statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
smul_coe_torsion_by (x : torsion_by R M a) : a • (x : M) = 0 | x.prop | lemma | submodule.smul_coe_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_torsion_by_iff (x : M) : x ∈ torsion_by R M a ↔ a • x = 0 | iff.rfl | lemma | submodule.mem_torsion_by_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_torsion_by_set_iff (x : M) :
x ∈ torsion_by_set R M s ↔ ∀ a : s, (a : R) • x = 0 | begin
refine ⟨λ h ⟨a, ha⟩, mem_Inf.mp h _ (set.mem_image_of_mem _ ha), λ h, mem_Inf.mpr _⟩,
rintro _ ⟨a, ha, rfl⟩, exact h ⟨a, ha⟩
end | lemma | submodule.mem_torsion_by_set_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_singleton_eq : torsion_by_set R M {a} = torsion_by R M a | begin
ext x,
simp only [mem_torsion_by_set_iff, set_coe.forall, subtype.coe_mk, set.mem_singleton_iff,
forall_eq, mem_torsion_by_iff]
end | lemma | submodule.torsion_by_singleton_eq | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"forall_eq",
"set.mem_singleton_iff",
"set_coe.forall",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set_le_torsion_by_set_of_subset {s t : set R} (st : s ⊆ t) :
torsion_by_set R M t ≤ torsion_by_set R M s | Inf_le_Inf $ λ _ ⟨a, ha, h⟩, ⟨a, st ha, h⟩ | lemma | submodule.torsion_by_set_le_torsion_by_set_of_subset | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"Inf_le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set_eq_torsion_by_span :
torsion_by_set R M s = torsion_by_set R M (ideal.span s) | begin
refine le_antisymm (λ x hx, _) (torsion_by_set_le_torsion_by_set_of_subset subset_span),
rw mem_torsion_by_set_iff at hx ⊢,
suffices : ideal.span s ≤ ideal.torsion_of R M x,
{ rintro ⟨a, ha⟩, exact this ha },
rw ideal.span_le, exact λ a ha, hx ⟨a, ha⟩
end | lemma | submodule.torsion_by_set_eq_torsion_by_span | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal.span",
"ideal.span_le",
"ideal.torsion_of"
] | Torsion by a set is torsion by the ideal generated by it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_by_span_singleton_eq : torsion_by_set R M (R ∙ a) = torsion_by R M a | ((torsion_by_set_eq_torsion_by_span _).symm.trans $ torsion_by_singleton_eq _) | lemma | submodule.torsion_by_span_singleton_eq | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_le_torsion_by_of_dvd (a b : R) (dvd : a ∣ b) :
torsion_by R M a ≤ torsion_by R M b | begin
rw [← torsion_by_span_singleton_eq, ← torsion_by_singleton_eq],
apply torsion_by_set_le_torsion_by_set_of_subset,
rintro c (rfl : c = b), exact ideal.mem_span_singleton.mpr dvd
end | lemma | submodule.torsion_by_le_torsion_by_of_dvd | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_one : torsion_by R M 1 = ⊥ | eq_bot_iff.mpr (λ _ h, by { rw [mem_torsion_by_iff, one_smul] at h, exact h }) | lemma | submodule.torsion_by_one | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_univ : torsion_by_set R M set.univ = ⊥ | by { rw [eq_bot_iff, ← torsion_by_one, ← torsion_by_singleton_eq],
exact torsion_by_set_le_torsion_by_set_of_subset (λ _ _, trivial) } | lemma | submodule.torsion_by_univ | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_by_singleton_iff : is_torsion_by_set R M {a} ↔ is_torsion_by R M a | begin
refine ⟨λ h x, @h _ ⟨_, set.mem_singleton _⟩, λ h x, _⟩,
rintro ⟨b, rfl : b = a⟩, exact @h _
end | lemma | module.is_torsion_by_singleton_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_by_set_iff_torsion_by_set_eq_top :
is_torsion_by_set R M s ↔ submodule.torsion_by_set R M s = ⊤ | ⟨λ h, eq_top_iff.mpr (λ _ _, (mem_torsion_by_set_iff _ _).mpr $ @h _),
λ h x, by { rw [← mem_torsion_by_set_iff, h], trivial }⟩ | lemma | module.is_torsion_by_set_iff_torsion_by_set_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submodule.torsion_by_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_by_iff_torsion_by_eq_top : is_torsion_by R M a ↔ torsion_by R M a = ⊤ | by rw [← torsion_by_singleton_eq, ← is_torsion_by_singleton_iff,
is_torsion_by_set_iff_torsion_by_set_eq_top] | lemma | module.is_torsion_by_iff_torsion_by_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | A `a`-torsion module is a module whose `a`-torsion submodule is the full space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by_set_iff_is_torsion_by_span :
is_torsion_by_set R M s ↔ is_torsion_by_set R M (ideal.span s) | by rw [is_torsion_by_set_iff_torsion_by_set_eq_top, is_torsion_by_set_iff_torsion_by_set_eq_top,
torsion_by_set_eq_torsion_by_span] | lemma | module.is_torsion_by_set_iff_is_torsion_by_span | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_by_span_singleton_iff : is_torsion_by_set R M (R ∙ a) ↔ is_torsion_by R M a | ((is_torsion_by_set_iff_is_torsion_by_span _).symm.trans $ is_torsion_by_singleton_iff _) | lemma | module.is_torsion_by_span_singleton_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set_is_torsion_by_set : is_torsion_by_set R (torsion_by_set R M s) s | λ ⟨x, hx⟩ a, subtype.ext $ (mem_torsion_by_set_iff _ _).mp hx a | lemma | submodule.torsion_by_set_is_torsion_by_set | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_is_torsion_by : is_torsion_by R (torsion_by R M a) a | λ _, smul_torsion_by _ _ | lemma | submodule.torsion_by_is_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | The `a`-torsion submodule is a `a`-torsion module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_by_torsion_by_eq_top : torsion_by R (torsion_by R M a) a = ⊤ | (is_torsion_by_iff_torsion_by_eq_top a).mp $ torsion_by_is_torsion_by a | lemma | submodule.torsion_by_torsion_by_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set_torsion_by_set_eq_top :
torsion_by_set R (torsion_by_set R M s) s = ⊤ | (is_torsion_by_set_iff_torsion_by_set_eq_top s).mp $ torsion_by_set_is_torsion_by_set s | lemma | submodule.torsion_by_set_torsion_by_set_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_gc : @galois_connection (submodule R M) (ideal R)ᵒᵈ _ _
annihilator (λ I, torsion_by_set R M $ I.of_dual) | λ A I, ⟨λ h x hx, (mem_torsion_by_set_iff _ _).mpr $ λ ⟨a, ha⟩, mem_annihilator.mp (h ha) x hx,
λ h a ha, mem_annihilator.mpr $ λ x hx, (mem_torsion_by_set_iff _ _).mp (h hx) ⟨a, ha⟩⟩ | lemma | submodule.torsion_gc | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"galois_connection",
"ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_torsion_by_ideal_eq_torsion_by_infi :
(⨆ i ∈ S, torsion_by_set R M $ p i) = torsion_by_set R M ↑(⨅ i ∈ S, p i) | begin
cases S.eq_empty_or_nonempty with h h,
{ rw h, convert supr_emptyset, convert torsion_by_univ, convert top_coe, exact infi_emptyset },
apply le_antisymm,
{ apply supr_le _, intro i, apply supr_le _, intro is,
apply torsion_by_set_le_torsion_by_set_of_subset,
exact (infi_le (λ i, ⨅ (H : i ∈ S), p i... | lemma | submodule.supr_torsion_by_ideal_eq_torsion_by_infi | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"finset.sum_smul",
"ideal.eq_top_iff_one",
"ideal.mul_mem_left",
"ideal.mul_mem_right",
"ideal.supr_infi_eq_top_iff_pairwise",
"infi_emptyset",
"infi_le",
"one_smul",
"smul_smul",
"supr_emptyset",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_indep_torsion_by_ideal : S.sup_indep (λ i, torsion_by_set R M $ p i) | λ T hT i hi hiT, begin
rw [disjoint_iff, finset.sup_eq_supr,
supr_torsion_by_ideal_eq_torsion_by_infi $ λ i hi j hj ij, hp (hT hi) (hT hj) ij],
have := @galois_connection.u_inf _ _ (order_dual.to_dual _) (order_dual.to_dual _) _ _ _ _
(torsion_gc R M), dsimp at this ⊢,
rw [← this, ideal.sup_infi_eq_top, t... | lemma | submodule.sup_indep_torsion_by_ideal | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"disjoint_iff",
"finset.sup_eq_supr",
"galois_connection.u_inf",
"ideal.sup_infi_eq_top",
"order_dual.to_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_torsion_by_eq_torsion_by_prod :
(⨆ i ∈ S, torsion_by R M $ q i) = torsion_by R M (∏ i in S, q i) | begin
rw [← torsion_by_span_singleton_eq, ideal.submodule_span_eq,
← ideal.finset_inf_span_singleton _ _ hq, finset.inf_eq_infi,
← supr_torsion_by_ideal_eq_torsion_by_infi],
{ congr, ext : 1, congr, ext : 1, exact (torsion_by_span_singleton_eq _).symm },
{ exact λ i hi j hj ij, (ideal.sup_eq_top_iff_is_co... | lemma | submodule.supr_torsion_by_eq_torsion_by_prod | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"finset.inf_eq_infi",
"ideal.finset_inf_span_singleton",
"ideal.submodule_span_eq",
"ideal.sup_eq_top_iff_is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_indep_torsion_by : S.sup_indep (λ i, torsion_by R M $ q i) | begin
convert sup_indep_torsion_by_ideal
(λ i hi j hj ij, (ideal.sup_eq_top_iff_is_coprime (q i) _).mpr $ hq hi hj ij),
ext : 1, exact (torsion_by_span_singleton_eq _).symm,
end | lemma | submodule.sup_indep_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal.sup_eq_top_iff_is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set_is_internal {p : ι → ideal R}
(hp : (S : set ι).pairwise $ λ i j, p i ⊔ p j = ⊤)
(hM : module.is_torsion_by_set R M (⨅ i ∈ S, p i : ideal R)) :
direct_sum.is_internal (λ i : S, torsion_by_set R M $ p i) | direct_sum.is_internal_submodule_of_independent_of_supr_eq_top
(complete_lattice.independent_iff_sup_indep.mpr $ sup_indep_torsion_by_ideal hp)
((supr_subtype'' ↑S $ λ i, torsion_by_set R M $ p i).trans $
(supr_torsion_by_ideal_eq_torsion_by_infi hp).trans $
(module.is_torsion_by_set_iff_torsion_by_set_eq_t... | lemma | submodule.torsion_by_set_is_internal | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"direct_sum.is_internal",
"direct_sum.is_internal_submodule_of_independent_of_supr_eq_top",
"ideal",
"module.is_torsion_by_set",
"module.is_torsion_by_set_iff_torsion_by_set_eq_top",
"pairwise",
"supr_subtype''"
] | If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
its `p i`-torsion submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_by_is_internal {q : ι → R} (hq : (S : set ι).pairwise $ is_coprime on q)
(hM : module.is_torsion_by R M $ ∏ i in S, q i) :
direct_sum.is_internal (λ i : S, torsion_by R M $ q i) | begin
rw [← module.is_torsion_by_span_singleton_iff, ideal.submodule_span_eq,
← ideal.finset_inf_span_singleton _ _ hq, finset.inf_eq_infi] at hM,
convert torsion_by_set_is_internal
(λ i hi j hj ij, (ideal.sup_eq_top_iff_is_coprime (q i) _).mpr $ hq hi hj ij) hM,
ext : 1, exact (torsion_by_span_singleton_... | lemma | submodule.torsion_by_is_internal | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"direct_sum.is_internal",
"finset.inf_eq_infi",
"ideal.finset_inf_span_singleton",
"ideal.submodule_span_eq",
"ideal.sup_eq_top_iff_is_coprime",
"is_coprime",
"module.is_torsion_by",
"module.is_torsion_by_span_singleton_iff",
"pairwise"
] | If the `q i` are pairwise coprime, a `∏ i, q i`-torsion module is the internal direct sum of
its `q i`-torsion submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by_set.has_smul : has_smul (R ⧸ I) M | { smul := λ b x, quotient.lift_on' b (• x) $ λ b₁ b₂ h, begin
show b₁ • x = b₂ • x,
have : (-b₁ + b₂) • x = 0 := @hM x ⟨_, quotient_add_group.left_rel_apply.mp h⟩,
rw [add_smul, neg_smul, neg_add_eq_zero] at this,
exact this
end } | def | module.is_torsion_by_set.has_smul | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"add_smul",
"has_smul",
"neg_smul",
"quotient.lift_on'"
] | can't be an instance because hM can't be inferred | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by_set.mk_smul (b : R) (x : M) :
by haveI | hM.has_smul; exact ideal.quotient.mk I b • x = b • x := rfl | lemma | module.is_torsion_by_set.mk_smul | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_by_set.module : module (R ⧸ I) M | @function.surjective.module_left _ _ _ _ _ _ _ hM.has_smul
_ ideal.quotient.mk_surjective (is_torsion_by_set.mk_smul hM) | def | module.is_torsion_by_set.module | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"function.surjective.module_left",
"ideal.quotient.mk_surjective",
"module"
] | A `(R ⧸ I)`-module is a `R`-module which `is_torsion_by_set R M I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_torsion_by_set.is_scalar_tower {S : Type*} [has_smul S R] [has_smul S M]
[is_scalar_tower S R M] [is_scalar_tower S R R] :
@@is_scalar_tower S (R ⧸ I) M _ (is_torsion_by_set.module hM).to_has_smul _ | { smul_assoc := λ b d x, quotient.induction_on' d $ λ c, (smul_assoc b c x : _) } | instance | module.is_torsion_by_set.is_scalar_tower | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"has_smul",
"is_scalar_tower",
"quotient.induction_on'",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_set.mk_smul (I : ideal R) (b : R) (x : torsion_by_set R M I) :
ideal.quotient.mk I b • x = b • x | rfl | lemma | submodule.torsion_by_set.mk_smul | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by.mk_smul (a b : R) (x : torsion_by R M a) :
ideal.quotient.mk (R ∙ a) b • x = b • x | rfl | lemma | submodule.torsion_by.mk_smul | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x = 0 | iff.rfl | lemma | submodule.mem_torsion'_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 | iff.rfl | lemma | submodule.mem_torsion_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion'_iff_torsion'_eq_top : is_torsion' M S ↔ torsion' R M S = ⊤ | ⟨λ h, eq_top_iff.mpr (λ _ _, @h _), λ h x, by { rw [← @mem_torsion'_iff R, h], trivial }⟩ | lemma | submodule.is_torsion'_iff_torsion'_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | A `S`-torsion module is a module whose `S`-torsion submodule is the full space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion'_is_torsion' : is_torsion' (torsion' R M S) S | λ ⟨x, ⟨a, h⟩⟩, ⟨a, subtype.ext h⟩ | lemma | submodule.torsion'_is_torsion' | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"subtype.ext"
] | The `S`-torsion submodule is a `S`-torsion module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ | (is_torsion'_iff_torsion'_eq_top S).mp $ torsion'_is_torsion' S | lemma | submodule.torsion'_torsion'_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_torsion_eq_top : torsion R (torsion R M) = ⊤ | torsion'_torsion'_eq_top R⁰ | lemma | submodule.torsion_torsion_eq_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [] | The torsion submodule of the torsion submodule (viewed as a module) is the full
torsion module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_is_torsion : module.is_torsion R (torsion R M) | torsion'_is_torsion' R⁰ | lemma | submodule.torsion_is_torsion | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"module.is_torsion"
] | The torsion submodule is always a torsion module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.module.is_torsion_by_set_annihilator_top :
module.is_torsion_by_set R M (⊤ : submodule R M).annihilator | λ x ha, mem_annihilator.mp ha.prop x mem_top | lemma | module.is_torsion_by_set_annihilator_top | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"module.is_torsion_by_set",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.annihilator_top_inter_non_zero_divisors [module.finite R M]
(hM : module.is_torsion R M) :
((⊤ : submodule R M).annihilator : set R) ∩ R⁰ ≠ ∅ | begin
obtain ⟨S, hS⟩ := ‹module.finite R M›.out,
refine set.nonempty.ne_empty ⟨_, _, (∏ x in S, (@hM x).some : R⁰).prop⟩,
rw [submonoid.coe_finset_prod, set_like.mem_coe, ←hS, mem_annihilator_span],
intro n,
letI := classical.dec_eq M,
rw [←finset.prod_erase_mul _ _ n.prop, mul_smul, ←submonoid.smul_def, (@... | lemma | submodule.annihilator_top_inter_non_zero_divisors | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"classical.dec_eq",
"module.finite",
"module.is_torsion",
"set_like.mem_coe",
"smul_zero",
"submodule",
"submonoid.coe_finset_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_torsion_eq_annihilator_ne_bot :
(torsion R M : set M) = { x : M | (R ∙ x).annihilator ≠ ⊥ } | begin
ext x, simp_rw [submodule.ne_bot_iff, mem_annihilator, mem_span_singleton],
exact ⟨λ ⟨a, hax⟩, ⟨a, λ _ ⟨b, hb⟩, by rw [← hb, smul_comm, ← submonoid.smul_def, hax, smul_zero],
non_zero_divisors.coe_ne_zero _⟩,
λ ⟨a, hax, ha⟩, ⟨⟨_, mem_non_zero_divisors_of_ne_zero ha⟩, hax x ⟨1, one_smul _ _⟩⟩⟩
end | lemma | submodule.coe_torsion_eq_annihilator_ne_bot | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"mem_non_zero_divisors_of_ne_zero",
"non_zero_divisors.coe_ne_zero",
"one_smul",
"smul_zero",
"submodule.ne_bot_iff",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors_iff_torsion_eq_bot :
no_zero_smul_divisors R M ↔ torsion R M = ⊥ | begin
split; intro h,
{ haveI : no_zero_smul_divisors R M := h,
rw eq_bot_iff, rintro x ⟨a, hax⟩,
change (a : R) • x = 0 at hax,
cases eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0,
{ exfalso, exact non_zero_divisors.coe_ne_zero a h0 }, { exact h0 } },
{ exact { eq_zero_or_eq_zero_of_smul_eq_z... | lemma | submodule.no_zero_smul_divisors_iff_torsion_eq_bot | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"eq_bot_iff",
"mem_non_zero_divisors_of_ne_zero",
"no_zero_smul_divisors",
"non_zero_divisors.coe_ne_zero"
] | A module over a domain has `no_zero_smul_divisors` iff its torsion submodule is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ | eq_bot_iff.mpr $ λ z, quotient.induction_on' z $ λ x ⟨a, hax⟩,
begin
rw [quotient.mk'_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero] at hax,
rw [mem_bot, quotient.mk'_eq_mk, quotient.mk_eq_zero],
cases hax with b h,
exact ⟨b * a, (mul_smul _ _ _).trans h⟩
end | lemma | submodule.quotient_torsion.torsion_eq_bot | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"quotient.induction_on'",
"quotient.mk'_eq_mk"
] | Quotienting by the torsion submodule gives a torsion-free module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_smul_divisors [is_domain R] : no_zero_smul_divisors R (M ⧸ torsion R M) | no_zero_smul_divisors_iff_torsion_eq_bot.mpr torsion_eq_bot | instance | submodule.quotient_torsion.no_zero_smul_divisors | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"is_domain",
"no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion'_powers_iff (p : R) :
is_torsion' M (submonoid.powers p) ↔ ∀ x : M, ∃ n : ℕ, p ^ n • x = 0 | ⟨λ h x, let ⟨⟨a, ⟨n, rfl⟩⟩, hx⟩ := @h x in ⟨n, hx⟩,
λ h x, let ⟨n, hn⟩ := h x in ⟨⟨_, ⟨n, rfl⟩⟩, hn⟩⟩ | lemma | submodule.is_torsion'_powers_iff | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_order {p : R} (hM : is_torsion' M $ submonoid.powers p) (x : M)
[Π n : ℕ, decidable (p ^ n • x = 0)] | nat.find $ (is_torsion'_powers_iff p).mp hM x | def | submodule.p_order | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submonoid.powers"
] | In a `p ^ ∞`-torsion module (that is, a module where all elements are cancelled by scalar
multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_p_order_smul {p : R} (hM : is_torsion' M $ submonoid.powers p) (x : M)
[Π n : ℕ, decidable (p ^ n • x = 0)] : p ^ p_order hM x • x = 0 | nat.find_spec $ (is_torsion'_powers_iff p).mp hM x | lemma | submodule.pow_p_order_smul | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_torsion_by {p : R} (hM : is_torsion' M $ submonoid.powers p)
(d : ℕ) (hd : d ≠ 0) (s : fin d → M) (hs : span R (set.range s) = ⊤) :
∃ j : fin d, module.is_torsion_by R M (p ^ p_order hM (s j)) | begin
let oj := list.argmax (λ i, p_order hM $ s i) (list.fin_range d),
have hoj : oj.is_some := (option.ne_none_iff_is_some.mp $
λ eq_none, hd $ list.fin_range_eq_nil.mp $ list.argmax_eq_none.mp eq_none),
use option.get hoj,
rw [is_torsion_by_iff_torsion_by_eq_top, eq_top_iff, ← hs, submodule.span_le,
... | lemma | submodule.exists_is_torsion_by | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"eq_top_iff",
"list.argmax",
"list.fin_range",
"list.le_of_mem_argmax",
"list.mem_fin_range",
"module.is_torsion_by",
"option.get_mem",
"pow_add",
"set.range",
"set.range_subset_iff",
"smul_zero",
"submodule.span_le",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
torsion_by_eq_span_singleton {R : Type*} [comm_ring R] (a b : R) (ha : a ∈ R⁰) :
torsion_by R (R ⧸ R ∙ a * b) a = R ∙ (mk _ b) | begin
ext x, rw [mem_torsion_by_iff, mem_span_singleton],
obtain ⟨x, rfl⟩ := mk_surjective x, split; intro h,
{ rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h,
obtain ⟨c, h⟩ := h, rw [smul_eq_mul, smul_eq_mul, mul_comm, mul_assoc,
mul_cancel_left_mem_non_zero_divisor h... | lemma | ideal.quotient.torsion_by_eq_span_singleton | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"comm_ring",
"mul_assoc",
"mul_cancel_left_mem_non_zero_divisor",
"mul_comm",
"smul_eq_mul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_iff_is_torsion_nat [add_comm_monoid M] :
add_monoid.is_torsion M ↔ module.is_torsion ℕ M | begin
refine ⟨λ h x, _, λ h x, _⟩,
{ obtain ⟨n, h0, hn⟩ := (is_of_fin_add_order_iff_nsmul_eq_zero x).mp (h x),
exact ⟨⟨n, mem_non_zero_divisors_of_ne_zero $ ne_of_gt h0⟩, hn⟩ },
{ rw is_of_fin_add_order_iff_nsmul_eq_zero,
obtain ⟨n, hn⟩ := @h x,
refine ⟨n, nat.pos_of_ne_zero (non_zero_divisors.coe_ne_... | theorem | add_monoid.is_torsion_iff_is_torsion_nat | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"add_comm_monoid",
"mem_non_zero_divisors_of_ne_zero",
"module.is_torsion",
"non_zero_divisors.coe_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_torsion_iff_is_torsion_int [add_comm_group M] :
add_monoid.is_torsion M ↔ module.is_torsion ℤ M | begin
refine ⟨λ h x, _, λ h x, _⟩,
{ obtain ⟨n, h0, hn⟩ := (is_of_fin_add_order_iff_nsmul_eq_zero x).mp (h x),
exact ⟨⟨n, mem_non_zero_divisors_of_ne_zero $ ne_of_gt $ int.coe_nat_pos.mpr h0⟩,
(coe_nat_zsmul _ _).trans hn⟩ },
{ rw is_of_fin_add_order_iff_nsmul_eq_zero,
obtain ⟨n, hn⟩ := @h x,
ex... | theorem | add_monoid.is_torsion_iff_is_torsion_int | algebra.module | src/algebra/module/torsion.lean | [
"algebra.direct_sum.module",
"algebra.module.big_operators",
"linear_algebra.isomorphisms",
"group_theory.torsion",
"ring_theory.coprime.ideal",
"ring_theory.finiteness"
] | [
"add_comm_group",
"mem_non_zero_divisors_of_ne_zero",
"module.is_torsion",
"non_zero_divisors.coe_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul_left [has_smul R M] :
has_smul (ulift R) M | ⟨λ s x, s.down • x⟩ | instance | ulift.has_smul_left | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def [has_smul R M] (s : ulift R) (x : M) : s • x = s.down • x | rfl | lemma | ulift.smul_def | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower [has_smul R M] [has_smul M N] [has_smul R N]
[is_scalar_tower R M N] : is_scalar_tower (ulift R) M N | ⟨λ x y z, show (x.down • y) • z = x.down • y • z, from smul_assoc _ _ _⟩ | instance | ulift.is_scalar_tower | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"has_smul",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower' [has_smul R M] [has_smul M N] [has_smul R N]
[is_scalar_tower R M N] : is_scalar_tower R (ulift M) N | ⟨λ x y z, show (x • y.down) • z = x • y.down • z, from smul_assoc _ _ _⟩ | instance | ulift.is_scalar_tower' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"has_smul",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower'' [has_smul R M] [has_smul M N] [has_smul R N]
[is_scalar_tower R M N] : is_scalar_tower R M (ulift N) | ⟨λ x y z, show up ((x • y) • z.down) = ⟨x • y • z.down⟩, by rw smul_assoc⟩ | instance | ulift.is_scalar_tower'' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action [monoid R] [mul_action R M] : mul_action (ulift R) M | { smul := (•),
mul_smul := λ _ _, mul_smul _ _,
one_smul := one_smul _ } | instance | ulift.mul_action | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid",
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action' [monoid R] [mul_action R M] :
mul_action R (ulift M) | { smul := (•),
mul_smul := λ r s ⟨f⟩, ext _ _ $ mul_smul _ _ _,
one_smul := λ ⟨f⟩, ext _ _ $ one_smul _ _ } | instance | ulift.mul_action' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid",
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_zero_class [has_zero M] [smul_zero_class R M] :
smul_zero_class (ulift R) M | { smul_zero := λ _, smul_zero _,
.. ulift.has_smul_left } | instance | ulift.smul_zero_class | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"smul_zero",
"smul_zero_class",
"ulift.has_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_zero_class' [has_zero M] [smul_zero_class R M] :
smul_zero_class R (ulift M) | { smul_zero := λ c, by { ext, simp [smul_zero], } } | instance | ulift.smul_zero_class' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"smul_zero",
"smul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_smul [add_zero_class M] [distrib_smul R M] :
distrib_smul (ulift R) M | { smul_add := λ _, smul_add _ } | instance | ulift.distrib_smul | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_zero_class",
"distrib_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_smul' [add_zero_class M] [distrib_smul R M] :
distrib_smul R (ulift M) | { smul_add := λ c f g, by { ext, simp [smul_add], } } | instance | ulift.distrib_smul' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_zero_class",
"distrib_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_mul_action [monoid R] [add_monoid M] [distrib_mul_action R M] :
distrib_mul_action (ulift R) M | { ..ulift.mul_action,
..ulift.distrib_smul } | instance | ulift.distrib_mul_action | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_monoid",
"distrib_mul_action",
"monoid",
"ulift.distrib_smul",
"ulift.mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_mul_action' [monoid R] [add_monoid M] [distrib_mul_action R M] :
distrib_mul_action R (ulift M) | { ..ulift.mul_action',
..ulift.distrib_smul' } | instance | ulift.distrib_mul_action' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_monoid",
"distrib_mul_action",
"monoid",
"ulift.distrib_smul'",
"ulift.mul_action'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_distrib_mul_action [monoid R] [monoid M] [mul_distrib_mul_action R M] :
mul_distrib_mul_action (ulift R) M | { smul_one := λ _, smul_one _,
smul_mul := λ _, smul_mul' _ } | instance | ulift.mul_distrib_mul_action | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid",
"mul_distrib_mul_action",
"smul_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_distrib_mul_action' [monoid R] [monoid M] [mul_distrib_mul_action R M] :
mul_distrib_mul_action R (ulift M) | { smul_one := λ _, by { ext, simp [smul_one], },
smul_mul := λ c f g, by { ext, simp [smul_mul'], },
..ulift.mul_action' } | instance | ulift.mul_distrib_mul_action' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid",
"mul_distrib_mul_action",
"smul_mul'",
"ulift.mul_action'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero [has_zero R] [has_zero M] [smul_with_zero R M] :
smul_with_zero (ulift R) M | { smul_zero := λ _, smul_zero _,
zero_smul := zero_smul _,
..ulift.has_smul_left } | instance | ulift.smul_with_zero | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"smul_with_zero",
"smul_zero",
"ulift.has_smul_left",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero' [has_zero R] [has_zero M] [smul_with_zero R M] :
smul_with_zero R (ulift M) | { smul_zero := λ _, ulift.ext _ _ $ smul_zero _,
zero_smul := λ _, ulift.ext _ _ $ zero_smul _ _ } | instance | ulift.smul_with_zero' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] :
mul_action_with_zero (ulift R) M | { ..ulift.smul_with_zero } | instance | ulift.mul_action_with_zero | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid_with_zero",
"mul_action_with_zero",
"ulift.smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero' [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] :
mul_action_with_zero R (ulift M) | { ..ulift.smul_with_zero' } | instance | ulift.mul_action_with_zero' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"monoid_with_zero",
"mul_action_with_zero",
"ulift.smul_with_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module [semiring R] [add_comm_monoid M] [module R M] : module (ulift R) M | { add_smul := λ _ _, add_smul _ _,
..ulift.smul_with_zero } | instance | ulift.module | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_comm_monoid",
"add_smul",
"module",
"semiring",
"ulift.smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' [semiring R] [add_comm_monoid M] [module R M] : module R (ulift M) | { add_smul := λ _ _ _, ulift.ext _ _ $ add_smul _ _ _,
..ulift.smul_with_zero' } | instance | ulift.module' | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_comm_monoid",
"add_smul",
"module",
"semiring",
"ulift.smul_with_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_equiv [semiring R] [add_comm_monoid M] [module R M] : ulift M ≃ₗ[R] M | { to_fun := ulift.down,
inv_fun := ulift.up,
map_smul' := λ r x, rfl,
map_add' := λ x y, rfl,
left_inv := by tidy,
right_inv := by tidy, } | def | ulift.module_equiv | algebra.module | src/algebra/module/ulift.lean | [
"algebra.ring.ulift",
"algebra.module.equiv"
] | [
"add_comm_monoid",
"inv_fun",
"module",
"semiring"
] | The `R`-linear equivalence between `ulift M` and `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fundamental_domain : set E | { m | ∀ i, b.repr m i ∈ set.Ico (0 : K) 1 } | def | zspan.fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"set.Ico"
] | The fundamental domain of the ℤ-lattice spanned by `b`. See `zspan.is_add_fundamental_domain`
for the proof that it is the fundamental domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_fundamental_domain {m : E} :
m ∈ fundamental_domain b ↔ ∀ i, b.repr m i ∈ set.Ico (0 : K) 1 | iff.rfl | lemma | zspan.mem_fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"set.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor (m : E) : span ℤ (set.range b) | ∑ i, ⌊b.repr m i⌋ • b.restrict_scalars ℤ i | def | zspan.floor | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"set.range"
] | The map that sends a vector of `E` to the element of the ℤ-lattice spanned by `b` obtained
by rounding down its coordinates on the basis `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ceil (m : E) : span ℤ (set.range b) | ∑ i, ⌈b.repr m i⌉ • b.restrict_scalars ℤ i | def | zspan.ceil | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"set.range"
] | The map that sends a vector of `E` to the element of the ℤ-lattice spanned by `b` obtained
by rounding up its coordinates on the basis `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_floor_apply (m : E) (i : ι) :
b.repr (floor b m) i = ⌊b.repr m i⌋ | by { classical ; simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, finsupp.single_apply,
finset.sum_apply', basis.repr_self, finsupp.smul_single', mul_one, finset.sum_ite_eq', coe_sum,
finset.mem_univ, if_true, coe_smul_of_tower, basis.restrict_scalars_apply, linear_equiv.map_sum] } | lemma | zspan.repr_floor_apply | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"basis.repr_self",
"basis.restrict_scalars_apply",
"finset.mem_univ",
"finset.sum_apply'",
"finsupp.single_apply",
"finsupp.smul_single'",
"linear_equiv.map_sum",
"mul_one",
"zsmul_eq_smul_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_ceil_apply (m : E) (i : ι) :
b.repr (ceil b m) i = ⌈b.repr m i⌉ | by { classical ; simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, finsupp.single_apply,
finset.sum_apply', basis.repr_self, finsupp.smul_single', mul_one, finset.sum_ite_eq', coe_sum,
finset.mem_univ, if_true, coe_smul_of_tower, basis.restrict_scalars_apply, linear_equiv.map_sum] } | lemma | zspan.repr_ceil_apply | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"basis.repr_self",
"basis.restrict_scalars_apply",
"finset.mem_univ",
"finset.sum_apply'",
"finsupp.single_apply",
"finsupp.smul_single'",
"linear_equiv.map_sum",
"mul_one",
"zsmul_eq_smul_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_self_of_mem (m : E) (h : m ∈ span ℤ (set.range b)) : (floor b m : E) = m | begin
apply b.ext_elem,
simp_rw [repr_floor_apply b],
intro i,
obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i,
rw [← hz],
exact congr_arg (coe : ℤ → K) (int.floor_int_cast z),
end | lemma | zspan.floor_eq_self_of_mem | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"int.floor_int_cast",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_self_of_mem (m : E) (h : m ∈ span ℤ (set.range b)) : (ceil b m : E) = m | begin
apply b.ext_elem,
simp_rw [repr_ceil_apply b],
intro i,
obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i,
rw [← hz],
exact congr_arg (coe : ℤ → K) (int.ceil_int_cast z),
end | lemma | zspan.ceil_eq_self_of_mem | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"int.ceil_int_cast",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract (m : E) : E | m - floor b m | def | zspan.fract | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [] | The map that sends a vector `E` to the fundamental domain of the lattice,
see `zspan.fract_mem_fundamental_domain`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fract_apply (m : E) : fract b m = m - floor b m | rfl | lemma | zspan.fract_apply | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_fract_apply (m : E) (i : ι):
b.repr (fract b m) i = int.fract (b.repr m i) | by rw [fract, map_sub, finsupp.coe_sub, pi.sub_apply, repr_floor_apply, int.fract] | lemma | zspan.repr_fract_apply | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"finsupp.coe_sub",
"int.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_fract (m : E) : fract b (fract b m) = fract b m | basis.ext_elem b (λ _, by { classical ; simp only [repr_fract_apply, int.fract_fract] }) | lemma | zspan.fract_fract | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"int.fract_fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_zspan_add (m : E) {v : E} (h : v ∈ span ℤ (set.range b)) :
fract b (v + m) = fract b m | begin
classical,
refine (basis.ext_elem_iff b).mpr (λ i, _),
simp_rw [repr_fract_apply, int.fract_eq_fract],
use (b.restrict_scalars ℤ).repr ⟨v, h⟩ i,
rw [map_add, finsupp.coe_add, pi.add_apply, add_tsub_cancel_right,
← (eq_int_cast (algebra_map ℤ K) _), basis.restrict_scalars_repr_apply, coe_mk],
end | lemma | zspan.fract_zspan_add | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"add_tsub_cancel_right",
"algebra_map",
"basis.ext_elem_iff",
"basis.restrict_scalars_repr_apply",
"eq_int_cast",
"finsupp.coe_add",
"int.fract_eq_fract",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_add_zspan (m : E) {v : E} (h : v ∈ span ℤ (set.range b)) :
fract b (m + v) = fract b m | by { rw [add_comm, fract_zspan_add b m h] } | lemma | zspan.fract_add_zspan | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_eq_self {x : E} :
fract b x = x ↔ x ∈ fundamental_domain b | by { classical ; simp only [basis.ext_elem_iff b, repr_fract_apply, int.fract_eq_self,
mem_fundamental_domain, set.mem_Ico] } | lemma | zspan.fract_eq_self | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"basis.ext_elem_iff",
"int.fract_eq_self",
"set.mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_mem_fundamental_domain (x : E) :
fract b x ∈ fundamental_domain b | fract_eq_self.mp (fract_fract b _) | lemma | zspan.fract_mem_fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_eq_fract (m n : E) :
fract b m = fract b n ↔ -m + n ∈ span ℤ (set.range b) | begin
classical,
rw [eq_comm, basis.ext_elem_iff b],
simp_rw [repr_fract_apply, int.fract_eq_fract, eq_comm, basis.mem_span_iff_repr_mem,
sub_eq_neg_add, map_add, linear_equiv.map_neg, finsupp.coe_add, finsupp.coe_neg, pi.add_apply,
pi.neg_apply, ← (eq_int_cast (algebra_map ℤ K) _), set.mem_range],
end | lemma | zspan.fract_eq_fract | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"algebra_map",
"basis.ext_elem_iff",
"basis.mem_span_iff_repr_mem",
"eq_int_cast",
"finsupp.coe_add",
"finsupp.coe_neg",
"int.fract_eq_fract",
"linear_equiv.map_neg",
"set.mem_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fract_le [has_solid_norm K] (m : E) :
‖fract b m‖ ≤ ∑ i, ‖b i‖ | begin
classical,
calc
‖fract b m‖ = ‖∑ i, b.repr (fract b m) i • b i‖ : by rw b.sum_repr
... = ‖∑ i, int.fract (b.repr m i) • b i‖ : by simp_rw repr_fract_apply
... ≤ ∑ i, ‖int.fract (b.repr m i) • b i‖ : norm_sum_le _ _
... ≤ ∑ i, ‖int.fract (b.repr m i)‖ * ‖b i‖ : by simp_r... | lemma | zspan.norm_fract_le | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"abs_one",
"has_solid_norm",
"int.abs_fract",
"int.fract",
"int.fract_lt_one",
"mul_le_mul_of_nonneg_right",
"norm_le_norm_of_abs_le_abs",
"norm_smul",
"norm_sum_le",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_floor_self (k : K) : (floor (basis.singleton ι K) k : K) = ⌊k⌋ | basis.ext_elem _ (λ _, by rw [repr_floor_apply, basis.singleton_repr, basis.singleton_repr]) | lemma | zspan.coe_floor_self | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"basis.singleton",
"basis.singleton_repr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fract_self (k : K) : (fract (basis.singleton ι K) k : K) = int.fract k | basis.ext_elem _ (λ _, by rw [repr_fract_apply, basis.singleton_repr, basis.singleton_repr]) | lemma | zspan.coe_fract_self | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"basis.singleton",
"basis.singleton_repr",
"int.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fundamental_domain_bounded [finite ι] [has_solid_norm K] :
metric.bounded (fundamental_domain b) | begin
casesI nonempty_fintype ι,
use 2 * ∑ j, ‖b j‖,
intros x hx y hy,
refine le_trans (dist_le_norm_add_norm x y) _,
rw [← fract_eq_self.mpr hx, ← fract_eq_self.mpr hy],
refine (add_le_add (norm_fract_le b x) (norm_fract_le b y)).trans _,
rw ← two_mul,
end | lemma | zspan.fundamental_domain_bounded | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"finite",
"has_solid_norm",
"metric.bounded",
"nonempty_fintype",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_mem_fundamental_domain [fintype ι] (y : span ℤ (set.range b)) (x : E) :
y +ᵥ x ∈ fundamental_domain b ↔ y = -floor b x | by rw [subtype.ext_iff, ← add_right_inj x, add_subgroup_class.coe_neg, ← sub_eq_add_neg,
← fract_apply, ← fract_zspan_add b _ (subtype.mem y), add_comm, ← vadd_eq_add, ← vadd_def,
eq_comm, ← fract_eq_self] | lemma | zspan.vadd_mem_fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"fintype",
"set.range",
"subtype.ext_iff",
"subtype.mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exist_unique_vadd_mem_fundamental_domain [finite ι] (x : E) :
∃! v : span ℤ (set.range b), v +ᵥ x ∈ fundamental_domain b | begin
casesI nonempty_fintype ι,
refine ⟨-floor b x, _, λ y h, _⟩,
{ exact (vadd_mem_fundamental_domain b (-floor b x) x).mpr rfl, },
{ exact (vadd_mem_fundamental_domain b y x).mp h, },
end | lemma | zspan.exist_unique_vadd_mem_fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"finite",
"nonempty_fintype",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fundamental_domain_measurable_set [measurable_space E] [opens_measurable_space E]
[finite ι] :
measurable_set (fundamental_domain b) | begin
haveI : finite_dimensional ℝ E := finite_dimensional.of_fintype_basis b,
let f := (finsupp.linear_equiv_fun_on_finite ℝ ℝ ι).to_linear_map.comp b.repr.to_linear_map,
let D : set (ι → ℝ) := set.pi set.univ (λ i : ι, (set.Ico (0 : ℝ) 1)),
rw ( _ : fundamental_domain b = f⁻¹' D),
{ refine measurable_set_pr... | lemma | zspan.fundamental_domain_measurable_set | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"finite",
"finite_dimensional",
"finite_dimensional.of_fintype_basis",
"finsupp.linear_equiv_fun_on_finite",
"linear_equiv.coe_to_linear_map",
"linear_map.coe_comp",
"linear_map.continuous_of_finite_dimensional",
"measurable",
"measurable_set",
"measurable_set_Ico",
"measurable_set_preimage",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_add_fundamental_domain [finite ι] [measurable_space E]
[opens_measurable_space E] (μ : measure E) :
is_add_fundamental_domain (span ℤ (set.range b)).to_add_subgroup (fundamental_domain b) μ | begin
casesI nonempty_fintype ι,
exact is_add_fundamental_domain.mk' (null_measurable_set (fundamental_domain_measurable_set b))
(λ x, exist_unique_vadd_mem_fundamental_domain b x),
end | lemma | zspan.is_add_fundamental_domain | algebra.module | src/algebra/module/zlattice.lean | [
"measure_theory.group.fundamental_domain"
] | [
"finite",
"measurable_space",
"nonempty_fintype",
"opens_measurable_space",
"set.range"
] | For a ℤ-lattice `submodule.span ℤ (set.range b)`, proves that the set defined
by `zspan.fundamental_domain` is a fundamental domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule (R : Type u) (M : Type v) [semiring R]
[add_comm_monoid M] [module R M] extends add_submonoid M, sub_mul_action R M : Type v. | structure | submodule | algebra.module.submodule | src/algebra/module/submodule/basic.lean | [
"algebra.module.linear_map",
"algebra.module.equiv",
"group_theory.group_action.sub_mul_action",
"group_theory.submonoid.membership"
] | [
"add_comm_monoid",
"add_submonoid",
"module",
"semiring",
"sub_mul_action"
] | A submodule of a module is one which is closed under vector operations.
This is a sufficient condition for the subset of vectors in the submodule
to themselves form a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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