {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g :=
ext h fun _ _ _ ↦ h' rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | dExt | A dependent version of `ext`. |
map_zero (f : E →ₗ.[R] F) : f 0 = 0 :=
f.toFun.map_zero | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | map_zero | null |
ext_iff {f g : E →ₗ.[R] F} :
f = g ↔
f.domain = g.domain ∧
∀ ⦃x : E⦄ ⦃hf : x ∈ f.domain⦄ ⦃hg : x ∈ g.domain⦄, f ⟨x, hf⟩ = g ⟨x, hg⟩ :=
⟨by rintro rfl; simp, fun ⟨deq, feq⟩ ↦ ext deq feq⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | ext_iff | null |
dExt_iff {f g : E →ₗ.[R] F} :
f = g ↔
∃ _domain_eq : f.domain = g.domain,
∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y :=
⟨fun EQ =>
EQ ▸
⟨rfl, fun x y h => by
congr
exact mod_cast h⟩,
fun ⟨deq, feq⟩ => dExt deq feq⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | dExt_iff | null |
ext' {s : Submodule R E} {f g : s →ₗ[R] F} (h : f = g) : mk s f = mk s g :=
h ▸ rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | ext' | null |
map_add (f : E →ₗ.[R] F) (x y : f.domain) : f (x + y) = f x + f y :=
f.toFun.map_add x y | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | map_add | null |
map_neg (f : E →ₗ.[R] F) (x : f.domain) : f (-x) = -f x :=
f.toFun.map_neg x | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | map_neg | null |
map_sub (f : E →ₗ.[R] F) (x y : f.domain) : f (x - y) = f x - f y :=
f.toFun.map_sub x y | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | map_sub | null |
map_smul (f : E →ₗ.[R] F) (c : R) (x : f.domain) : f (c • x) = c • f x :=
f.toFun.map_smul c x
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | map_smul | null |
mk_apply (p : Submodule R E) (f : p →ₗ[R] F) (x : p) : mk p f x = f x :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mk_apply | null |
noncomputable mkSpanSingleton' (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) :
E →ₗ.[R] F where
domain := R ∙ x
toFun :=
have H : ∀ c₁ c₂ : R, c₁ • x = c₂ • x → c₁ • y = c₂ • y := by
intro c₁ c₂ h
rw [← sub_eq_zero, ← sub_smul] at h ⊢
exact H _ h
{ toFun z := Classical.choose (mem_span_singleton.1 z.prop) • y
map_add' y z := by
rw [← add_smul, H]
have (w : R ∙ x) := Classical.choose_spec (mem_span_singleton.1 w.prop)
simp only [add_smul, this, ← coe_add]
map_smul' c z := by
rw [smul_smul, H]
have (w : R ∙ x) := Classical.choose_spec (mem_span_singleton.1 w.prop)
simp only [mul_smul, this]
apply coe_smul }
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mkSpanSingleton' | The unique `LinearPMap` on `R ∙ x` that sends `x` to `y`. This version works for modules
over rings, and requires a proof of `∀ c, c • x = 0 → c • y = 0`. |
domain_mkSpanSingleton (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) :
(mkSpanSingleton' x y H).domain = R ∙ x :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | domain_mkSpanSingleton | null |
mkSpanSingleton'_apply (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (c : R) (h) :
mkSpanSingleton' x y H ⟨c • x, h⟩ = c • y := by
dsimp [mkSpanSingleton']
rw [← sub_eq_zero, ← sub_smul]
apply H
simp only [sub_smul, sub_eq_zero]
apply Classical.choose_spec (mem_span_singleton.1 h)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mkSpanSingleton'_apply | null |
mkSpanSingleton'_apply_self (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (h) :
mkSpanSingleton' x y H ⟨x, h⟩ = y := by
conv_rhs => rw [← one_smul R y]
rw [← mkSpanSingleton'_apply x y H 1 ?_]
· congr
rw [one_smul]
· rwa [one_smul] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mkSpanSingleton'_apply_self | null |
noncomputable mkSpanSingleton {K E F : Type*} [DivisionRing K] [AddCommGroup E] [Module K E]
[AddCommGroup F] [Module K F] (x : E) (y : F) (hx : x ≠ 0) : E →ₗ.[K] F :=
mkSpanSingleton' x y fun c hc =>
(smul_eq_zero.1 hc).elim (fun hc => by rw [hc, zero_smul]) fun hx' => absurd hx' hx | abbrev | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mkSpanSingleton | The unique `LinearPMap` on `span R {x}` that sends a non-zero vector `x` to `y`.
This version works for modules over division rings. |
mkSpanSingleton_apply (K : Type*) {E F : Type*} [DivisionRing K] [AddCommGroup E]
[Module K E] [AddCommGroup F] [Module K F] {x : E} (hx : x ≠ 0) (y : F) :
mkSpanSingleton x y hx ⟨x, (Submodule.mem_span_singleton_self x : x ∈ Submodule.span K {x})⟩ =
y :=
LinearPMap.mkSpanSingleton'_apply_self _ _ _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | mkSpanSingleton_apply | null |
protected fst (p : Submodule R E) (p' : Submodule R F) : E × F →ₗ.[R] E where
domain := p.prod p'
toFun := (LinearMap.fst R E F).comp (p.prod p').subtype
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | fst | Projection to the first coordinate as a `LinearPMap` |
fst_apply (p : Submodule R E) (p' : Submodule R F) (x : p.prod p') :
LinearPMap.fst p p' x = (x : E × F).1 :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | fst_apply | null |
protected snd (p : Submodule R E) (p' : Submodule R F) : E × F →ₗ.[R] F where
domain := p.prod p'
toFun := (LinearMap.snd R E F).comp (p.prod p').subtype
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | snd | Projection to the second coordinate as a `LinearPMap` |
snd_apply (p : Submodule R E) (p' : Submodule R F) (x : p.prod p') :
LinearPMap.snd p p' x = (x : E × F).2 :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | snd_apply | null |
le : LE (E →ₗ.[R] F) :=
⟨fun f g => f.domain ≤ g.domain ∧ ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | le | null |
apply_comp_inclusion {T S : E →ₗ.[R] F} (h : T ≤ S) (x : T.domain) :
T x = S (Submodule.inclusion h.1 x) :=
h.2 rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | apply_comp_inclusion | null |
exists_of_le {T S : E →ₗ.[R] F} (h : T ≤ S) (x : T.domain) :
∃ y : S.domain, (x : E) = y ∧ T x = S y :=
⟨⟨x.1, h.1 x.2⟩, ⟨rfl, h.2 rfl⟩⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | exists_of_le | null |
eq_of_le_of_domain_eq {f g : E →ₗ.[R] F} (hle : f ≤ g) (heq : f.domain = g.domain) :
f = g :=
dExt heq hle.2 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | eq_of_le_of_domain_eq | null |
eqLocus (f g : E →ₗ.[R] F) : Submodule R E where
carrier := { x | ∃ (hf : x ∈ f.domain) (hg : x ∈ g.domain), f ⟨x, hf⟩ = g ⟨x, hg⟩ }
zero_mem' := ⟨zero_mem _, zero_mem _, f.map_zero.trans g.map_zero.symm⟩
add_mem' {x y} := fun ⟨hfx, hgx, hx⟩ ⟨hfy, hgy, hy⟩ ↦
⟨add_mem hfx hfy, add_mem hgx hgy, by
simp_all [← AddMemClass.mk_add_mk, f.map_add, g.map_add]⟩
smul_mem' c x := fun ⟨hfx, hgx, hx⟩ ↦
⟨smul_mem _ c hfx, smul_mem _ c hgx, by
have {f : E →ₗ.[R] F} (hfx) : (⟨c • x, smul_mem _ c hfx⟩ : f.domain) = c • ⟨x, hfx⟩ := by simp
rw [this hfx, this hgx, f.map_smul, g.map_smul, hx]⟩ | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | eqLocus | Given two partial linear maps `f`, `g`, the set of points `x` such that
both `f` and `g` are defined at `x` and `f x = g x` form a submodule. |
bot : Bot (E →ₗ.[R] F) :=
⟨⟨⊥, 0⟩⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | bot | null |
inhabited : Inhabited (E →ₗ.[R] F) :=
⟨⊥⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | inhabited | null |
semilatticeInf : SemilatticeInf (E →ₗ.[R] F) where
le := (· ≤ ·)
le_refl f := ⟨le_refl f.domain, fun _ _ h => Subtype.eq h ▸ rfl⟩
le_trans := fun _ _ _ ⟨fg_le, fg_eq⟩ ⟨gh_le, gh_eq⟩ =>
⟨le_trans fg_le gh_le, fun x _ hxz =>
have hxy : (x : E) = inclusion fg_le x := rfl
(fg_eq hxy).trans (gh_eq <| hxy.symm.trans hxz)⟩
le_antisymm _ _ fg gf := eq_of_le_of_domain_eq fg (le_antisymm fg.1 gf.1)
inf f g := ⟨f.eqLocus g, f.toFun.comp <| inclusion fun _x hx => hx.fst⟩
le_inf := by
intro f g h ⟨fg_le, fg_eq⟩ ⟨fh_le, fh_eq⟩
exact ⟨fun x hx =>
⟨fg_le hx, fh_le hx,
(fg_eq (x := ⟨x, hx⟩) rfl).symm.trans (fh_eq rfl)⟩,
fun x ⟨y, yg, hy⟩ h => fg_eq h⟩
inf_le_left f _ := ⟨fun _ hx => hx.fst, fun _ _ h => congr_arg f <| Subtype.eq <| h⟩
inf_le_right _ g :=
⟨fun _ hx => hx.snd.fst, fun ⟨_, _, _, hx⟩ _ h => hx.trans <| congr_arg g <| Subtype.eq <| h⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | semilatticeInf | null |
orderBot : OrderBot (E →ₗ.[R] F) where
bot := ⊥
bot_le f :=
⟨bot_le, fun x y h => by
have hx : x = 0 := Subtype.eq ((mem_bot R).1 x.2)
have hy : y = 0 := Subtype.eq (h.symm.trans (congr_arg _ hx))
rw [hx, hy, map_zero, map_zero]⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | orderBot | null |
le_of_eqLocus_ge {f g : E →ₗ.[R] F} (H : f.domain ≤ f.eqLocus g) : f ≤ g :=
suffices f ≤ f ⊓ g from le_trans this inf_le_right
⟨H, fun _x _y hxy => ((inf_le_left : f ⊓ g ≤ f).2 hxy.symm).symm⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | le_of_eqLocus_ge | null |
domain_mono : StrictMono (@domain R _ E _ _ F _ _) := fun _f _g hlt =>
lt_of_le_of_ne hlt.1.1 fun heq => ne_of_lt hlt <| eq_of_le_of_domain_eq (le_of_lt hlt) heq | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | domain_mono | null |
private sup_aux (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) :
∃ fg : ↥(f.domain ⊔ g.domain) →ₗ[R] F,
∀ (x : f.domain) (y : g.domain) (z : ↥(f.domain ⊔ g.domain)),
(x : E) + y = ↑z → fg z = f x + g y := by
choose x hx y hy hxy using fun z : ↥(f.domain ⊔ g.domain) => mem_sup.1 z.prop
set fg := fun z => f ⟨x z, hx z⟩ + g ⟨y z, hy z⟩
have fg_eq : ∀ (x' : f.domain) (y' : g.domain) (z' : ↥(f.domain ⊔ g.domain))
(_H : (x' : E) + y' = z'), fg z' = f x' + g y' := by
intro x' y' z' H
dsimp [fg]
rw [add_comm, ← sub_eq_sub_iff_add_eq_add, eq_comm, ← map_sub, ← map_sub]
apply h
simp only [← eq_sub_iff_add_eq] at hxy
simp only [AddSubgroupClass.coe_sub, hxy, ← sub_add, ← sub_sub, sub_self,
zero_sub, ← H]
apply neg_add_eq_sub
use { toFun := fg, map_add' := ?_, map_smul' := ?_ }, fg_eq
· rintro ⟨z₁, hz₁⟩ ⟨z₂, hz₂⟩
rw [← add_assoc, add_right_comm (f _), ← map_add, add_assoc, ← map_add]
apply fg_eq
simp only [coe_add, ← add_assoc]
rw [add_right_comm (x _), hxy, add_assoc, hxy, coe_mk, coe_mk]
· intro c z
rw [smul_add, ← map_smul, ← map_smul]
apply fg_eq
simp only [coe_smul, ← smul_add, hxy, RingHom.id_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | sup_aux | null |
protected noncomputable sup (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : E →ₗ.[R] F :=
⟨_, Classical.choose (sup_aux f g h)⟩
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | sup | Given two partial linear maps that agree on the intersection of their domains,
`f.sup g h` is the unique partial linear map on `f.domain ⊔ g.domain` that agrees
with `f` and `g`. |
domain_sup (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) :
(f.sup g h).domain = f.domain ⊔ g.domain :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | domain_sup | null |
sup_apply {f g : E →ₗ.[R] F} (H : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y)
(x : f.domain) (y : g.domain) (z : ↥(f.domain ⊔ g.domain)) (hz : (↑x : E) + ↑y = ↑z) :
f.sup g H z = f x + g y :=
Classical.choose_spec (sup_aux f g H) x y z hz | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | sup_apply | null |
protected left_le_sup (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : f ≤ f.sup g h := by
refine ⟨le_sup_left, fun z₁ z₂ hz => ?_⟩
rw [← add_zero (f _), ← g.map_zero]
refine (sup_apply h _ _ _ ?_).symm
simpa | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | left_le_sup | null |
protected right_le_sup (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : g ≤ f.sup g h := by
refine ⟨le_sup_right, fun z₁ z₂ hz => ?_⟩
rw [← zero_add (g _), ← f.map_zero]
refine (sup_apply h _ _ _ ?_).symm
simpa | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | right_le_sup | null |
protected sup_le {f g h : E →ₗ.[R] F}
(H : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) (fh : f ≤ h) (gh : g ≤ h) :
f.sup g H ≤ h :=
have Hf : f ≤ f.sup g H ⊓ h := le_inf (f.left_le_sup g H) fh
have Hg : g ≤ f.sup g H ⊓ h := le_inf (f.right_le_sup g H) gh
le_of_eqLocus_ge <| sup_le Hf.1 Hg.1 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | sup_le | null |
sup_h_of_disjoint (f g : E →ₗ.[R] F) (h : Disjoint f.domain g.domain) (x : f.domain)
(y : g.domain) (hxy : (x : E) = y) : f x = g y := by
rw [disjoint_def] at h
have hy : y = 0 := Subtype.eq (h y (hxy ▸ x.2) y.2)
have hx : x = 0 := Subtype.eq (hxy.trans <| congr_arg _ hy)
simp [*]
/-! ### Algebraic operations -/ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | sup_h_of_disjoint | Hypothesis for `LinearPMap.sup` holds, if `f.domain` is disjoint with `g.domain`. |
instZero : Zero (E →ₗ.[R] F) := ⟨⊤, 0⟩
@[simp] | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instZero | null |
zero_domain : (0 : E →ₗ.[R] F).domain = ⊤ := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | zero_domain | null |
zero_apply (x : (⊤ : Submodule R E)) : (0 : E →ₗ.[R] F) x = 0 := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | zero_apply | null |
instSMul : SMul M (E →ₗ.[R] F) :=
⟨fun a f =>
{ domain := f.domain
toFun := a • f.toFun }⟩
@[simp] | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instSMul | null |
smul_domain (a : M) (f : E →ₗ.[R] F) : (a • f).domain = f.domain :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | smul_domain | null |
smul_apply (a : M) (f : E →ₗ.[R] F) (x : (a • f).domain) : (a • f) x = a • f x :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | smul_apply | null |
coe_smul (a : M) (f : E →ₗ.[R] F) : ⇑(a • f) = a • ⇑f :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | coe_smul | null |
instSMulCommClass [SMulCommClass M N F] : SMulCommClass M N (E →ₗ.[R] F) :=
⟨fun a b f => ext' <| smul_comm a b f.toFun⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instSMulCommClass | null |
instIsScalarTower [SMul M N] [IsScalarTower M N F] : IsScalarTower M N (E →ₗ.[R] F) :=
⟨fun a b f => ext' <| smul_assoc a b f.toFun⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instIsScalarTower | null |
instMulAction : MulAction M (E →ₗ.[R] F) where
smul := (· • ·)
one_smul := fun ⟨_s, f⟩ => ext' <| one_smul M f
mul_smul a b f := ext' <| mul_smul a b f.toFun | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instMulAction | null |
instNeg : Neg (E →ₗ.[R] F) :=
⟨fun f => ⟨f.domain, -f.toFun⟩⟩
@[simp] | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instNeg | null |
neg_domain (f : E →ₗ.[R] F) : (-f).domain = f.domain := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | neg_domain | null |
neg_apply (f : E →ₗ.[R] F) (x) : (-f) x = -f x :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | neg_apply | null |
instInvolutiveNeg : InvolutiveNeg (E →ₗ.[R] F) :=
⟨fun f => by
ext x y hxy
· rfl
· simp only [neg_apply, neg_neg]⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instInvolutiveNeg | null |
instAdd : Add (E →ₗ.[R] F) :=
⟨fun f g =>
{ domain := f.domain ⊓ g.domain
toFun := f.toFun.comp (inclusion (inf_le_left : f.domain ⊓ g.domain ≤ _))
+ g.toFun.comp (inclusion (inf_le_right : f.domain ⊓ g.domain ≤ _)) }⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | instAdd | null |
add_domain (f g : E →ₗ.[R] F) : (f + g).domain = f.domain ⊓ g.domain := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Prod"
] | Mathlib/LinearAlgebra/LinearPMap.lean | add_domain | null |
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