Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
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import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : Type*} {S : Type*} {M : Type*} {N : Type*} {ι : Type*} {κ : Type*}
[CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M]
[IsScalarTower R S M] [AddCommMonoid N] [Module R N]
def Basis.tensorProduct (b : Basis ι S M) (c : Basis κ R N) :
Basis (ι × κ) S (TensorProduct R M N) :=
Finsupp.basisSingleOne.map
((TensorProduct.AlgebraTensorModule.congr b.repr c.repr).trans <|
(finsuppTensorFinsupp R S _ _ _ _).trans <|
Finsupp.lcongr (Equiv.refl _) (TensorProduct.AlgebraTensorModule.rid R S S)).symm
#align basis.tensor_product Basis.tensorProduct
@[simp]
theorem Basis.tensorProduct_apply (b : Basis ι R M) (c : Basis κ R N) (i : ι) (j : κ) :
Basis.tensorProduct b c (i, j) = b i ⊗ₜ c j := by
simp [Basis.tensorProduct]
#align basis.tensor_product_apply Basis.tensorProduct_apply
theorem Basis.tensorProduct_apply' (b : Basis ι R M) (c : Basis κ R N) (i : ι × κ) :
Basis.tensorProduct b c i = b i.1 ⊗ₜ c i.2 := by
simp [Basis.tensorProduct]
#align basis.tensor_product_apply' Basis.tensorProduct_apply'
@[simp]
| Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 50 | 53 | theorem Basis.tensorProduct_repr_tmul_apply (b : Basis ι R M) (c : Basis κ R N) (m : M) (n : N)
(i : ι) (j : κ) :
(Basis.tensorProduct b c).repr (m ⊗ₜ n) (i, j) = b.repr m i * c.repr n j := by |
simp [Basis.tensorProduct, mul_comm]
| 1,667 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.finiteness from "leanprover-community/mathlib"@"c813ed7de0f5115f956239124e9b30f3a621966f"
open Function (Surjective)
namespace Submodule
variable {R : Type*} {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
open Set
def FG (N : Submodule R M) : Prop :=
∃ S : Finset M, Submodule.span R ↑S = N
#align submodule.fg Submodule.FG
theorem fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
exact ⟨t, rfl⟩⟩
#align submodule.fg_def Submodule.fg_def
theorem fg_iff_addSubmonoid_fg (P : Submodule ℕ M) : P.FG ↔ P.toAddSubmonoid.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_nat_eq_addSubmonoid_closure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_nat_eq_addSubmonoid_closure] using hS⟩⟩
#align submodule.fg_iff_add_submonoid_fg Submodule.fg_iff_addSubmonoid_fg
theorem fg_iff_add_subgroup_fg {G : Type*} [AddCommGroup G] (P : Submodule ℤ G) :
P.FG ↔ P.toAddSubgroup.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_int_eq_addSubgroup_closure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_int_eq_addSubgroup_closure] using hS⟩⟩
#align submodule.fg_iff_add_subgroup_fg Submodule.fg_iff_add_subgroup_fg
| Mathlib/RingTheory/Finiteness.lean | 69 | 77 | theorem fg_iff_exists_fin_generating_family {N : Submodule R M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), span R (range s) = N := by |
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· rintro ⟨n, s, hs⟩
exact ⟨range s, finite_range s, hs⟩
| 1,668 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.finiteness from "leanprover-community/mathlib"@"c813ed7de0f5115f956239124e9b30f3a621966f"
open Function (Surjective)
namespace Submodule
variable {R : Type*} {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
open Set
def FG (N : Submodule R M) : Prop :=
∃ S : Finset M, Submodule.span R ↑S = N
#align submodule.fg Submodule.FG
theorem fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
exact ⟨t, rfl⟩⟩
#align submodule.fg_def Submodule.fg_def
theorem fg_iff_addSubmonoid_fg (P : Submodule ℕ M) : P.FG ↔ P.toAddSubmonoid.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_nat_eq_addSubmonoid_closure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_nat_eq_addSubmonoid_closure] using hS⟩⟩
#align submodule.fg_iff_add_submonoid_fg Submodule.fg_iff_addSubmonoid_fg
theorem fg_iff_add_subgroup_fg {G : Type*} [AddCommGroup G] (P : Submodule ℤ G) :
P.FG ↔ P.toAddSubgroup.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_int_eq_addSubgroup_closure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_int_eq_addSubgroup_closure] using hS⟩⟩
#align submodule.fg_iff_add_subgroup_fg Submodule.fg_iff_add_subgroup_fg
theorem fg_iff_exists_fin_generating_family {N : Submodule R M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), span R (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· rintro ⟨n, s, hs⟩
exact ⟨range s, finite_range s, hs⟩
#align submodule.fg_iff_exists_fin_generating_family Submodule.fg_iff_exists_fin_generating_family
| Mathlib/RingTheory/Finiteness.lean | 82 | 134 | theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by |
rw [fg_def] at hn
rcases hn with ⟨s, hfs, hs⟩
have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
refine ⟨1, ?_, ?_, ?_⟩
· rw [sub_self]
exact I.zero_mem
· rw [hs]
intro n hn
rw [mem_comap]
change (1 : R) • n ∈ I • N
rw [one_smul]
exact hin hn
· rw [← span_le, hs]
clear hin hs
revert this
refine Set.Finite.dinduction_on _ hfs (fun H => ?_) @fun i s _ _ ih H => ?_
· rcases H with ⟨r, hr1, hrn, _⟩
refine ⟨r, hr1, fun n hn => ?_⟩
specialize hrn hn
rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
apply ih
rcases H with ⟨r, hr1, hrn, hs⟩
rw [← Set.singleton_union, span_union, smul_sup] at hrn
rw [Set.insert_subset_iff] at hs
have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
specialize hrn hs.1
rw [mem_comap, mem_sup] at hrn
rcases hrn with ⟨y, hy, z, hz, hyz⟩
dsimp at hyz
rw [mem_smul_span_singleton] at hy
rcases hy with ⟨c, hci, rfl⟩
use r - c
constructor
· rw [sub_right_comm]
exact I.sub_mem hr1 hci
· rw [sub_smul, ← hyz, add_sub_cancel_left]
exact hz
rcases this with ⟨c, hc1, hci⟩
refine ⟨c * r, ?_, ?_, hs.2⟩
· simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
· intro n hn
specialize hrn hn
rw [mem_comap, mem_sup] at hrn
rcases hrn with ⟨y, hy, z, hz, hyz⟩
dsimp at hyz
rw [mem_smul_span_singleton] at hy
rcases hy with ⟨d, _, rfl⟩
simp only [mem_comap, LinearMap.lsmul_apply]
rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
exact add_mem (smul_mem _ _ hci) (smul_mem _ _ hz)
| 1,668 |
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
#align_import ring_theory.ideal.idempotent_fg from "leanprover-community/mathlib"@"25cf7631da8ddc2d5f957c388bf5e4b25a77d8dc"
namespace Ideal
| Mathlib/RingTheory/Ideal/IdempotentFG.lean | 20 | 35 | theorem isIdempotentElem_iff_of_fg {R : Type*} [CommRing R] (I : Ideal R) (h : I.FG) :
IsIdempotentElem I ↔ ∃ e : R, IsIdempotentElem e ∧ I = R ∙ e := by |
constructor
· intro e
obtain ⟨r, hr, hr'⟩ :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h
(by
rw [smul_eq_mul]
exact e.ge)
simp_rw [smul_eq_mul] at hr'
refine ⟨r, hr' r hr, antisymm ?_ ((Submodule.span_singleton_le_iff_mem _ _).mpr hr)⟩
intro x hx
rw [← hr' x hx]
exact Ideal.mem_span_singleton'.mpr ⟨_, mul_comm _ _⟩
· rintro ⟨e, he, rfl⟩
simp [IsIdempotentElem, Ideal.span_singleton_mul_span_singleton, he.eq]
| 1,669 |
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
#align_import ring_theory.ideal.idempotent_fg from "leanprover-community/mathlib"@"25cf7631da8ddc2d5f957c388bf5e4b25a77d8dc"
namespace Ideal
theorem isIdempotentElem_iff_of_fg {R : Type*} [CommRing R] (I : Ideal R) (h : I.FG) :
IsIdempotentElem I ↔ ∃ e : R, IsIdempotentElem e ∧ I = R ∙ e := by
constructor
· intro e
obtain ⟨r, hr, hr'⟩ :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h
(by
rw [smul_eq_mul]
exact e.ge)
simp_rw [smul_eq_mul] at hr'
refine ⟨r, hr' r hr, antisymm ?_ ((Submodule.span_singleton_le_iff_mem _ _).mpr hr)⟩
intro x hx
rw [← hr' x hx]
exact Ideal.mem_span_singleton'.mpr ⟨_, mul_comm _ _⟩
· rintro ⟨e, he, rfl⟩
simp [IsIdempotentElem, Ideal.span_singleton_mul_span_singleton, he.eq]
#align ideal.is_idempotent_elem_iff_of_fg Ideal.isIdempotentElem_iff_of_fg
| Mathlib/RingTheory/Ideal/IdempotentFG.lean | 38 | 47 | theorem isIdempotentElem_iff_eq_bot_or_top {R : Type*} [CommRing R] [IsDomain R] (I : Ideal R)
(h : I.FG) : IsIdempotentElem I ↔ I = ⊥ ∨ I = ⊤ := by |
constructor
· intro H
obtain ⟨e, he, rfl⟩ := (I.isIdempotentElem_iff_of_fg h).mp H
simp only [Ideal.submodule_span_eq, Ideal.span_singleton_eq_bot]
apply Or.imp id _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)
rintro rfl
simp
· rintro (rfl | rfl) <;> simp [IsIdempotentElem]
| 1,669 |
import Mathlib.RingTheory.Finiteness
import Mathlib.Logic.Equiv.TransferInstance
universe u v w
open Function
variable (R : Type u) [Semiring R]
@[mk_iff]
class OrzechProperty : Prop where
injective_of_surjective_of_submodule' : ∀ {M : Type u} [AddCommMonoid M] [Module R M]
[Module.Finite R M] {N : Submodule R M} (f : N →ₗ[R] M), Surjective f → Injective f
namespace OrzechProperty
variable {R}
variable [OrzechProperty R] {M : Type v} [AddCommMonoid M] [Module R M] [Module.Finite R M]
| Mathlib/RingTheory/OrzechProperty.lean | 69 | 82 | theorem injective_of_surjective_of_injective
{N : Type w} [AddCommMonoid N] [Module R N]
(i f : N →ₗ[R] M) (hi : Injective i) (hf : Surjective f) : Injective f := by |
obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M
haveI := small_of_surjective hg
letI := Equiv.addCommMonoid (equivShrink M).symm
letI := Equiv.module R (equivShrink M).symm
let j : Shrink.{u} M ≃ₗ[R] M := Equiv.linearEquiv R (equivShrink M).symm
haveI := Module.Finite.equiv j.symm
let i' := j.symm.toLinearMap ∘ₗ i
replace hi : Injective i' := by simpa [i'] using hi
let f' := j.symm.toLinearMap ∘ₗ f ∘ₗ (LinearEquiv.ofInjective i' hi).symm.toLinearMap
replace hf : Surjective f' := by simpa [f'] using hf
simpa [f'] using injective_of_surjective_of_submodule' f' hf
| 1,670 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
section
variable (R : Type u) [Semiring R]
@[mk_iff]
class StrongRankCondition : Prop where
le_of_fin_injective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Injective f → n ≤ m
#align strong_rank_condition StrongRankCondition
theorem le_of_fin_injective [StrongRankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Injective f → n ≤ m :=
StrongRankCondition.le_of_fin_injective f
#align le_of_fin_injective le_of_fin_injective
| Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 130 | 139 | theorem strongRankCondition_iff_succ :
StrongRankCondition R ↔
∀ (n : ℕ) (f : (Fin (n + 1) → R) →ₗ[R] Fin n → R), ¬Function.Injective f := by |
refine ⟨fun h n => fun f hf => ?_, fun h => ⟨@fun n m f hf => ?_⟩⟩
· letI : StrongRankCondition R := h
exact Nat.not_succ_le_self n (le_of_fin_injective R f hf)
· by_contra H
exact
h m (f.comp (Function.ExtendByZero.linearMap R (Fin.castLE (not_le.1 H))))
(hf.comp (Function.extend_injective (Fin.strictMono_castLE _).injective _))
| 1,671 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
section
variable (R : Type u) [Semiring R]
@[mk_iff]
class StrongRankCondition : Prop where
le_of_fin_injective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Injective f → n ≤ m
#align strong_rank_condition StrongRankCondition
theorem le_of_fin_injective [StrongRankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Injective f → n ≤ m :=
StrongRankCondition.le_of_fin_injective f
#align le_of_fin_injective le_of_fin_injective
theorem strongRankCondition_iff_succ :
StrongRankCondition R ↔
∀ (n : ℕ) (f : (Fin (n + 1) → R) →ₗ[R] Fin n → R), ¬Function.Injective f := by
refine ⟨fun h n => fun f hf => ?_, fun h => ⟨@fun n m f hf => ?_⟩⟩
· letI : StrongRankCondition R := h
exact Nat.not_succ_le_self n (le_of_fin_injective R f hf)
· by_contra H
exact
h m (f.comp (Function.ExtendByZero.linearMap R (Fin.castLE (not_le.1 H))))
(hf.comp (Function.extend_injective (Fin.strictMono_castLE _).injective _))
#align strong_rank_condition_iff_succ strongRankCondition_iff_succ
instance (priority := 100) strongRankCondition_of_orzechProperty
[Nontrivial R] [OrzechProperty R] : StrongRankCondition R := by
refine (strongRankCondition_iff_succ R).2 fun n i hi ↦ ?_
let f : (Fin (n + 1) → R) →ₗ[R] Fin n → R := {
toFun := fun x ↦ x ∘ Fin.castSucc
map_add' := fun _ _ ↦ rfl
map_smul' := fun _ _ ↦ rfl
}
have h : (0 : Fin (n + 1) → R) = update (0 : Fin (n + 1) → R) (Fin.last n) 1 := by
apply OrzechProperty.injective_of_surjective_of_injective i f hi
(Fin.castSucc_injective _).surjective_comp_right
ext m
simp [f, update_apply, (Fin.castSucc_lt_last m).ne]
simpa using congr_fun h (Fin.last n)
| Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 158 | 164 | theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by |
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
| 1,671 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
section
variable (R : Type u) [Semiring R]
@[mk_iff]
class StrongRankCondition : Prop where
le_of_fin_injective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Injective f → n ≤ m
#align strong_rank_condition StrongRankCondition
theorem le_of_fin_injective [StrongRankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Injective f → n ≤ m :=
StrongRankCondition.le_of_fin_injective f
#align le_of_fin_injective le_of_fin_injective
theorem strongRankCondition_iff_succ :
StrongRankCondition R ↔
∀ (n : ℕ) (f : (Fin (n + 1) → R) →ₗ[R] Fin n → R), ¬Function.Injective f := by
refine ⟨fun h n => fun f hf => ?_, fun h => ⟨@fun n m f hf => ?_⟩⟩
· letI : StrongRankCondition R := h
exact Nat.not_succ_le_self n (le_of_fin_injective R f hf)
· by_contra H
exact
h m (f.comp (Function.ExtendByZero.linearMap R (Fin.castLE (not_le.1 H))))
(hf.comp (Function.extend_injective (Fin.strictMono_castLE _).injective _))
#align strong_rank_condition_iff_succ strongRankCondition_iff_succ
instance (priority := 100) strongRankCondition_of_orzechProperty
[Nontrivial R] [OrzechProperty R] : StrongRankCondition R := by
refine (strongRankCondition_iff_succ R).2 fun n i hi ↦ ?_
let f : (Fin (n + 1) → R) →ₗ[R] Fin n → R := {
toFun := fun x ↦ x ∘ Fin.castSucc
map_add' := fun _ _ ↦ rfl
map_smul' := fun _ _ ↦ rfl
}
have h : (0 : Fin (n + 1) → R) = update (0 : Fin (n + 1) → R) (Fin.last n) 1 := by
apply OrzechProperty.injective_of_surjective_of_injective i f hi
(Fin.castSucc_injective _).surjective_comp_right
ext m
simp [f, update_apply, (Fin.castSucc_lt_last m).ne]
simpa using congr_fun h (Fin.last n)
theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
#align card_le_of_injective card_le_of_injective
| Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 167 | 173 | theorem card_le_of_injective' [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by |
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_injective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.injective.comp i).comp Q.injective)
| 1,671 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
section
variable (R : Type u) [Semiring R]
@[mk_iff]
class StrongRankCondition : Prop where
le_of_fin_injective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Injective f → n ≤ m
#align strong_rank_condition StrongRankCondition
theorem le_of_fin_injective [StrongRankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Injective f → n ≤ m :=
StrongRankCondition.le_of_fin_injective f
#align le_of_fin_injective le_of_fin_injective
theorem strongRankCondition_iff_succ :
StrongRankCondition R ↔
∀ (n : ℕ) (f : (Fin (n + 1) → R) →ₗ[R] Fin n → R), ¬Function.Injective f := by
refine ⟨fun h n => fun f hf => ?_, fun h => ⟨@fun n m f hf => ?_⟩⟩
· letI : StrongRankCondition R := h
exact Nat.not_succ_le_self n (le_of_fin_injective R f hf)
· by_contra H
exact
h m (f.comp (Function.ExtendByZero.linearMap R (Fin.castLE (not_le.1 H))))
(hf.comp (Function.extend_injective (Fin.strictMono_castLE _).injective _))
#align strong_rank_condition_iff_succ strongRankCondition_iff_succ
instance (priority := 100) strongRankCondition_of_orzechProperty
[Nontrivial R] [OrzechProperty R] : StrongRankCondition R := by
refine (strongRankCondition_iff_succ R).2 fun n i hi ↦ ?_
let f : (Fin (n + 1) → R) →ₗ[R] Fin n → R := {
toFun := fun x ↦ x ∘ Fin.castSucc
map_add' := fun _ _ ↦ rfl
map_smul' := fun _ _ ↦ rfl
}
have h : (0 : Fin (n + 1) → R) = update (0 : Fin (n + 1) → R) (Fin.last n) 1 := by
apply OrzechProperty.injective_of_surjective_of_injective i f hi
(Fin.castSucc_injective _).surjective_comp_right
ext m
simp [f, update_apply, (Fin.castSucc_lt_last m).ne]
simpa using congr_fun h (Fin.last n)
theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
#align card_le_of_injective card_le_of_injective
theorem card_le_of_injective' [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_injective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.injective.comp i).comp Q.injective)
#align card_le_of_injective' card_le_of_injective'
class RankCondition : Prop where
le_of_fin_surjective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Surjective f → m ≤ n
#align rank_condition RankCondition
theorem le_of_fin_surjective [RankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Surjective f → m ≤ n :=
RankCondition.le_of_fin_surjective f
#align le_of_fin_surjective le_of_fin_surjective
| Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 188 | 194 | theorem card_le_of_surjective [RankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Surjective f) : Fintype.card β ≤ Fintype.card α := by |
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_surjective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).surjective.comp i).comp (LinearEquiv.surjective P))
| 1,671 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
section
variable (R : Type u) [Semiring R]
@[mk_iff]
class StrongRankCondition : Prop where
le_of_fin_injective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Injective f → n ≤ m
#align strong_rank_condition StrongRankCondition
theorem le_of_fin_injective [StrongRankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Injective f → n ≤ m :=
StrongRankCondition.le_of_fin_injective f
#align le_of_fin_injective le_of_fin_injective
theorem strongRankCondition_iff_succ :
StrongRankCondition R ↔
∀ (n : ℕ) (f : (Fin (n + 1) → R) →ₗ[R] Fin n → R), ¬Function.Injective f := by
refine ⟨fun h n => fun f hf => ?_, fun h => ⟨@fun n m f hf => ?_⟩⟩
· letI : StrongRankCondition R := h
exact Nat.not_succ_le_self n (le_of_fin_injective R f hf)
· by_contra H
exact
h m (f.comp (Function.ExtendByZero.linearMap R (Fin.castLE (not_le.1 H))))
(hf.comp (Function.extend_injective (Fin.strictMono_castLE _).injective _))
#align strong_rank_condition_iff_succ strongRankCondition_iff_succ
instance (priority := 100) strongRankCondition_of_orzechProperty
[Nontrivial R] [OrzechProperty R] : StrongRankCondition R := by
refine (strongRankCondition_iff_succ R).2 fun n i hi ↦ ?_
let f : (Fin (n + 1) → R) →ₗ[R] Fin n → R := {
toFun := fun x ↦ x ∘ Fin.castSucc
map_add' := fun _ _ ↦ rfl
map_smul' := fun _ _ ↦ rfl
}
have h : (0 : Fin (n + 1) → R) = update (0 : Fin (n + 1) → R) (Fin.last n) 1 := by
apply OrzechProperty.injective_of_surjective_of_injective i f hi
(Fin.castSucc_injective _).surjective_comp_right
ext m
simp [f, update_apply, (Fin.castSucc_lt_last m).ne]
simpa using congr_fun h (Fin.last n)
theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
#align card_le_of_injective card_le_of_injective
theorem card_le_of_injective' [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_injective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.injective.comp i).comp Q.injective)
#align card_le_of_injective' card_le_of_injective'
class RankCondition : Prop where
le_of_fin_surjective : ∀ {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R), Surjective f → m ≤ n
#align rank_condition RankCondition
theorem le_of_fin_surjective [RankCondition R] {n m : ℕ} (f : (Fin n → R) →ₗ[R] Fin m → R) :
Surjective f → m ≤ n :=
RankCondition.le_of_fin_surjective f
#align le_of_fin_surjective le_of_fin_surjective
theorem card_le_of_surjective [RankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Surjective f) : Fintype.card β ≤ Fintype.card α := by
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_surjective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).surjective.comp i).comp (LinearEquiv.surjective P))
#align card_le_of_surjective card_le_of_surjective
| Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 197 | 203 | theorem card_le_of_surjective' [RankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Surjective f) : Fintype.card β ≤ Fintype.card α := by |
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_surjective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.surjective.comp i).comp Q.surjective)
| 1,671 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section InvariantBasisNumber
variable [InvariantBasisNumber R]
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 58 | 83 | theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
Cardinal.lift.{w'} #ι = Cardinal.lift.{w} #ι' := by |
classical
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite ι
· -- `v` is a finite basis, so by `basis_finite_of_finite_spans` so is `v'`.
-- haveI : Finite (range v) := Set.finite_range v
haveI := basis_finite_of_finite_spans _ (Set.finite_range v) v.span_eq v'
cases nonempty_fintype ι'
-- We clean up a little:
rw [Cardinal.mk_fintype, Cardinal.mk_fintype]
simp only [Cardinal.lift_natCast, Cardinal.natCast_inj]
-- Now we can use invariant basis number to show they have the same cardinality.
apply card_eq_of_linearEquiv R
exact
(Finsupp.linearEquivFunOnFinite R R ι).symm.trans v.repr.symm ≪≫ₗ v'.repr ≪≫ₗ
Finsupp.linearEquivFunOnFinite R R ι'
· -- `v` is an infinite basis,
-- so by `infinite_basis_le_maximal_linearIndependent`, `v'` is at least as big,
-- and then applying `infinite_basis_le_maximal_linearIndependent` again
-- we see they have the same cardinality.
have w₁ := infinite_basis_le_maximal_linearIndependent' v _ v'.linearIndependent v'.maximal
rcases Cardinal.lift_mk_le'.mp w₁ with ⟨f⟩
haveI : Infinite ι' := Infinite.of_injective f f.2
have w₂ := infinite_basis_le_maximal_linearIndependent' v' _ v.linearIndependent v.maximal
exact le_antisymm w₁ w₂
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section RankCondition
variable [RankCondition R]
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 109 | 118 | theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
(s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w := by |
-- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
-- by expressing a linear combination in `w` as a linear combination in `ι`.
fapply card_le_of_surjective' R
· exact b.repr.toLinearMap.comp (Finsupp.total w M R (↑))
· apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.surjective
rw [← LinearMap.range_eq_top, Finsupp.range_total]
simpa using s
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section RankCondition
variable [RankCondition R]
theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
(s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w := by
-- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
-- by expressing a linear combination in `w` as a linear combination in `ι`.
fapply card_le_of_surjective' R
· exact b.repr.toLinearMap.comp (Finsupp.total w M R (↑))
· apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.surjective
rw [← LinearMap.range_eq_top, Finsupp.range_total]
simpa using s
#align basis.le_span'' Basis.le_span''
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 125 | 132 | theorem basis_le_span' {ι : Type*} (b : Basis ι R M) {w : Set M} [Fintype w] (s : span R w = ⊤) :
#ι ≤ Fintype.card w := by |
haveI := nontrivial_of_invariantBasisNumber R
haveI := basis_finite_of_finite_spans w (toFinite _) s b
cases nonempty_fintype ι
rw [Cardinal.mk_fintype ι]
simp only [Cardinal.natCast_le]
exact Basis.le_span'' b s
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section RankCondition
variable [RankCondition R]
theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
(s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w := by
-- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
-- by expressing a linear combination in `w` as a linear combination in `ι`.
fapply card_le_of_surjective' R
· exact b.repr.toLinearMap.comp (Finsupp.total w M R (↑))
· apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.surjective
rw [← LinearMap.range_eq_top, Finsupp.range_total]
simpa using s
#align basis.le_span'' Basis.le_span''
theorem basis_le_span' {ι : Type*} (b : Basis ι R M) {w : Set M} [Fintype w] (s : span R w = ⊤) :
#ι ≤ Fintype.card w := by
haveI := nontrivial_of_invariantBasisNumber R
haveI := basis_finite_of_finite_spans w (toFinite _) s b
cases nonempty_fintype ι
rw [Cardinal.mk_fintype ι]
simp only [Cardinal.natCast_le]
exact Basis.le_span'' b s
#align basis_le_span' basis_le_span'
-- Note that if `R` satisfies the strong rank condition,
-- this also follows from `linearIndependent_le_span` below.
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 140 | 164 | theorem Basis.le_span {J : Set M} (v : Basis ι R M) (hJ : span R J = ⊤) : #(range v) ≤ #J := by |
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite J
· rw [← Cardinal.lift_le, Cardinal.mk_range_eq_of_injective v.injective, Cardinal.mk_fintype J]
convert Cardinal.lift_le.{v}.2 (basis_le_span' v hJ)
simp
· let S : J → Set ι := fun j => ↑(v.repr j).support
let S' : J → Set M := fun j => v '' S j
have hs : range v ⊆ ⋃ j, S' j := by
intro b hb
rcases mem_range.1 hb with ⟨i, hi⟩
have : span R J ≤ comap v.repr.toLinearMap (Finsupp.supported R R (⋃ j, S j)) :=
span_le.2 fun j hj x hx => ⟨_, ⟨⟨j, hj⟩, rfl⟩, hx⟩
rw [hJ] at this
replace : v.repr (v i) ∈ Finsupp.supported R R (⋃ j, S j) := this trivial
rw [v.repr_self, Finsupp.mem_supported, Finsupp.support_single_ne_zero _ one_ne_zero] at this
· subst b
rcases mem_iUnion.1 (this (Finset.mem_singleton_self _)) with ⟨j, hj⟩
exact mem_iUnion.2 ⟨j, (mem_image _ _ _).2 ⟨i, hj, rfl⟩⟩
refine le_of_not_lt fun IJ => ?_
suffices #(⋃ j, S' j) < #(range v) by exact not_le_of_lt this ⟨Set.embeddingOfSubset _ _ hs⟩
refine lt_of_le_of_lt (le_trans Cardinal.mk_iUnion_le_sum_mk
(Cardinal.sum_le_sum _ (fun _ => ℵ₀) ?_)) ?_
· exact fun j => (Cardinal.lt_aleph0_of_finite _).le
· simpa
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 177 | 191 | theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by |
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 214 | 220 | theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by |
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
#align linear_independent_le_span' linearIndependent_le_span'
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 228 | 232 | theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by |
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
#align linear_independent_le_span' linearIndependent_le_span'
theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
#align linear_independent_le_span linearIndependent_le_span
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 236 | 238 | theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by |
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
#align linear_independent_le_span' linearIndependent_le_span'
theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
#align linear_independent_le_span linearIndependent_le_span
theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
#align linear_independent_le_span_finset linearIndependent_le_span_finset
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 244 | 258 | theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι := by |
classical
by_contra h
rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h
let Φ := fun k : κ => (b.repr (v k)).support
obtain ⟨s, w : Infinite ↑(Φ ⁻¹' {s})⟩ := Cardinal.exists_infinite_fiber Φ h (by infer_instance)
let v' := fun k : Φ ⁻¹' {s} => v k
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have w' : Finite (Φ ⁻¹' {s}) := by
apply i'.finite_of_le_span_finite v' (s.image b)
rintro m ⟨⟨p, ⟨rfl⟩⟩, rfl⟩
simp only [SetLike.mem_coe, Subtype.coe_mk, Finset.coe_image]
apply Basis.mem_span_repr_support
exact w.false
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
#align linear_independent_le_span' linearIndependent_le_span'
theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
#align linear_independent_le_span linearIndependent_le_span
theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
#align linear_independent_le_span_finset linearIndependent_le_span_finset
theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι := by
classical
by_contra h
rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h
let Φ := fun k : κ => (b.repr (v k)).support
obtain ⟨s, w : Infinite ↑(Φ ⁻¹' {s})⟩ := Cardinal.exists_infinite_fiber Φ h (by infer_instance)
let v' := fun k : Φ ⁻¹' {s} => v k
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have w' : Finite (Φ ⁻¹' {s}) := by
apply i'.finite_of_le_span_finite v' (s.image b)
rintro m ⟨⟨p, ⟨rfl⟩⟩, rfl⟩
simp only [SetLike.mem_coe, Subtype.coe_mk, Finset.coe_image]
apply Basis.mem_span_repr_support
exact w.false
#align linear_independent_le_infinite_basis linearIndependent_le_infinite_basis
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 266 | 276 | theorem linearIndependent_le_basis {ι : Type w} (b : Basis ι R M) {κ : Type w} (v : κ → M)
(i : LinearIndependent R v) : #κ ≤ #ι := by |
classical
-- We split into cases depending on whether `ι` is infinite.
cases fintypeOrInfinite ι
· rw [Cardinal.mk_fintype ι] -- When `ι` is finite, we have `linearIndependent_le_span`,
haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R
rw [Fintype.card_congr (Equiv.ofInjective b b.injective)]
exact linearIndependent_le_span v i (range b) b.span_eq
· -- and otherwise we have `linearIndependent_le_infinite_basis`.
exact linearIndependent_le_infinite_basis b v i
| 1,672 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type w} {ι' : Type w'}
open Cardinal Basis Submodule Function Set
attribute [local instance] nontrivial_of_invariantBasisNumber
section StrongRankCondition
variable [StrongRankCondition R]
open Submodule
-- An auxiliary lemma for `linearIndependent_le_span'`,
-- with the additional assumption that the linearly independent family is finite.
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_of_injective' R
· apply Finsupp.total
exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
· intro f g h
apply_fun Finsupp.total w M R (↑) at h
simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
rw [← sub_eq_zero, ← LinearMap.map_sub] at h
exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
#align linear_independent_le_span_aux' linearIndependent_le_span_aux'
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι :=
letI := Fintype.ofFinite w
Fintype.finite <| fintypeOfFinsetCardLe (Fintype.card w) fun t => by
let v' := fun x : (t : Set ι) => v x
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have s' : range v' ≤ span R w := (range_comp_subset_range _ _).trans s
simpa using linearIndependent_le_span_aux' v' i' w s'
#align linear_independent_fintype_of_le_span_fintype LinearIndependent.finite_of_le_span_finite
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
#align linear_independent_le_span' linearIndependent_le_span'
theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
#align linear_independent_le_span linearIndependent_le_span
theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
#align linear_independent_le_span_finset linearIndependent_le_span_finset
theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι := by
classical
by_contra h
rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h
let Φ := fun k : κ => (b.repr (v k)).support
obtain ⟨s, w : Infinite ↑(Φ ⁻¹' {s})⟩ := Cardinal.exists_infinite_fiber Φ h (by infer_instance)
let v' := fun k : Φ ⁻¹' {s} => v k
have i' : LinearIndependent R v' := i.comp _ Subtype.val_injective
have w' : Finite (Φ ⁻¹' {s}) := by
apply i'.finite_of_le_span_finite v' (s.image b)
rintro m ⟨⟨p, ⟨rfl⟩⟩, rfl⟩
simp only [SetLike.mem_coe, Subtype.coe_mk, Finset.coe_image]
apply Basis.mem_span_repr_support
exact w.false
#align linear_independent_le_infinite_basis linearIndependent_le_infinite_basis
theorem linearIndependent_le_basis {ι : Type w} (b : Basis ι R M) {κ : Type w} (v : κ → M)
(i : LinearIndependent R v) : #κ ≤ #ι := by
classical
-- We split into cases depending on whether `ι` is infinite.
cases fintypeOrInfinite ι
· rw [Cardinal.mk_fintype ι] -- When `ι` is finite, we have `linearIndependent_le_span`,
haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R
rw [Fintype.card_congr (Equiv.ofInjective b b.injective)]
exact linearIndependent_le_span v i (range b) b.span_eq
· -- and otherwise we have `linearIndependent_le_infinite_basis`.
exact linearIndependent_le_infinite_basis b v i
#align linear_independent_le_basis linearIndependent_le_basis
theorem Basis.card_le_card_of_linearIndependent_aux {R : Type*} [Ring R] [StrongRankCondition R]
(n : ℕ) {m : ℕ} (v : Fin m → Fin n → R) : LinearIndependent R v → m ≤ n := fun h => by
simpa using linearIndependent_le_basis (Pi.basisFun R (Fin n)) v h
#align basis.card_le_card_of_linear_independent_aux Basis.card_le_card_of_linearIndependent_aux
-- When the basis is not infinite this need not be true!
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 294 | 299 | theorem maximal_linearIndependent_eq_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι]
{κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) : #κ = #ι := by |
apply le_antisymm
· exact linearIndependent_le_basis b v i
· haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R
exact infinite_basis_le_maximal_linearIndependent b v i m
| 1,672 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.FreeAlgebra
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.free_algebra from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
universe u v
namespace FreeAlgebra
variable (R : Type u) (X : Type v)
section
variable [CommSemiring R]
-- @[simps]
noncomputable def basisFreeMonoid : Basis (FreeMonoid X) R (FreeAlgebra R X) :=
Finsupp.basisSingleOne.map (equivMonoidAlgebraFreeMonoid (R := R) (X := X)).symm.toLinearEquiv
#align free_algebra.basis_free_monoid FreeAlgebra.basisFreeMonoid
instance : Module.Free R (FreeAlgebra R X) :=
have : Module.Free R (MonoidAlgebra R (FreeMonoid X)) := Module.Free.finsupp _ _ _
Module.Free.of_equiv (equivMonoidAlgebraFreeMonoid (R := R) (X := X)).symm.toLinearEquiv
end
| Mathlib/LinearAlgebra/FreeAlgebra.lean | 44 | 47 | theorem rank_eq [CommRing R] [Nontrivial R] :
Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u} (Cardinal.mk (List X)) := by |
rw [← (Basis.mk_eq_rank'.{_,_,_,u} (basisFreeMonoid R X)).trans (Cardinal.lift_id _),
Cardinal.lift_umax'.{v,u}, FreeMonoid]
| 1,673 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type*} (b : Basis ι R M)
open Submodule.IsPrincipal Submodule
| Mathlib/LinearAlgebra/FreeModule/PID.lean | 59 | 69 | theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
{ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
(hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine b.ext_elem fun i ↦ ?_
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
exact
(Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
⟨x, hx, rfl⟩
| 1,674 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {ι : Type*} (b : Basis ι R M)
open Submodule.IsPrincipal Submodule
theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
{ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
(hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ := by
rw [Submodule.eq_bot_iff]
intro x hx
refine b.ext_elem fun i ↦ ?_
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
exact
(Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
⟨x, hx, rfl⟩
#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
| Mathlib/LinearAlgebra/FreeModule/PID.lean | 72 | 81 | theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
(hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
[(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine (mk_eq_zero _ _).mp (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i ↦ ?_)
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
refine (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _ ?_
exact (LinearMap.mem_submoduleImage_of_le hNO).mpr ⟨x, hx, rfl⟩
| 1,674 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section IsDomain
variable {ι : Type*} {R : Type*} [CommRing R] [IsDomain R]
variable {M : Type*} [AddCommGroup M] [Module R M] {b : ι → M}
open Submodule.IsPrincipal Set Submodule
| Mathlib/LinearAlgebra/FreeModule/PID.lean | 93 | 98 | theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
x ∣ generator I ↔ I = Ideal.span {x} := by |
conv_rhs => rw [← span_singleton_generator I]
rw [Ideal.submodule_span_eq, Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated,
← mem_iff_generator_dvd]
exact ⟨fun h ↦ ⟨hx, h⟩, fun h ↦ h.2⟩
| 1,674 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 45 | 50 | theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by |
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 54 | 59 | theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by |
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_comp_to_free_abelian_group FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 63 | 68 | theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by |
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_comp_to_free_abelian_group FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_comp_to_finsupp Finsupp.toFreeAbelianGroup_comp_toFinsupp
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 72 | 74 | theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by |
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_comp_to_free_abelian_group FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_comp_to_finsupp Finsupp.toFreeAbelianGroup_comp_toFinsupp
@[simp]
theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_to_finsupp Finsupp.toFreeAbelianGroup_toFinsupp
namespace FreeAbelianGroup
open Finsupp
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 82 | 83 | theorem toFinsupp_of (x : X) : toFinsupp (of x) = Finsupp.single x 1 := by |
simp only [toFinsupp, lift.of]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_comp_to_free_abelian_group FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_comp_to_finsupp Finsupp.toFreeAbelianGroup_comp_toFinsupp
@[simp]
theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_to_finsupp Finsupp.toFreeAbelianGroup_toFinsupp
namespace FreeAbelianGroup
open Finsupp
@[simp]
theorem toFinsupp_of (x : X) : toFinsupp (of x) = Finsupp.single x 1 := by
simp only [toFinsupp, lift.of]
#align free_abelian_group.to_finsupp_of FreeAbelianGroup.toFinsupp_of
@[simp]
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 87 | 89 | theorem toFinsupp_toFreeAbelianGroup (f : X →₀ ℤ) :
FreeAbelianGroup.toFinsupp (Finsupp.toFreeAbelianGroup f) = f := by |
rw [← AddMonoidHom.comp_apply, toFinsupp_comp_toFreeAbelianGroup, AddMonoidHom.id_apply]
| 1,675 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
noncomputable section
variable {X : Type*}
def FreeAbelianGroup.toFinsupp : FreeAbelianGroup X →+ X →₀ ℤ :=
FreeAbelianGroup.lift fun x => Finsupp.single x (1 : ℤ)
#align free_abelian_group.to_finsupp FreeAbelianGroup.toFinsupp
def Finsupp.toFreeAbelianGroup : (X →₀ ℤ) →+ FreeAbelianGroup X :=
Finsupp.liftAddHom fun x => (smulAddHom ℤ (FreeAbelianGroup X)).flip (FreeAbelianGroup.of x)
#align finsupp.to_free_abelian_group Finsupp.toFreeAbelianGroup
open Finsupp FreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
#align finsupp.to_free_abelian_group_comp_single_add_hom Finsupp.toFreeAbelianGroup_comp_singleAddHom
@[simp]
theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_comp_to_free_abelian_group FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup
@[simp]
theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_comp_to_finsupp Finsupp.toFreeAbelianGroup_comp_toFinsupp
@[simp]
theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
#align finsupp.to_free_abelian_group_to_finsupp Finsupp.toFreeAbelianGroup_toFinsupp
namespace FreeAbelianGroup
open Finsupp
@[simp]
theorem toFinsupp_of (x : X) : toFinsupp (of x) = Finsupp.single x 1 := by
simp only [toFinsupp, lift.of]
#align free_abelian_group.to_finsupp_of FreeAbelianGroup.toFinsupp_of
@[simp]
theorem toFinsupp_toFreeAbelianGroup (f : X →₀ ℤ) :
FreeAbelianGroup.toFinsupp (Finsupp.toFreeAbelianGroup f) = f := by
rw [← AddMonoidHom.comp_apply, toFinsupp_comp_toFreeAbelianGroup, AddMonoidHom.id_apply]
#align free_abelian_group.to_finsupp_to_free_abelian_group FreeAbelianGroup.toFinsupp_toFreeAbelianGroup
variable (X)
@[simps!]
def equivFinsupp : FreeAbelianGroup X ≃+ (X →₀ ℤ) where
toFun := toFinsupp
invFun := toFreeAbelianGroup
left_inv := toFreeAbelianGroup_toFinsupp
right_inv := toFinsupp_toFreeAbelianGroup
map_add' := toFinsupp.map_add
#align free_abelian_group.equiv_finsupp FreeAbelianGroup.equivFinsupp
noncomputable def basis (α : Type*) : Basis α ℤ (FreeAbelianGroup α) :=
⟨(FreeAbelianGroup.equivFinsupp α).toIntLinearEquiv⟩
#align free_abelian_group.basis FreeAbelianGroup.basis
def Equiv.ofFreeAbelianGroupLinearEquiv {α β : Type*}
(e : FreeAbelianGroup α ≃ₗ[ℤ] FreeAbelianGroup β) : α ≃ β :=
let t : Basis α ℤ (FreeAbelianGroup β) := (FreeAbelianGroup.basis α).map e
t.indexEquiv <| FreeAbelianGroup.basis _
#align free_abelian_group.equiv.of_free_abelian_group_linear_equiv FreeAbelianGroup.Equiv.ofFreeAbelianGroupLinearEquiv
def Equiv.ofFreeAbelianGroupEquiv {α β : Type*} (e : FreeAbelianGroup α ≃+ FreeAbelianGroup β) :
α ≃ β :=
Equiv.ofFreeAbelianGroupLinearEquiv e.toIntLinearEquiv
#align free_abelian_group.equiv.of_free_abelian_group_equiv FreeAbelianGroup.Equiv.ofFreeAbelianGroupEquiv
def Equiv.ofFreeGroupEquiv {α β : Type*} (e : FreeGroup α ≃* FreeGroup β) : α ≃ β :=
Equiv.ofFreeAbelianGroupEquiv (MulEquiv.toAdditive e.abelianizationCongr)
#align free_abelian_group.equiv.of_free_group_equiv FreeAbelianGroup.Equiv.ofFreeGroupEquiv
open IsFreeGroup
def Equiv.ofIsFreeGroupEquiv {G H : Type*} [Group G] [Group H] [IsFreeGroup G] [IsFreeGroup H]
(e : G ≃* H) : Generators G ≃ Generators H :=
Equiv.ofFreeGroupEquiv <| MulEquiv.trans (toFreeGroup G).symm <| MulEquiv.trans e <| toFreeGroup H
#align free_abelian_group.equiv.of_is_free_group_equiv FreeAbelianGroup.Equiv.ofIsFreeGroupEquiv
variable {X}
def coeff (x : X) : FreeAbelianGroup X →+ ℤ :=
(Finsupp.applyAddHom x).comp toFinsupp
#align free_abelian_group.coeff FreeAbelianGroup.coeff
def support (a : FreeAbelianGroup X) : Finset X :=
a.toFinsupp.support
#align free_abelian_group.support FreeAbelianGroup.support
| Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 149 | 151 | theorem mem_support_iff (x : X) (a : FreeAbelianGroup X) : x ∈ a.support ↔ coeff x a ≠ 0 := by |
rw [support, Finsupp.mem_support_iff]
exact Iff.rfl
| 1,675 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
namespace Ideal
section TorsionOf
variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
@[simps!]
def torsionOf (x : M) : Ideal R :=
-- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
LinearMap.ker (LinearMap.toSpanSingleton R M x)
#align ideal.torsion_of Ideal.torsionOf
@[simp]
| Mathlib/Algebra/Module/Torsion.lean | 79 | 79 | theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by | simp [torsionOf]
| 1,676 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
namespace Ideal
section TorsionOf
variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
@[simps!]
def torsionOf (x : M) : Ideal R :=
-- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
LinearMap.ker (LinearMap.toSpanSingleton R M x)
#align ideal.torsion_of Ideal.torsionOf
@[simp]
theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsionOf]
#align ideal.torsion_of_zero Ideal.torsionOf_zero
variable {R M}
@[simp]
theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
Iff.rfl
#align ideal.mem_torsion_of_iff Ideal.mem_torsionOf_iff
variable (R)
@[simp]
| Mathlib/Algebra/Module/Torsion.lean | 92 | 95 | theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 := by |
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top
| 1,676 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
namespace Ideal
section TorsionOf
variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
@[simps!]
def torsionOf (x : M) : Ideal R :=
-- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
LinearMap.ker (LinearMap.toSpanSingleton R M x)
#align ideal.torsion_of Ideal.torsionOf
@[simp]
theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsionOf]
#align ideal.torsion_of_zero Ideal.torsionOf_zero
variable {R M}
@[simp]
theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
Iff.rfl
#align ideal.mem_torsion_of_iff Ideal.mem_torsionOf_iff
variable (R)
@[simp]
theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 := by
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top
#align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iff
@[simp]
| Mathlib/Algebra/Module/Torsion.lean | 99 | 105 | theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
torsionOf R M m = ⊥ ↔ m ≠ 0 := by |
refine ⟨fun h contra => ?_, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => ?_⟩
· rw [contra, torsionOf_zero] at h
exact bot_ne_top.symm h
· rw [mem_torsionOf_iff, smul_eq_zero] at hr
tauto
| 1,676 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
namespace Ideal
section TorsionOf
variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
@[simps!]
def torsionOf (x : M) : Ideal R :=
-- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
LinearMap.ker (LinearMap.toSpanSingleton R M x)
#align ideal.torsion_of Ideal.torsionOf
@[simp]
theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsionOf]
#align ideal.torsion_of_zero Ideal.torsionOf_zero
variable {R M}
@[simp]
theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
Iff.rfl
#align ideal.mem_torsion_of_iff Ideal.mem_torsionOf_iff
variable (R)
@[simp]
theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 := by
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top
#align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iff
@[simp]
theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
torsionOf R M m = ⊥ ↔ m ≠ 0 := by
refine ⟨fun h contra => ?_, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => ?_⟩
· rw [contra, torsionOf_zero] at h
exact bot_ne_top.symm h
· rw [mem_torsionOf_iff, smul_eq_zero] at hr
tauto
#align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
| Mathlib/Algebra/Module/Torsion.lean | 110 | 123 | theorem CompleteLattice.Independent.linear_independent' {ι R M : Type*} {v : ι → M} [Ring R]
[AddCommGroup M] [Module R M] (hv : CompleteLattice.Independent fun i => R ∙ v i)
(h_ne_zero : ∀ i, Ideal.torsionOf R M (v i) = ⊥) : LinearIndependent R v := by |
refine linearIndependent_iff_not_smul_mem_span.mpr fun i r hi => ?_
replace hv := CompleteLattice.independent_def.mp hv i
simp only [iSup_subtype', ← Submodule.span_range_eq_iSup (ι := Subtype _), disjoint_iff] at hv
have : r • v i ∈ (⊥ : Submodule R M) := by
rw [← hv, Submodule.mem_inf]
refine ⟨Submodule.mem_span_singleton.mpr ⟨r, rfl⟩, ?_⟩
convert hi
ext
simp
rw [← Submodule.mem_bot R, ← h_ne_zero i]
simpa using this
| 1,676 |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap_ideal_mk {m n : ℕ} (hmn : m ≤ n) (x : R) :
transitionMap I R hmn (Ideal.Quotient.mk (I ^ n • ⊤ : Ideal R) x) =
Ideal.Quotient.mk (I ^ m • ⊤ : Ideal R) x :=
rfl
@[local simp]
theorem transitionMap_map_one {m n : ℕ} (hmn : m ≤ n) : transitionMap I R hmn 1 = 1 :=
rfl
@[local simp]
theorem transitionMap_map_mul {m n : ℕ} (hmn : m ≤ n) (x y : R ⧸ (I ^ n • ⊤ : Ideal R)) :
transitionMap I R hmn (x * y) = transitionMap I R hmn x * transitionMap I R hmn y :=
Quotient.inductionOn₂' x y (fun _ _ ↦ rfl)
def transitionMapₐ {m n : ℕ} (hmn : m ≤ n) :
R ⧸ (I ^ n • ⊤ : Ideal R) →ₐ[R] R ⧸ (I ^ m • ⊤ : Ideal R) :=
AlgHom.ofLinearMap (transitionMap I R hmn) rfl (transitionMap_map_mul I hmn)
def subalgebra : Subalgebra R (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Submodule.toSubalgebra (submodule I R) (fun _ ↦ by simp)
(fun x y hx hy m n hmn ↦ by simp [hx hmn, hy hmn])
def subring : Subring (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Subalgebra.toSubring (subalgebra I)
instance : CommRing (AdicCompletion I R) :=
inferInstanceAs <| CommRing (subring I)
instance : Algebra R (AdicCompletion I R) :=
inferInstanceAs <| Algebra R (subalgebra I)
@[simp]
theorem val_one (n : ℕ) : (1 : AdicCompletion I R).val n = 1 :=
rfl
@[simp]
theorem val_mul (n : ℕ) (x y : AdicCompletion I R) : (x * y).val n = x.val n * y.val n :=
rfl
def evalₐ (n : ℕ) : AdicCompletion I R →ₐ[R] R ⧸ I ^ n :=
have h : (I ^ n • ⊤ : Ideal R) = I ^ n := by ext x; simp
AlgHom.comp
(Ideal.quotientEquivAlgOfEq R h)
(AlgHom.ofLinearMap (eval I R n) rfl (fun _ _ ↦ rfl))
@[simp]
| Mathlib/RingTheory/AdicCompletion/Algebra.lean | 87 | 89 | theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by |
simp [evalₐ]
| 1,677 |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap_ideal_mk {m n : ℕ} (hmn : m ≤ n) (x : R) :
transitionMap I R hmn (Ideal.Quotient.mk (I ^ n • ⊤ : Ideal R) x) =
Ideal.Quotient.mk (I ^ m • ⊤ : Ideal R) x :=
rfl
@[local simp]
theorem transitionMap_map_one {m n : ℕ} (hmn : m ≤ n) : transitionMap I R hmn 1 = 1 :=
rfl
@[local simp]
theorem transitionMap_map_mul {m n : ℕ} (hmn : m ≤ n) (x y : R ⧸ (I ^ n • ⊤ : Ideal R)) :
transitionMap I R hmn (x * y) = transitionMap I R hmn x * transitionMap I R hmn y :=
Quotient.inductionOn₂' x y (fun _ _ ↦ rfl)
def transitionMapₐ {m n : ℕ} (hmn : m ≤ n) :
R ⧸ (I ^ n • ⊤ : Ideal R) →ₐ[R] R ⧸ (I ^ m • ⊤ : Ideal R) :=
AlgHom.ofLinearMap (transitionMap I R hmn) rfl (transitionMap_map_mul I hmn)
def subalgebra : Subalgebra R (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Submodule.toSubalgebra (submodule I R) (fun _ ↦ by simp)
(fun x y hx hy m n hmn ↦ by simp [hx hmn, hy hmn])
def subring : Subring (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Subalgebra.toSubring (subalgebra I)
instance : CommRing (AdicCompletion I R) :=
inferInstanceAs <| CommRing (subring I)
instance : Algebra R (AdicCompletion I R) :=
inferInstanceAs <| Algebra R (subalgebra I)
@[simp]
theorem val_one (n : ℕ) : (1 : AdicCompletion I R).val n = 1 :=
rfl
@[simp]
theorem val_mul (n : ℕ) (x y : AdicCompletion I R) : (x * y).val n = x.val n * y.val n :=
rfl
def evalₐ (n : ℕ) : AdicCompletion I R →ₐ[R] R ⧸ I ^ n :=
have h : (I ^ n • ⊤ : Ideal R) = I ^ n := by ext x; simp
AlgHom.comp
(Ideal.quotientEquivAlgOfEq R h)
(AlgHom.ofLinearMap (eval I R n) rfl (fun _ _ ↦ rfl))
@[simp]
theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by
simp [evalₐ]
def AdicCauchySequence.subalgebra : Subalgebra R (ℕ → R) :=
Submodule.toSubalgebra (AdicCauchySequence.submodule I R)
(fun {m n} _ ↦ by simp; rfl)
(fun x y hx hy {m n} hmn ↦ by
simp only [Pi.mul_apply]
exact SModEq.mul (hx hmn) (hy hmn))
def AdicCauchySequence.subring : Subring (ℕ → R) :=
Subalgebra.toSubring (AdicCauchySequence.subalgebra I)
instance : CommRing (AdicCauchySequence I R) :=
inferInstanceAs <| CommRing (AdicCauchySequence.subring I)
instance : Algebra R (AdicCauchySequence I R) :=
inferInstanceAs <| Algebra R (AdicCauchySequence.subalgebra I)
@[simp]
theorem one_apply (n : ℕ) : (1 : AdicCauchySequence I R) n = 1 :=
rfl
@[simp]
theorem mul_apply (n : ℕ) (f g : AdicCauchySequence I R) : (f * g) n = f n * g n :=
rfl
@[simps!]
def mkₐ : AdicCauchySequence I R →ₐ[R] AdicCompletion I R :=
AlgHom.ofLinearMap (mk I R) rfl (fun _ _ ↦ rfl)
@[simp]
| Mathlib/RingTheory/AdicCompletion/Algebra.lean | 123 | 125 | theorem evalₐ_mkₐ (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mkₐ I x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by |
simp [mkₐ]
| 1,677 |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap_ideal_mk {m n : ℕ} (hmn : m ≤ n) (x : R) :
transitionMap I R hmn (Ideal.Quotient.mk (I ^ n • ⊤ : Ideal R) x) =
Ideal.Quotient.mk (I ^ m • ⊤ : Ideal R) x :=
rfl
@[local simp]
theorem transitionMap_map_one {m n : ℕ} (hmn : m ≤ n) : transitionMap I R hmn 1 = 1 :=
rfl
@[local simp]
theorem transitionMap_map_mul {m n : ℕ} (hmn : m ≤ n) (x y : R ⧸ (I ^ n • ⊤ : Ideal R)) :
transitionMap I R hmn (x * y) = transitionMap I R hmn x * transitionMap I R hmn y :=
Quotient.inductionOn₂' x y (fun _ _ ↦ rfl)
def transitionMapₐ {m n : ℕ} (hmn : m ≤ n) :
R ⧸ (I ^ n • ⊤ : Ideal R) →ₐ[R] R ⧸ (I ^ m • ⊤ : Ideal R) :=
AlgHom.ofLinearMap (transitionMap I R hmn) rfl (transitionMap_map_mul I hmn)
def subalgebra : Subalgebra R (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Submodule.toSubalgebra (submodule I R) (fun _ ↦ by simp)
(fun x y hx hy m n hmn ↦ by simp [hx hmn, hy hmn])
def subring : Subring (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Subalgebra.toSubring (subalgebra I)
instance : CommRing (AdicCompletion I R) :=
inferInstanceAs <| CommRing (subring I)
instance : Algebra R (AdicCompletion I R) :=
inferInstanceAs <| Algebra R (subalgebra I)
@[simp]
theorem val_one (n : ℕ) : (1 : AdicCompletion I R).val n = 1 :=
rfl
@[simp]
theorem val_mul (n : ℕ) (x y : AdicCompletion I R) : (x * y).val n = x.val n * y.val n :=
rfl
def evalₐ (n : ℕ) : AdicCompletion I R →ₐ[R] R ⧸ I ^ n :=
have h : (I ^ n • ⊤ : Ideal R) = I ^ n := by ext x; simp
AlgHom.comp
(Ideal.quotientEquivAlgOfEq R h)
(AlgHom.ofLinearMap (eval I R n) rfl (fun _ _ ↦ rfl))
@[simp]
theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by
simp [evalₐ]
def AdicCauchySequence.subalgebra : Subalgebra R (ℕ → R) :=
Submodule.toSubalgebra (AdicCauchySequence.submodule I R)
(fun {m n} _ ↦ by simp; rfl)
(fun x y hx hy {m n} hmn ↦ by
simp only [Pi.mul_apply]
exact SModEq.mul (hx hmn) (hy hmn))
def AdicCauchySequence.subring : Subring (ℕ → R) :=
Subalgebra.toSubring (AdicCauchySequence.subalgebra I)
instance : CommRing (AdicCauchySequence I R) :=
inferInstanceAs <| CommRing (AdicCauchySequence.subring I)
instance : Algebra R (AdicCauchySequence I R) :=
inferInstanceAs <| Algebra R (AdicCauchySequence.subalgebra I)
@[simp]
theorem one_apply (n : ℕ) : (1 : AdicCauchySequence I R) n = 1 :=
rfl
@[simp]
theorem mul_apply (n : ℕ) (f g : AdicCauchySequence I R) : (f * g) n = f n * g n :=
rfl
@[simps!]
def mkₐ : AdicCauchySequence I R →ₐ[R] AdicCompletion I R :=
AlgHom.ofLinearMap (mk I R) rfl (fun _ _ ↦ rfl)
@[simp]
theorem evalₐ_mkₐ (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mkₐ I x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by
simp [mkₐ]
| Mathlib/RingTheory/AdicCompletion/Algebra.lean | 127 | 131 | theorem Ideal.mk_eq_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R) :
Ideal.Quotient.mk (I ^ m) (r.val n) = Ideal.Quotient.mk (I ^ m) (r.val m) := by |
have h : I ^ m = I ^ m • ⊤ := by simp
rw [h, ← Ideal.Quotient.mk_eq_mk, ← Ideal.Quotient.mk_eq_mk]
exact (r.property hmn).symm
| 1,677 |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap_ideal_mk {m n : ℕ} (hmn : m ≤ n) (x : R) :
transitionMap I R hmn (Ideal.Quotient.mk (I ^ n • ⊤ : Ideal R) x) =
Ideal.Quotient.mk (I ^ m • ⊤ : Ideal R) x :=
rfl
@[local simp]
theorem transitionMap_map_one {m n : ℕ} (hmn : m ≤ n) : transitionMap I R hmn 1 = 1 :=
rfl
@[local simp]
theorem transitionMap_map_mul {m n : ℕ} (hmn : m ≤ n) (x y : R ⧸ (I ^ n • ⊤ : Ideal R)) :
transitionMap I R hmn (x * y) = transitionMap I R hmn x * transitionMap I R hmn y :=
Quotient.inductionOn₂' x y (fun _ _ ↦ rfl)
def transitionMapₐ {m n : ℕ} (hmn : m ≤ n) :
R ⧸ (I ^ n • ⊤ : Ideal R) →ₐ[R] R ⧸ (I ^ m • ⊤ : Ideal R) :=
AlgHom.ofLinearMap (transitionMap I R hmn) rfl (transitionMap_map_mul I hmn)
def subalgebra : Subalgebra R (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Submodule.toSubalgebra (submodule I R) (fun _ ↦ by simp)
(fun x y hx hy m n hmn ↦ by simp [hx hmn, hy hmn])
def subring : Subring (∀ n, R ⧸ (I ^ n • ⊤ : Ideal R)) :=
Subalgebra.toSubring (subalgebra I)
instance : CommRing (AdicCompletion I R) :=
inferInstanceAs <| CommRing (subring I)
instance : Algebra R (AdicCompletion I R) :=
inferInstanceAs <| Algebra R (subalgebra I)
@[simp]
theorem val_one (n : ℕ) : (1 : AdicCompletion I R).val n = 1 :=
rfl
@[simp]
theorem val_mul (n : ℕ) (x y : AdicCompletion I R) : (x * y).val n = x.val n * y.val n :=
rfl
def evalₐ (n : ℕ) : AdicCompletion I R →ₐ[R] R ⧸ I ^ n :=
have h : (I ^ n • ⊤ : Ideal R) = I ^ n := by ext x; simp
AlgHom.comp
(Ideal.quotientEquivAlgOfEq R h)
(AlgHom.ofLinearMap (eval I R n) rfl (fun _ _ ↦ rfl))
@[simp]
theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by
simp [evalₐ]
def AdicCauchySequence.subalgebra : Subalgebra R (ℕ → R) :=
Submodule.toSubalgebra (AdicCauchySequence.submodule I R)
(fun {m n} _ ↦ by simp; rfl)
(fun x y hx hy {m n} hmn ↦ by
simp only [Pi.mul_apply]
exact SModEq.mul (hx hmn) (hy hmn))
def AdicCauchySequence.subring : Subring (ℕ → R) :=
Subalgebra.toSubring (AdicCauchySequence.subalgebra I)
instance : CommRing (AdicCauchySequence I R) :=
inferInstanceAs <| CommRing (AdicCauchySequence.subring I)
instance : Algebra R (AdicCauchySequence I R) :=
inferInstanceAs <| Algebra R (AdicCauchySequence.subalgebra I)
@[simp]
theorem one_apply (n : ℕ) : (1 : AdicCauchySequence I R) n = 1 :=
rfl
@[simp]
theorem mul_apply (n : ℕ) (f g : AdicCauchySequence I R) : (f * g) n = f n * g n :=
rfl
@[simps!]
def mkₐ : AdicCauchySequence I R →ₐ[R] AdicCompletion I R :=
AlgHom.ofLinearMap (mk I R) rfl (fun _ _ ↦ rfl)
@[simp]
theorem evalₐ_mkₐ (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mkₐ I x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by
simp [mkₐ]
theorem Ideal.mk_eq_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R) :
Ideal.Quotient.mk (I ^ m) (r.val n) = Ideal.Quotient.mk (I ^ m) (r.val m) := by
have h : I ^ m = I ^ m • ⊤ := by simp
rw [h, ← Ideal.Quotient.mk_eq_mk, ← Ideal.Quotient.mk_eq_mk]
exact (r.property hmn).symm
| Mathlib/RingTheory/AdicCompletion/Algebra.lean | 133 | 139 | theorem smul_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R)
(x : AdicCauchySequence I M) :
r.val n • Submodule.Quotient.mk (p := (I ^ m • ⊤ : Submodule R M)) (x.val n) =
r.val m • Submodule.Quotient.mk (p := (I ^ m • ⊤ : Submodule R M)) (x.val m) := by |
rw [← Submodule.Quotient.mk_smul, ← Module.Quotient.mk_smul_mk,
AdicCauchySequence.mk_eq_mk hmn, Ideal.mk_eq_mk I hmn, Module.Quotient.mk_smul_mk,
Submodule.Quotient.mk_smul]
| 1,677 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 52 | 60 | theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 65 | 75 | theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 80 | 84 | theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by |
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 88 | 93 | theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by |
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 98 | 116 | theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by |
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_singleton x),
hN'.sup (fg_span_singleton y), Set.insert_subset ?_ fun z hz ↦ ?_⟩
· exact ⟨⟨x, mem_sup_right (mem_span_singleton_self x)⟩ ⊗ₜ
⟨y, mem_sup_right (mem_span_singleton_self y)⟩, rfl⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h hz)
· obtain ⟨M₁', N₁', hM₁', hN₁', h₁⟩ := hx
obtain ⟨M₂', N₂', hM₂', hN₂', h₂⟩ := hy
refine ⟨_, _, hM₁'.sup hM₂', hN₁'.sup hN₂', Set.insert_subset (add_mem ?_ ?_) fun z hz ↦ ?_⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.mem_insert x s))
· exact range_mapIncl_mono le_sup_right le_sup_right (h₂ (Set.mem_insert y s))
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.subset_insert x s hz))
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_singleton x),
hN'.sup (fg_span_singleton y), Set.insert_subset ?_ fun z hz ↦ ?_⟩
· exact ⟨⟨x, mem_sup_right (mem_span_singleton_self x)⟩ ⊗ₜ
⟨y, mem_sup_right (mem_span_singleton_self y)⟩, rfl⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h hz)
· obtain ⟨M₁', N₁', hM₁', hN₁', h₁⟩ := hx
obtain ⟨M₂', N₂', hM₂', hN₂', h₂⟩ := hy
refine ⟨_, _, hM₁'.sup hM₂', hN₁'.sup hN₂', Set.insert_subset (add_mem ?_ ?_) fun z hz ↦ ?_⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.mem_insert x s))
· exact range_mapIncl_mono le_sup_right le_sup_right (h₂ (Set.mem_insert y s))
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.subset_insert x s hz))
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 121 | 126 | theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by |
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_singleton x),
hN'.sup (fg_span_singleton y), Set.insert_subset ?_ fun z hz ↦ ?_⟩
· exact ⟨⟨x, mem_sup_right (mem_span_singleton_self x)⟩ ⊗ₜ
⟨y, mem_sup_right (mem_span_singleton_self y)⟩, rfl⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h hz)
· obtain ⟨M₁', N₁', hM₁', hN₁', h₁⟩ := hx
obtain ⟨M₂', N₂', hM₂', hN₂', h₂⟩ := hy
refine ⟨_, _, hM₁'.sup hM₂', hN₁'.sup hN₂', Set.insert_subset (add_mem ?_ ?_) fun z hz ↦ ?_⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.mem_insert x s))
· exact range_mapIncl_mono le_sup_right le_sup_right (h₂ (Set.mem_insert y s))
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.subset_insert x s hz))
theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 131 | 136 | theorem exists_finite_submodule_right_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ N' : Submodule R N, Module.Finite R N' ∧ s ⊆ LinearMap.range (N'.subtype.lTensor M) := by |
obtain ⟨_, N', _, hfin, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨N', hfin, ?_⟩
rw [mapIncl, ← LinearMap.lTensor_comp_rTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 1,678 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace TensorProduct
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_singleton x),
hN'.sup (fg_span_singleton y), Set.insert_subset ?_ fun z hz ↦ ?_⟩
· exact ⟨⟨x, mem_sup_right (mem_span_singleton_self x)⟩ ⊗ₜ
⟨y, mem_sup_right (mem_span_singleton_self y)⟩, rfl⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h hz)
· obtain ⟨M₁', N₁', hM₁', hN₁', h₁⟩ := hx
obtain ⟨M₂', N₂', hM₂', hN₂', h₂⟩ := hy
refine ⟨_, _, hM₁'.sup hM₂', hN₁'.sup hN₂', Set.insert_subset (add_mem ?_ ?_) fun z hz ↦ ?_⟩
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.mem_insert x s))
· exact range_mapIncl_mono le_sup_right le_sup_right (h₂ (Set.mem_insert y s))
· exact range_mapIncl_mono le_sup_left le_sup_left (h₁ (Set.subset_insert x s hz))
theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
theorem exists_finite_submodule_right_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ N' : Submodule R N, Module.Finite R N' ∧ s ⊆ LinearMap.range (N'.subtype.lTensor M) := by
obtain ⟨_, N', _, hfin, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨N', hfin, ?_⟩
rw [mapIncl, ← LinearMap.lTensor_comp_rTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 140 | 152 | theorem exists_finite_submodule_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N) (hM : M' ≤ M₁) (hN : N' ≤ N₁),
Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (TensorProduct.map (inclusion hM) (inclusion hN)) := by |
obtain ⟨M', N', _, _, h⟩ := exists_finite_submodule_of_finite s hs
have hM := map_subtype_le M₁ M'
have hN := map_subtype_le N₁ N'
refine ⟨_, _, hM, hN, .map _ _, .map _ _, ?_⟩
rw [mapIncl,
show M'.subtype = inclusion hM ∘ₗ M₁.subtype.submoduleMap M' by ext; simp,
show N'.subtype = inclusion hN ∘ₗ N₁.subtype.submoduleMap N' by ext; simp,
map_comp] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 1,678 |
import Mathlib.RingTheory.Finiteness
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.finite.basic from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
universe u v w
variable (R : Type u) (M : Type v) (N : Type w)
namespace Module.Free
section CommRing
variable [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M]
variable [AddCommGroup N] [Module R N] [Module.Free R N]
variable {R}
| Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean | 53 | 58 | theorem _root_.Module.Finite.of_basis {R M ι : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
[_root_.Finite ι] (b : Basis ι R M) : Module.Finite R M := by |
cases nonempty_fintype ι
classical
refine ⟨⟨Finset.univ.image b, ?_⟩⟩
simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]
| 1,679 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
section Tower
variable (F : Type u) (K : Type v) (A : Type w)
variable [Ring F] [Ring K] [AddCommGroup A]
variable [Module F K] [Module K A] [Module F A] [IsScalarTower F K A]
variable [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A]
| Mathlib/LinearAlgebra/Dimension/Free.lean | 41 | 48 | theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by |
let b := Module.Free.chooseBasis F K
let c := Module.Free.chooseBasis K A
rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank,
← lift_umax.{w, v}, ← (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift,
lift_lift, lift_umax.{v, w}]
| 1,680 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
section Tower
variable (F : Type u) (K : Type v) (A : Type w)
variable [Ring F] [Ring K] [AddCommGroup A]
variable [Module F K] [Module K A] [Module F A] [IsScalarTower F K A]
variable [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A]
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by
let b := Module.Free.chooseBasis F K
let c := Module.Free.chooseBasis K A
rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank,
← lift_umax.{w, v}, ← (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift,
lift_lift, lift_umax.{v, w}]
#align lift_rank_mul_lift_rank lift_rank_mul_lift_rank
| Mathlib/LinearAlgebra/Dimension/Free.lean | 55 | 58 | theorem rank_mul_rank (A : Type v) [AddCommGroup A]
[Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
Module.rank F K * Module.rank K A = Module.rank F A := by |
convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]
| 1,680 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
section Tower
variable (F : Type u) (K : Type v) (A : Type w)
variable [Ring F] [Ring K] [AddCommGroup A]
variable [Module F K] [Module K A] [Module F A] [IsScalarTower F K A]
variable [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A]
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by
let b := Module.Free.chooseBasis F K
let c := Module.Free.chooseBasis K A
rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank,
← lift_umax.{w, v}, ← (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift,
lift_lift, lift_umax.{v, w}]
#align lift_rank_mul_lift_rank lift_rank_mul_lift_rank
theorem rank_mul_rank (A : Type v) [AddCommGroup A]
[Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
Module.rank F K * Module.rank K A = Module.rank F A := by
convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]
#align rank_mul_rank rank_mul_rank
| Mathlib/LinearAlgebra/Dimension/Free.lean | 63 | 66 | theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by |
simp_rw [finrank]
rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
lift_rank_mul_lift_rank, toNat_lift]
| 1,680 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
variable {R : Type u} {M M₁ : Type v} {M' : Type v'}
variable [Ring R] [StrongRankCondition R]
variable [AddCommGroup M] [Module R M] [Module.Free R M]
variable [AddCommGroup M'] [Module R M'] [Module.Free R M']
variable [AddCommGroup M₁] [Module R M₁] [Module.Free R M₁]
namespace Module.Free
variable (R M)
theorem rank_eq_card_chooseBasisIndex : Module.rank R M = #(ChooseBasisIndex R M) :=
(chooseBasis R M).mk_eq_rank''.symm
#align module.free.rank_eq_card_choose_basis_index Module.Free.rank_eq_card_chooseBasisIndex
| Mathlib/LinearAlgebra/Dimension/Free.lean | 88 | 90 | theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
finrank R M = Fintype.card (ChooseBasisIndex R M) := by |
simp [finrank, rank_eq_card_chooseBasisIndex]
| 1,680 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
variable {R : Type u} {M M₁ : Type v} {M' : Type v'}
variable [Ring R] [StrongRankCondition R]
variable [AddCommGroup M] [Module R M] [Module.Free R M]
variable [AddCommGroup M'] [Module R M'] [Module.Free R M']
variable [AddCommGroup M₁] [Module R M₁] [Module.Free R M₁]
open Module.Free
open Cardinal
| Mathlib/LinearAlgebra/Dimension/Free.lean | 111 | 118 | theorem nonempty_linearEquiv_of_lift_rank_eq
(cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
Nonempty (M ≃ₗ[R] M') := by |
obtain ⟨⟨α, B⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
obtain ⟨⟨β, B'⟩⟩ := Module.Free.exists_basis (R := R) (M := M')
have : Cardinal.lift.{v', v} #α = Cardinal.lift.{v, v'} #β := by
rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']
exact (Cardinal.lift_mk_eq.{v, v', 0}.1 this).map (B.equiv B')
| 1,680 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearIndependent
variable (hv : LinearIndependent R v) {u : ι ⊕ ι' → S.X₂}
(hw : LinearIndependent R (S.g ∘ u ∘ Sum.inr))
(hm : Mono S.f) (huv : u ∘ Sum.inl = S.f ∘ v)
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 44 | 49 | theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl)))
(span R (range (u ∘ Sum.inr))) := by |
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.range_coe, span_eq (LinearMap.range S.f), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
| 1,681 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearIndependent
variable (hv : LinearIndependent R v) {u : ι ⊕ ι' → S.X₂}
(hw : LinearIndependent R (S.g ∘ u ∘ Sum.inr))
(hm : Mono S.f) (huv : u ∘ Sum.inl = S.f ∘ v)
theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl)))
(span R (range (u ∘ Sum.inr))) := by
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.range_coe, span_eq (LinearMap.range S.f), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 62 | 68 | theorem linearIndependent_leftExact : LinearIndependent R u := by |
rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv
| 1,681 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearIndependent
variable (hv : LinearIndependent R v) {u : ι ⊕ ι' → S.X₂}
(hw : LinearIndependent R (S.g ∘ u ∘ Sum.inr))
(hm : Mono S.f) (huv : u ∘ Sum.inl = S.f ∘ v)
theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl)))
(span R (range (u ∘ Sum.inr))) := by
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.range_coe, span_eq (LinearMap.range S.f), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
theorem linearIndependent_leftExact : LinearIndependent R u := by
rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 72 | 78 | theorem linearIndependent_shortExact {w : ι' → S.X₃} (hw : LinearIndependent R w) :
LinearIndependent R (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w)) := by |
apply linearIndependent_leftExact hS'.exact hv _ hS'.mono_f rfl
dsimp
convert hw
ext
apply Function.rightInverse_invFun ((epi_iff_surjective _).mp hS'.epi_g)
| 1,681 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section Span
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 94 | 125 | theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl = S.f ∘ v)
(hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range (S.g ∘ u ∘ Sum.inr))) :
⊤ ≤ span R (range u) := by |
intro m _
have hgm : S.g m ∈ span R (range (S.g ∘ u ∘ Sum.inr)) := hw mem_top
rw [Finsupp.mem_span_range_iff_exists_finsupp] at hgm
obtain ⟨cm, hm⟩ := hgm
let m' : S.X₂ := Finsupp.sum cm fun j a ↦ a • (u (Sum.inr j))
have hsub : m - m' ∈ LinearMap.range S.f := by
rw [hS.moduleCat_range_eq_ker]
simp only [LinearMap.mem_ker, map_sub, sub_eq_zero]
rw [← hm, map_finsupp_sum]
simp only [Function.comp_apply, map_smul]
obtain ⟨n, hnm⟩ := hsub
have hn : n ∈ span R (range v) := hv mem_top
rw [Finsupp.mem_span_range_iff_exists_finsupp] at hn
obtain ⟨cn, hn⟩ := hn
rw [← hn, map_finsupp_sum] at hnm
rw [← sub_add_cancel m m', ← hnm,]
simp only [map_smul]
have hn' : (Finsupp.sum cn fun a b ↦ b • S.f (v a)) =
(Finsupp.sum cn fun a b ↦ b • u (Sum.inl a)) := by
congr; ext a b; rw [← Function.comp_apply (f := S.f), ← huv, Function.comp_apply]
rw [hn']
apply add_mem
· rw [Finsupp.mem_span_range_iff_exists_finsupp]
use cn.mapDomain (Sum.inl)
rw [Finsupp.sum_mapDomain_index_inj Sum.inl_injective]
· rw [Finsupp.mem_span_range_iff_exists_finsupp]
use cm.mapDomain (Sum.inr)
rw [Finsupp.sum_mapDomain_index_inj Sum.inr_injective]
| 1,681 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section Span
theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl = S.f ∘ v)
(hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range (S.g ∘ u ∘ Sum.inr))) :
⊤ ≤ span R (range u) := by
intro m _
have hgm : S.g m ∈ span R (range (S.g ∘ u ∘ Sum.inr)) := hw mem_top
rw [Finsupp.mem_span_range_iff_exists_finsupp] at hgm
obtain ⟨cm, hm⟩ := hgm
let m' : S.X₂ := Finsupp.sum cm fun j a ↦ a • (u (Sum.inr j))
have hsub : m - m' ∈ LinearMap.range S.f := by
rw [hS.moduleCat_range_eq_ker]
simp only [LinearMap.mem_ker, map_sub, sub_eq_zero]
rw [← hm, map_finsupp_sum]
simp only [Function.comp_apply, map_smul]
obtain ⟨n, hnm⟩ := hsub
have hn : n ∈ span R (range v) := hv mem_top
rw [Finsupp.mem_span_range_iff_exists_finsupp] at hn
obtain ⟨cn, hn⟩ := hn
rw [← hn, map_finsupp_sum] at hnm
rw [← sub_add_cancel m m', ← hnm,]
simp only [map_smul]
have hn' : (Finsupp.sum cn fun a b ↦ b • S.f (v a)) =
(Finsupp.sum cn fun a b ↦ b • u (Sum.inl a)) := by
congr; ext a b; rw [← Function.comp_apply (f := S.f), ← huv, Function.comp_apply]
rw [hn']
apply add_mem
· rw [Finsupp.mem_span_range_iff_exists_finsupp]
use cn.mapDomain (Sum.inl)
rw [Finsupp.sum_mapDomain_index_inj Sum.inl_injective]
· rw [Finsupp.mem_span_range_iff_exists_finsupp]
use cm.mapDomain (Sum.inr)
rw [Finsupp.sum_mapDomain_index_inj Sum.inr_injective]
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 129 | 138 | theorem span_rightExact {w : ι' → S.X₃} (hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range w)) (hE : Epi S.g) :
⊤ ≤ span R (range (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w))) := by |
refine span_exact hS ?_ hv ?_
· simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inl]
· convert hw
simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inr]
rw [ModuleCat.epi_iff_surjective] at hE
rw [← Function.comp.assoc, Function.RightInverse.comp_eq_id (Function.rightInverse_invFun hE),
Function.id_comp]
| 1,681 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Quotient
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 47 | 56 | theorem LinearIndependent.sum_elim_of_quotient
{M' : Submodule R M} {ι₁ ι₂} {f : ι₁ → M'} (hf : LinearIndependent R f) (g : ι₂ → M)
(hg : LinearIndependent R (Submodule.Quotient.mk (p := M') ∘ g)) :
LinearIndependent R (Sum.elim (f · : ι₁ → M) g) := by |
refine .sum_type (hf.map' M'.subtype M'.ker_subtype) (.of_comp M'.mkQ hg) ?_
refine disjoint_def.mpr fun x h₁ h₂ ↦ ?_
have : x ∈ M' := span_le.mpr (Set.range_subset_iff.mpr fun i ↦ (f i).prop) h₁
obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp h₂
simp_rw [← Quotient.mk_eq_zero, ← mkQ_apply, map_finsupp_sum, map_smul, mkQ_apply] at this
rw [linearIndependent_iff.mp hg _ this, Finsupp.sum_zero_index]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Quotient
theorem LinearIndependent.sum_elim_of_quotient
{M' : Submodule R M} {ι₁ ι₂} {f : ι₁ → M'} (hf : LinearIndependent R f) (g : ι₂ → M)
(hg : LinearIndependent R (Submodule.Quotient.mk (p := M') ∘ g)) :
LinearIndependent R (Sum.elim (f · : ι₁ → M) g) := by
refine .sum_type (hf.map' M'.subtype M'.ker_subtype) (.of_comp M'.mkQ hg) ?_
refine disjoint_def.mpr fun x h₁ h₂ ↦ ?_
have : x ∈ M' := span_le.mpr (Set.range_subset_iff.mpr fun i ↦ (f i).prop) h₁
obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp h₂
simp_rw [← Quotient.mk_eq_zero, ← mkQ_apply, map_finsupp_sum, map_smul, mkQ_apply] at this
rw [linearIndependent_iff.mp hg _ this, Finsupp.sum_zero_index]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 58 | 64 | theorem LinearIndependent.union_of_quotient
{M' : Submodule R M} {s : Set M} (hs : s ⊆ M') (hs' : LinearIndependent (ι := s) R Subtype.val)
{t : Set M} (ht : LinearIndependent (ι := t) R (Submodule.Quotient.mk (p := M') ∘ Subtype.val)) :
LinearIndependent (ι := (s ∪ t : _)) R Subtype.val := by |
refine (LinearIndependent.sum_elim_of_quotient (f := Set.embeddingOfSubset s M' hs)
(of_comp M'.subtype (by simpa using hs')) Subtype.val ht).to_subtype_range' ?_
simp only [embeddingOfSubset_apply_coe, Sum.elim_range, Subtype.range_val]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Quotient
theorem LinearIndependent.sum_elim_of_quotient
{M' : Submodule R M} {ι₁ ι₂} {f : ι₁ → M'} (hf : LinearIndependent R f) (g : ι₂ → M)
(hg : LinearIndependent R (Submodule.Quotient.mk (p := M') ∘ g)) :
LinearIndependent R (Sum.elim (f · : ι₁ → M) g) := by
refine .sum_type (hf.map' M'.subtype M'.ker_subtype) (.of_comp M'.mkQ hg) ?_
refine disjoint_def.mpr fun x h₁ h₂ ↦ ?_
have : x ∈ M' := span_le.mpr (Set.range_subset_iff.mpr fun i ↦ (f i).prop) h₁
obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp h₂
simp_rw [← Quotient.mk_eq_zero, ← mkQ_apply, map_finsupp_sum, map_smul, mkQ_apply] at this
rw [linearIndependent_iff.mp hg _ this, Finsupp.sum_zero_index]
theorem LinearIndependent.union_of_quotient
{M' : Submodule R M} {s : Set M} (hs : s ⊆ M') (hs' : LinearIndependent (ι := s) R Subtype.val)
{t : Set M} (ht : LinearIndependent (ι := t) R (Submodule.Quotient.mk (p := M') ∘ Subtype.val)) :
LinearIndependent (ι := (s ∪ t : _)) R Subtype.val := by
refine (LinearIndependent.sum_elim_of_quotient (f := Set.embeddingOfSubset s M' hs)
(of_comp M'.subtype (by simpa using hs')) Subtype.val ht).to_subtype_range' ?_
simp only [embeddingOfSubset_apply_coe, Sum.elim_range, Subtype.range_val]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 66 | 75 | theorem rank_quotient_add_rank_le [Nontrivial R] (M' : Submodule R M) :
Module.rank R (M ⧸ M') + Module.rank R M' ≤ Module.rank R M := by |
conv_lhs => simp only [Module.rank_def]
have := nonempty_linearIndependent_set R (M ⧸ M')
have := nonempty_linearIndependent_set R M'
rw [Cardinal.ciSup_add_ciSup _ (bddAbove_range.{v, v} _) _ (bddAbove_range.{v, v} _)]
refine ciSup_le fun ⟨s, hs⟩ ↦ ciSup_le fun ⟨t, ht⟩ ↦ ?_
choose f hf using Quotient.mk_surjective M'
simpa [add_comm] using (LinearIndependent.sum_elim_of_quotient ht (fun (i : s) ↦ f i)
(by simpa [Function.comp, hf] using hs)).cardinal_le_rank
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section ULift
@[simp]
theorem rank_ulift : Module.rank R (ULift.{w} M) = Cardinal.lift.{w} (Module.rank R M) :=
Cardinal.lift_injective.{v} <| Eq.symm <| (lift_lift _).trans ULift.moduleEquiv.symm.lift_rank_eq
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 104 | 105 | theorem finrank_ulift : finrank R (ULift M) = finrank R M := by |
simp_rw [finrank, rank_ulift, toNat_lift]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 164 | 168 | theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by |
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 171 | 172 | theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by |
simp [rank_finsupp]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 178 | 179 | theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by |
simp [rank_finsupp]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 183 | 183 | theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by | simp
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 188 | 193 | theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by |
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 198 | 205 | theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by |
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 211 | 213 | theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by |
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 219 | 220 | theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by | simp
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
namespace FiniteDimensional
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 230 | 231 | theorem finrank_finsupp {ι : Type v} [Fintype ι] : finrank R (ι →₀ M) = card ι * finrank R M := by |
rw [finrank, finrank, rank_finsupp, ← mk_toNat_eq_card, toNat_mul, toNat_lift, toNat_lift]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
namespace FiniteDimensional
@[simp]
theorem finrank_finsupp {ι : Type v} [Fintype ι] : finrank R (ι →₀ M) = card ι * finrank R M := by
rw [finrank, finrank, rank_finsupp, ← mk_toNat_eq_card, toNat_mul, toNat_lift, toNat_lift]
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 235 | 236 | theorem finrank_finsupp_self {ι : Type v} [Fintype ι] : finrank R (ι →₀ R) = card ι := by |
rw [finrank, rank_finsupp_self, ← mk_toNat_eq_card, toNat_lift]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
namespace FiniteDimensional
@[simp]
theorem finrank_finsupp {ι : Type v} [Fintype ι] : finrank R (ι →₀ M) = card ι * finrank R M := by
rw [finrank, finrank, rank_finsupp, ← mk_toNat_eq_card, toNat_mul, toNat_lift, toNat_lift]
@[simp]
theorem finrank_finsupp_self {ι : Type v} [Fintype ι] : finrank R (ι →₀ R) = card ι := by
rw [finrank, rank_finsupp_self, ← mk_toNat_eq_card, toNat_lift]
#align finite_dimensional.finrank_finsupp FiniteDimensional.finrank_finsupp_self
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 241 | 246 | theorem finrank_directSum {ι : Type v} [Fintype ι] (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (⨁ i, M i) = ∑ i, finrank R (M i) := by |
letI := nontrivial_of_invariantBasisNumber R
simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,
mk_toNat_eq_card, card_sigma]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
namespace FiniteDimensional
@[simp]
theorem finrank_finsupp {ι : Type v} [Fintype ι] : finrank R (ι →₀ M) = card ι * finrank R M := by
rw [finrank, finrank, rank_finsupp, ← mk_toNat_eq_card, toNat_mul, toNat_lift, toNat_lift]
@[simp]
theorem finrank_finsupp_self {ι : Type v} [Fintype ι] : finrank R (ι →₀ R) = card ι := by
rw [finrank, rank_finsupp_self, ← mk_toNat_eq_card, toNat_lift]
#align finite_dimensional.finrank_finsupp FiniteDimensional.finrank_finsupp_self
@[simp]
theorem finrank_directSum {ι : Type v} [Fintype ι] (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (⨁ i, M i) = ∑ i, finrank R (M i) := by
letI := nontrivial_of_invariantBasisNumber R
simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,
mk_toNat_eq_card, card_sigma]
#align finite_dimensional.finrank_direct_sum FiniteDimensional.finrank_directSum
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 251 | 252 | theorem finrank_matrix (m n : Type*) [Fintype m] [Fintype n] :
finrank R (Matrix m n R) = card m * card n := by | simp [finrank]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section Pi
variable [StrongRankCondition R] [Module.Free R M]
variable [∀ i, AddCommGroup (φ i)] [∀ i, Module R (φ i)] [∀ i, Module.Free R (φ i)]
open Module.Free
open LinearMap
-- this result is not true without the freeness assumption
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 271 | 276 | theorem rank_pi [Finite η] : Module.rank R (∀ i, φ i) =
Cardinal.sum fun i => Module.rank R (φ i) := by |
cases nonempty_fintype η
let B i := chooseBasis R (φ i)
let b : Basis _ R (∀ i, φ i) := Pi.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section TensorProduct
open TensorProduct
variable [StrongRankCondition S]
variable [Module S M] [Module.Free S M] [Module S M'] [Module.Free S M']
variable [Module S M₁] [Module.Free S M₁]
open Module.Free
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 359 | 364 | theorem rank_tensorProduct :
Module.rank S (M ⊗[S] M') =
Cardinal.lift.{v'} (Module.rank S M) * Cardinal.lift.{v} (Module.rank S M') := by |
obtain ⟨⟨_, bM⟩⟩ := Module.Free.exists_basis (R := S) (M := M)
obtain ⟨⟨_, bN⟩⟩ := Module.Free.exists_basis (R := S) (M := M')
rw [← bM.mk_eq_rank'', ← bN.mk_eq_rank'', ← (bM.tensorProduct bN).mk_eq_rank'', Cardinal.mk_prod]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section TensorProduct
open TensorProduct
variable [StrongRankCondition S]
variable [Module S M] [Module.Free S M] [Module S M'] [Module.Free S M']
variable [Module S M₁] [Module.Free S M₁]
open Module.Free
@[simp]
theorem rank_tensorProduct :
Module.rank S (M ⊗[S] M') =
Cardinal.lift.{v'} (Module.rank S M) * Cardinal.lift.{v} (Module.rank S M') := by
obtain ⟨⟨_, bM⟩⟩ := Module.Free.exists_basis (R := S) (M := M)
obtain ⟨⟨_, bN⟩⟩ := Module.Free.exists_basis (R := S) (M := M')
rw [← bM.mk_eq_rank'', ← bN.mk_eq_rank'', ← (bM.tensorProduct bN).mk_eq_rank'', Cardinal.mk_prod]
#align rank_tensor_product rank_tensorProduct
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 369 | 370 | theorem rank_tensorProduct' :
Module.rank S (M ⊗[S] M₁) = Module.rank S M * Module.rank S M₁ := by | simp
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section TensorProduct
open TensorProduct
variable [StrongRankCondition S]
variable [Module S M] [Module.Free S M] [Module S M'] [Module.Free S M']
variable [Module S M₁] [Module.Free S M₁]
open Module.Free
@[simp]
theorem rank_tensorProduct :
Module.rank S (M ⊗[S] M') =
Cardinal.lift.{v'} (Module.rank S M) * Cardinal.lift.{v} (Module.rank S M') := by
obtain ⟨⟨_, bM⟩⟩ := Module.Free.exists_basis (R := S) (M := M)
obtain ⟨⟨_, bN⟩⟩ := Module.Free.exists_basis (R := S) (M := M')
rw [← bM.mk_eq_rank'', ← bN.mk_eq_rank'', ← (bM.tensorProduct bN).mk_eq_rank'', Cardinal.mk_prod]
#align rank_tensor_product rank_tensorProduct
theorem rank_tensorProduct' :
Module.rank S (M ⊗[S] M₁) = Module.rank S M * Module.rank S M₁ := by simp
#align rank_tensor_product' rank_tensorProduct'
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 375 | 376 | theorem FiniteDimensional.finrank_tensorProduct :
finrank S (M ⊗[S] M') = finrank S M * finrank S M' := by | simp [finrank]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section Span
variable [StrongRankCondition R]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 439 | 443 | theorem rank_span_le (s : Set M) : Module.rank R (span R s) ≤ #s := by |
rw [Finsupp.span_eq_range_total, ← lift_strictMono.le_iff_le]
refine (lift_rank_range_le _).trans ?_
rw [rank_finsupp_self]
simp only [lift_lift, ge_iff_le, le_refl]
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section SubalgebraRank
open Module
variable {F E : Type*} [CommRing F] [Ring E] [Algebra F E]
@[simp]
theorem Subalgebra.rank_toSubmodule (S : Subalgebra F E) :
Module.rank F (Subalgebra.toSubmodule S) = Module.rank F S :=
rfl
#align subalgebra.rank_to_submodule Subalgebra.rank_toSubmodule
@[simp]
theorem Subalgebra.finrank_toSubmodule (S : Subalgebra F E) :
finrank F (Subalgebra.toSubmodule S) = finrank F S :=
rfl
#align subalgebra.finrank_to_submodule Subalgebra.finrank_toSubmodule
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 538 | 541 | theorem subalgebra_top_rank_eq_submodule_top_rank :
Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by |
rw [← Algebra.top_toSubmodule]
rfl
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section SubalgebraRank
open Module
variable {F E : Type*} [CommRing F] [Ring E] [Algebra F E]
@[simp]
theorem Subalgebra.rank_toSubmodule (S : Subalgebra F E) :
Module.rank F (Subalgebra.toSubmodule S) = Module.rank F S :=
rfl
#align subalgebra.rank_to_submodule Subalgebra.rank_toSubmodule
@[simp]
theorem Subalgebra.finrank_toSubmodule (S : Subalgebra F E) :
finrank F (Subalgebra.toSubmodule S) = finrank F S :=
rfl
#align subalgebra.finrank_to_submodule Subalgebra.finrank_toSubmodule
theorem subalgebra_top_rank_eq_submodule_top_rank :
Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by
rw [← Algebra.top_toSubmodule]
rfl
#align subalgebra_top_rank_eq_submodule_top_rank subalgebra_top_rank_eq_submodule_top_rank
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 544 | 547 | theorem subalgebra_top_finrank_eq_submodule_top_finrank :
finrank F (⊤ : Subalgebra F E) = finrank F (⊤ : Submodule F E) := by |
rw [← Algebra.top_toSubmodule]
rfl
| 1,682 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
section Finsupp
variable (R M M')
variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M']
open Module.Free
@[simp]
theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
#align rank_finsupp rank_finsupp
theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by
simp [rank_finsupp]
#align rank_finsupp' rank_finsupp'
-- Porting note, this should not be `@[simp]`, as simp can prove it.
-- @[simp]
theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by
simp [rank_finsupp]
#align rank_finsupp_self rank_finsupp_self
theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by simp
#align rank_finsupp_self' rank_finsupp_self'
@[simp]
theorem rank_directSum {ι : Type v} (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] :
Module.rank R (⨁ i, M i) = Cardinal.sum fun i => Module.rank R (M i) := by
let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
#align rank_direct_sum rank_directSum
@[simp]
theorem rank_matrix (m : Type v) (n : Type w) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) =
Cardinal.lift.{max v w u, v} #m * Cardinal.lift.{max v w u, w} #n := by
cases nonempty_fintype m
cases nonempty_fintype n
have h := (Matrix.stdBasis R m n).mk_eq_rank
rw [← lift_lift.{max v w u, max v w}, lift_inj] at h
simpa using h.symm
#align rank_matrix rank_matrix
@[simp high]
theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
#align rank_matrix' rank_matrix'
-- @[simp] -- Porting note (#10618): simp can prove this
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
#align rank_matrix'' rank_matrix''
variable [Module.Finite R M] [Module.Finite R M']
open Fintype
section SubalgebraRank
open Module
variable {F E : Type*} [CommRing F] [Ring E] [Algebra F E]
@[simp]
theorem Subalgebra.rank_toSubmodule (S : Subalgebra F E) :
Module.rank F (Subalgebra.toSubmodule S) = Module.rank F S :=
rfl
#align subalgebra.rank_to_submodule Subalgebra.rank_toSubmodule
@[simp]
theorem Subalgebra.finrank_toSubmodule (S : Subalgebra F E) :
finrank F (Subalgebra.toSubmodule S) = finrank F S :=
rfl
#align subalgebra.finrank_to_submodule Subalgebra.finrank_toSubmodule
theorem subalgebra_top_rank_eq_submodule_top_rank :
Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by
rw [← Algebra.top_toSubmodule]
rfl
#align subalgebra_top_rank_eq_submodule_top_rank subalgebra_top_rank_eq_submodule_top_rank
theorem subalgebra_top_finrank_eq_submodule_top_finrank :
finrank F (⊤ : Subalgebra F E) = finrank F (⊤ : Submodule F E) := by
rw [← Algebra.top_toSubmodule]
rfl
#align subalgebra_top_finrank_eq_submodule_top_finrank subalgebra_top_finrank_eq_submodule_top_finrank
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 550 | 552 | theorem Subalgebra.rank_top : Module.rank F (⊤ : Subalgebra F E) = Module.rank F E := by |
rw [subalgebra_top_rank_eq_submodule_top_rank]
exact _root_.rank_top F E
| 1,682 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open Finset
open Polynomial
structure LinearRecurrence (α : Type*) [CommSemiring α] where
order : ℕ
coeffs : Fin order → α
#align linear_recurrence LinearRecurrence
instance (α : Type*) [CommSemiring α] : Inhabited (LinearRecurrence α) :=
⟨⟨0, default⟩⟩
namespace LinearRecurrence
section CommSemiring
variable {α : Type*} [CommSemiring α] (E : LinearRecurrence α)
def IsSolution (u : ℕ → α) :=
∀ n, u (n + E.order) = ∑ i, E.coeffs i * u (n + i)
#align linear_recurrence.is_solution LinearRecurrence.IsSolution
def mkSol (init : Fin E.order → α) : ℕ → α
| n =>
if h : n < E.order then init ⟨n, h⟩
else
∑ k : Fin E.order,
have _ : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h)
simp only [zero_add]
E.coeffs k * mkSol init (n - E.order + k)
#align linear_recurrence.mk_sol LinearRecurrence.mkSol
| Mathlib/Algebra/LinearRecurrence.lean | 85 | 88 | theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by |
intro n
rw [mkSol]
simp
| 1,683 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open Finset
open Polynomial
structure LinearRecurrence (α : Type*) [CommSemiring α] where
order : ℕ
coeffs : Fin order → α
#align linear_recurrence LinearRecurrence
instance (α : Type*) [CommSemiring α] : Inhabited (LinearRecurrence α) :=
⟨⟨0, default⟩⟩
namespace LinearRecurrence
section CommSemiring
variable {α : Type*} [CommSemiring α] (E : LinearRecurrence α)
def IsSolution (u : ℕ → α) :=
∀ n, u (n + E.order) = ∑ i, E.coeffs i * u (n + i)
#align linear_recurrence.is_solution LinearRecurrence.IsSolution
def mkSol (init : Fin E.order → α) : ℕ → α
| n =>
if h : n < E.order then init ⟨n, h⟩
else
∑ k : Fin E.order,
have _ : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h)
simp only [zero_add]
E.coeffs k * mkSol init (n - E.order + k)
#align linear_recurrence.mk_sol LinearRecurrence.mkSol
theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by
intro n
rw [mkSol]
simp
#align linear_recurrence.is_sol_mk_sol LinearRecurrence.is_sol_mkSol
| Mathlib/Algebra/LinearRecurrence.lean | 92 | 95 | theorem mkSol_eq_init (init : Fin E.order → α) : ∀ n : Fin E.order, E.mkSol init n = init n := by |
intro n
rw [mkSol]
simp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]
| 1,683 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open Finset
open Polynomial
structure LinearRecurrence (α : Type*) [CommSemiring α] where
order : ℕ
coeffs : Fin order → α
#align linear_recurrence LinearRecurrence
instance (α : Type*) [CommSemiring α] : Inhabited (LinearRecurrence α) :=
⟨⟨0, default⟩⟩
namespace LinearRecurrence
section CommSemiring
variable {α : Type*} [CommSemiring α] (E : LinearRecurrence α)
def IsSolution (u : ℕ → α) :=
∀ n, u (n + E.order) = ∑ i, E.coeffs i * u (n + i)
#align linear_recurrence.is_solution LinearRecurrence.IsSolution
def mkSol (init : Fin E.order → α) : ℕ → α
| n =>
if h : n < E.order then init ⟨n, h⟩
else
∑ k : Fin E.order,
have _ : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h)
simp only [zero_add]
E.coeffs k * mkSol init (n - E.order + k)
#align linear_recurrence.mk_sol LinearRecurrence.mkSol
theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by
intro n
rw [mkSol]
simp
#align linear_recurrence.is_sol_mk_sol LinearRecurrence.is_sol_mkSol
theorem mkSol_eq_init (init : Fin E.order → α) : ∀ n : Fin E.order, E.mkSol init n = init n := by
intro n
rw [mkSol]
simp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]
#align linear_recurrence.mk_sol_eq_init LinearRecurrence.mkSol_eq_init
| Mathlib/Algebra/LinearRecurrence.lean | 100 | 115 | theorem eq_mk_of_is_sol_of_eq_init {u : ℕ → α} {init : Fin E.order → α} (h : E.IsSolution u)
(heq : ∀ n : Fin E.order, u n = init n) : ∀ n, u n = E.mkSol init n := by |
intro n
rw [mkSol]
split_ifs with h'
· exact mod_cast heq ⟨n, h'⟩
simp only
rw [← tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]
congr with k
have : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h'), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h')
simp only [zero_add]
rw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]
simp
| 1,683 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open Finset
open Polynomial
structure LinearRecurrence (α : Type*) [CommSemiring α] where
order : ℕ
coeffs : Fin order → α
#align linear_recurrence LinearRecurrence
instance (α : Type*) [CommSemiring α] : Inhabited (LinearRecurrence α) :=
⟨⟨0, default⟩⟩
namespace LinearRecurrence
section CommSemiring
variable {α : Type*} [CommSemiring α] (E : LinearRecurrence α)
def IsSolution (u : ℕ → α) :=
∀ n, u (n + E.order) = ∑ i, E.coeffs i * u (n + i)
#align linear_recurrence.is_solution LinearRecurrence.IsSolution
def mkSol (init : Fin E.order → α) : ℕ → α
| n =>
if h : n < E.order then init ⟨n, h⟩
else
∑ k : Fin E.order,
have _ : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h)
simp only [zero_add]
E.coeffs k * mkSol init (n - E.order + k)
#align linear_recurrence.mk_sol LinearRecurrence.mkSol
theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by
intro n
rw [mkSol]
simp
#align linear_recurrence.is_sol_mk_sol LinearRecurrence.is_sol_mkSol
theorem mkSol_eq_init (init : Fin E.order → α) : ∀ n : Fin E.order, E.mkSol init n = init n := by
intro n
rw [mkSol]
simp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]
#align linear_recurrence.mk_sol_eq_init LinearRecurrence.mkSol_eq_init
theorem eq_mk_of_is_sol_of_eq_init {u : ℕ → α} {init : Fin E.order → α} (h : E.IsSolution u)
(heq : ∀ n : Fin E.order, u n = init n) : ∀ n, u n = E.mkSol init n := by
intro n
rw [mkSol]
split_ifs with h'
· exact mod_cast heq ⟨n, h'⟩
simp only
rw [← tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]
congr with k
have : n - E.order + k < n := by
rw [add_comm, ← add_tsub_assoc_of_le (not_lt.mp h'), tsub_lt_iff_left]
· exact add_lt_add_right k.is_lt n
· convert add_le_add (zero_le (k : ℕ)) (not_lt.mp h')
simp only [zero_add]
rw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]
simp
#align linear_recurrence.eq_mk_of_is_sol_of_eq_init LinearRecurrence.eq_mk_of_is_sol_of_eq_init
theorem eq_mk_of_is_sol_of_eq_init' {u : ℕ → α} {init : Fin E.order → α} (h : E.IsSolution u)
(heq : ∀ n : Fin E.order, u n = init n) : u = E.mkSol init :=
funext (E.eq_mk_of_is_sol_of_eq_init h heq)
#align linear_recurrence.eq_mk_of_is_sol_of_eq_init' LinearRecurrence.eq_mk_of_is_sol_of_eq_init'
def solSpace : Submodule α (ℕ → α) where
carrier := { u | E.IsSolution u }
zero_mem' n := by simp
add_mem' {u v} hu hv n := by simp [mul_add, sum_add_distrib, hu n, hv n]
smul_mem' a u hu n := by simp [hu n, mul_sum]; congr; ext; ac_rfl
#align linear_recurrence.sol_space LinearRecurrence.solSpace
theorem is_sol_iff_mem_solSpace (u : ℕ → α) : E.IsSolution u ↔ u ∈ E.solSpace :=
Iff.rfl
#align linear_recurrence.is_sol_iff_mem_sol_space LinearRecurrence.is_sol_iff_mem_solSpace
def toInit : E.solSpace ≃ₗ[α] Fin E.order → α where
toFun u x := (u : ℕ → α) x
map_add' u v := by
ext
simp
map_smul' a u := by
ext
simp
invFun u := ⟨E.mkSol u, E.is_sol_mkSol u⟩
left_inv u := by ext n; symm; apply E.eq_mk_of_is_sol_of_eq_init u.2; intro k; rfl
right_inv u := Function.funext_iff.mpr fun n ↦ E.mkSol_eq_init u n
#align linear_recurrence.to_init LinearRecurrence.toInit
| Mathlib/Algebra/LinearRecurrence.lean | 156 | 166 | theorem sol_eq_of_eq_init (u v : ℕ → α) (hu : E.IsSolution u) (hv : E.IsSolution v) :
u = v ↔ Set.EqOn u v ↑(range E.order) := by |
refine Iff.intro (fun h x _ ↦ h ▸ rfl) ?_
intro h
set u' : ↥E.solSpace := ⟨u, hu⟩
set v' : ↥E.solSpace := ⟨v, hv⟩
change u'.val = v'.val
suffices h' : u' = v' from h' ▸ rfl
rw [← E.toInit.toEquiv.apply_eq_iff_eq, LinearEquiv.coe_toEquiv]
ext x
exact mod_cast h (mem_range.mpr x.2)
| 1,683 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
variable {R : Type u} {M M₁ : Type v} {M' : Type v'} {ι : Type w}
variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
attribute [local instance] nontrivial_of_invariantBasisNumber
open Cardinal Basis Submodule Function Set FiniteDimensional
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 34 | 40 | theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by |
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
| 1,684 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
variable {R : Type u} {M M₁ : Type v} {M' : Type v'} {ι : Type w}
variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
attribute [local instance] nontrivial_of_invariantBasisNumber
open Cardinal Basis Submodule Function Set FiniteDimensional
theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
#align rank_le rank_le
section RankZero
lemma rank_eq_zero_iff :
Module.rank R M = 0 ↔ ∀ x : M, ∃ a : R, a ≠ 0 ∧ a • x = 0 := by
nontriviality R
constructor
· contrapose!
rintro ⟨x, hx⟩
rw [← Cardinal.one_le_iff_ne_zero]
have : LinearIndependent R (fun _ : Unit ↦ x) :=
linearIndependent_iff.mpr (fun l hl ↦ Finsupp.unique_ext <| not_not.mp fun H ↦
hx _ H ((Finsupp.total_unique _ _ _).symm.trans hl))
simpa using this.cardinal_lift_le_rank
· intro h
rw [← le_zero_iff, Module.rank_def]
apply ciSup_le'
intro ⟨s, hs⟩
rw [nonpos_iff_eq_zero, Cardinal.mk_eq_zero_iff, ← not_nonempty_iff]
rintro ⟨i : s⟩
obtain ⟨a, ha, ha'⟩ := h i
apply ha
simpa using DFunLike.congr_fun (linearIndependent_iff.mp hs (Finsupp.single i a) (by simpa)) i
variable [Nontrivial R]
variable [NoZeroSMulDivisors R M]
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 70 | 73 | theorem rank_zero_iff_forall_zero :
Module.rank R M = 0 ↔ ∀ x : M, x = 0 := by |
simp_rw [rank_eq_zero_iff, smul_eq_zero, and_or_left, not_and_self_iff, false_or,
exists_and_right, and_iff_right (exists_ne (0 : R))]
| 1,684 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
variable {R : Type u} {M M₁ : Type v} {M' : Type v'} {ι : Type w}
variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
attribute [local instance] nontrivial_of_invariantBasisNumber
open Cardinal Basis Submodule Function Set FiniteDimensional
theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
#align rank_le rank_le
section RankZero
lemma rank_eq_zero_iff :
Module.rank R M = 0 ↔ ∀ x : M, ∃ a : R, a ≠ 0 ∧ a • x = 0 := by
nontriviality R
constructor
· contrapose!
rintro ⟨x, hx⟩
rw [← Cardinal.one_le_iff_ne_zero]
have : LinearIndependent R (fun _ : Unit ↦ x) :=
linearIndependent_iff.mpr (fun l hl ↦ Finsupp.unique_ext <| not_not.mp fun H ↦
hx _ H ((Finsupp.total_unique _ _ _).symm.trans hl))
simpa using this.cardinal_lift_le_rank
· intro h
rw [← le_zero_iff, Module.rank_def]
apply ciSup_le'
intro ⟨s, hs⟩
rw [nonpos_iff_eq_zero, Cardinal.mk_eq_zero_iff, ← not_nonempty_iff]
rintro ⟨i : s⟩
obtain ⟨a, ha, ha'⟩ := h i
apply ha
simpa using DFunLike.congr_fun (linearIndependent_iff.mp hs (Finsupp.single i a) (by simpa)) i
variable [Nontrivial R]
variable [NoZeroSMulDivisors R M]
theorem rank_zero_iff_forall_zero :
Module.rank R M = 0 ↔ ∀ x : M, x = 0 := by
simp_rw [rank_eq_zero_iff, smul_eq_zero, and_or_left, not_and_self_iff, false_or,
exists_and_right, and_iff_right (exists_ne (0 : R))]
#align rank_zero_iff_forall_zero rank_zero_iff_forall_zero
theorem rank_zero_iff : Module.rank R M = 0 ↔ Subsingleton M :=
rank_zero_iff_forall_zero.trans (subsingleton_iff_forall_eq 0).symm
#align rank_zero_iff rank_zero_iff
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 82 | 84 | theorem rank_pos_iff_exists_ne_zero : 0 < Module.rank R M ↔ ∃ x : M, x ≠ 0 := by |
rw [← not_iff_not]
simpa using rank_zero_iff_forall_zero
| 1,684 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' w
variable {R : Type u} {M M₁ : Type v} {M' : Type v'} {ι : Type w}
variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
variable [Module R M] [Module R M'] [Module R M₁]
attribute [local instance] nontrivial_of_invariantBasisNumber
open Cardinal Basis Submodule Function Set FiniteDimensional
theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
#align rank_le rank_le
section Finite
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 125 | 131 | theorem Module.finite_of_rank_eq_nat [Module.Free R M] {n : ℕ} (h : Module.rank R M = n) :
Module.Finite R M := by |
nontriviality R
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt <| nat_lt_aleph0 n
exact Module.Finite.of_basis b
| 1,684 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
open FiniteDimensional
namespace Subalgebra
variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(A B : Subalgebra R S) [Module.Free R A] [Module.Free R B]
[Module.Free A (Algebra.adjoin A (B : Set S))]
[Module.Free B (Algebra.adjoin B (A : Set S))]
| Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 30 | 41 | theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by |
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _
letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul
haveI : IsScalarTower R A (Algebra.adjoin A (B : Set S)) :=
IsScalarTower.of_algebraMap_eq (congrFun rfl)
rw [rank_mul_rank R A (Algebra.adjoin A (B : Set S))]
change _ = Module.rank R ((Algebra.adjoin A (B : Set S)).restrictScalars R)
rw [Algebra.restrictScalars_adjoin]; rfl
| 1,685 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
open FiniteDimensional
namespace Subalgebra
variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(A B : Subalgebra R S) [Module.Free R A] [Module.Free R B]
[Module.Free A (Algebra.adjoin A (B : Set S))]
[Module.Free B (Algebra.adjoin B (A : Set S))]
theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _
letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul
haveI : IsScalarTower R A (Algebra.adjoin A (B : Set S)) :=
IsScalarTower.of_algebraMap_eq (congrFun rfl)
rw [rank_mul_rank R A (Algebra.adjoin A (B : Set S))]
change _ = Module.rank R ((Algebra.adjoin A (B : Set S)).restrictScalars R)
rw [Algebra.restrictScalars_adjoin]; rfl
| Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 43 | 45 | theorem rank_sup_eq_rank_right_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R B * Module.rank B (Algebra.adjoin B (A : Set S)) := by |
rw [sup_comm, rank_sup_eq_rank_left_mul_rank_of_free]
| 1,685 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
open FiniteDimensional
namespace Subalgebra
variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(A B : Subalgebra R S) [Module.Free R A] [Module.Free R B]
[Module.Free A (Algebra.adjoin A (B : Set S))]
[Module.Free B (Algebra.adjoin B (A : Set S))]
theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _
letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul
haveI : IsScalarTower R A (Algebra.adjoin A (B : Set S)) :=
IsScalarTower.of_algebraMap_eq (congrFun rfl)
rw [rank_mul_rank R A (Algebra.adjoin A (B : Set S))]
change _ = Module.rank R ((Algebra.adjoin A (B : Set S)).restrictScalars R)
rw [Algebra.restrictScalars_adjoin]; rfl
theorem rank_sup_eq_rank_right_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R B * Module.rank B (Algebra.adjoin B (A : Set S)) := by
rw [sup_comm, rank_sup_eq_rank_left_mul_rank_of_free]
| Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 47 | 49 | theorem finrank_sup_eq_finrank_left_mul_finrank_of_free :
finrank R ↥(A ⊔ B) = finrank R A * finrank A (Algebra.adjoin A (B : Set S)) := by |
simpa only [map_mul] using congr(Cardinal.toNat $(rank_sup_eq_rank_left_mul_rank_of_free A B))
| 1,685 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
open FiniteDimensional
namespace Subalgebra
variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(A B : Subalgebra R S) [Module.Free R A] [Module.Free R B]
[Module.Free A (Algebra.adjoin A (B : Set S))]
[Module.Free B (Algebra.adjoin B (A : Set S))]
theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _
letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul
haveI : IsScalarTower R A (Algebra.adjoin A (B : Set S)) :=
IsScalarTower.of_algebraMap_eq (congrFun rfl)
rw [rank_mul_rank R A (Algebra.adjoin A (B : Set S))]
change _ = Module.rank R ((Algebra.adjoin A (B : Set S)).restrictScalars R)
rw [Algebra.restrictScalars_adjoin]; rfl
theorem rank_sup_eq_rank_right_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R B * Module.rank B (Algebra.adjoin B (A : Set S)) := by
rw [sup_comm, rank_sup_eq_rank_left_mul_rank_of_free]
theorem finrank_sup_eq_finrank_left_mul_finrank_of_free :
finrank R ↥(A ⊔ B) = finrank R A * finrank A (Algebra.adjoin A (B : Set S)) := by
simpa only [map_mul] using congr(Cardinal.toNat $(rank_sup_eq_rank_left_mul_rank_of_free A B))
| Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 51 | 53 | theorem finrank_sup_eq_finrank_right_mul_finrank_of_free :
finrank R ↥(A ⊔ B) = finrank R B * finrank B (Algebra.adjoin B (A : Set S)) := by |
rw [sup_comm, finrank_sup_eq_finrank_left_mul_finrank_of_free]
| 1,685 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 46 | 60 | theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by |
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 63 | 71 | theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by |
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
#align le_rank_iff_exists_linear_independent_finset le_rank_iff_exists_linearIndependent_finset
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 76 | 100 | theorem rank_le_one_iff [Module.Free K V] :
Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v := by |
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
constructor
· intro hd
rw [← b.mk_eq_rank'', le_one_iff_subsingleton] at hd
rcases isEmpty_or_nonempty κ with hb | ⟨⟨i⟩⟩
· use 0
have h' : ∀ v : V, v = 0 := by
simpa [range_eq_empty, Submodule.eq_bot_iff] using b.span_eq.symm
intro v
simp [h' v]
· use b i
have h' : (K ∙ b i) = ⊤ :=
(subsingleton_range b).eq_singleton_of_mem (mem_range_self i) ▸ b.span_eq
intro v
have hv : v ∈ (⊤ : Submodule K V) := mem_top
rwa [← h', mem_span_singleton] at hv
· rintro ⟨v₀, hv₀⟩
have h : (K ∙ v₀) = ⊤ := by
ext
simp [mem_span_singleton, hv₀]
rw [← rank_top, ← h]
refine (rank_span_le _).trans_eq ?_
simp
| 1,686 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
#align le_rank_iff_exists_linear_independent_finset le_rank_iff_exists_linearIndependent_finset
theorem rank_le_one_iff [Module.Free K V] :
Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v := by
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
constructor
· intro hd
rw [← b.mk_eq_rank'', le_one_iff_subsingleton] at hd
rcases isEmpty_or_nonempty κ with hb | ⟨⟨i⟩⟩
· use 0
have h' : ∀ v : V, v = 0 := by
simpa [range_eq_empty, Submodule.eq_bot_iff] using b.span_eq.symm
intro v
simp [h' v]
· use b i
have h' : (K ∙ b i) = ⊤ :=
(subsingleton_range b).eq_singleton_of_mem (mem_range_self i) ▸ b.span_eq
intro v
have hv : v ∈ (⊤ : Submodule K V) := mem_top
rwa [← h', mem_span_singleton] at hv
· rintro ⟨v₀, hv₀⟩
have h : (K ∙ v₀) = ⊤ := by
ext
simp [mem_span_singleton, hv₀]
rw [← rank_top, ← h]
refine (rank_span_le _).trans_eq ?_
simp
#align rank_le_one_iff rank_le_one_iff
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 105 | 119 | theorem rank_eq_one_iff [Module.Free K V] :
Module.rank K V = 1 ↔ ∃ v₀ : V, v₀ ≠ 0 ∧ ∀ v, ∃ r : K, r • v₀ = v := by |
haveI := nontrivial_of_invariantBasisNumber K
refine ⟨fun h ↦ ?_, fun ⟨v₀, h, hv⟩ ↦ (rank_le_one_iff.2 ⟨v₀, hv⟩).antisymm ?_⟩
· obtain ⟨v₀, hv⟩ := rank_le_one_iff.1 h.le
refine ⟨v₀, fun hzero ↦ ?_, hv⟩
simp_rw [hzero, smul_zero, exists_const] at hv
haveI : Subsingleton V := .intro fun _ _ ↦ by simp_rw [← hv]
exact one_ne_zero (h ▸ rank_subsingleton' K V)
· by_contra H
rw [not_le, lt_one_iff_zero] at H
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (H ▸ b.mk_eq_rank'')
haveI := b.repr.toEquiv.subsingleton
exact h (Subsingleton.elim _ _)
| 1,686 |
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