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
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map₂_supr_right (f : M →ₗ[R] N →ₗ[R] P) (s : submodule R M) (t : ι → submodule R N) :
map₂ f s (⨆ i, t i) = ⨆ i, map₂ f s (t i) | begin
suffices :
map₂ f (span R s) (⨆ i, span R (t i : set N)) = (⨆ i, map₂ f (span R s) (span R (t i))),
{ simpa only [span_eq] using this },
simp_rw [map₂_span_span, ← span_Union, map₂_span_span, set.image2_Union_right],
end | lemma | submodule.map₂_supr_right | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2_Union_right",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_span_singleton_eq_map (f : M →ₗ[R] N →ₗ[R] P) (m : M) :
map₂ f (span R {m}) = map (f m) | begin
funext, rw map₂_eq_span_image2, apply le_antisymm,
{ rw [span_le, set.image2_subset_iff],
intros x hx y hy,
obtain ⟨a, rfl⟩ := mem_span_singleton.1 hx,
rw [f.map_smul],
exact smul_mem _ a (mem_map_of_mem hy) },
{ rintro _ ⟨n, hn, rfl⟩,
exact subset_span ⟨m, n, mem_span_singleton_self m, ... | theorem | submodule.map₂_span_singleton_eq_map | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_span_singleton_eq_map_flip (f : M →ₗ[R] N →ₗ[R] P) (s : submodule R M) (n : N) :
map₂ f s (span R {n}) = map (f.flip n) s | by rw [← map₂_span_singleton_eq_map, map₂_flip] | theorem | submodule.map₂_span_singleton_eq_map_flip | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited' : inhabited (submodule R M) | ⟨⊥⟩ | instance | submodule.inhabited' | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coe : ((⊥ : submodule R M) : set M) = {0} | rfl | lemma | submodule.bot_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_to_add_submonoid : (⊥ : submodule R M).to_add_submonoid = ⊥ | rfl | lemma | submodule.bot_to_add_submonoid | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_bot : restrict_scalars S (⊥ : submodule R M) = ⊥ | rfl | lemma | submodule.restrict_scalars_bot | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"restrict_scalars",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : M} : x ∈ (⊥ : submodule R M) ↔ x = 0 | set.mem_singleton_iff | lemma | submodule.mem_bot | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"set.mem_singleton_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_eq_bot_iff {p : submodule R M} :
restrict_scalars S p = ⊥ ↔ p = ⊥ | by simp [set_like.ext_iff] | lemma | submodule.restrict_scalars_eq_bot_iff | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"restrict_scalars",
"set_like.ext_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_bot : unique (⊥ : submodule R M) | ⟨infer_instance, λ x, subtype.ext $ (mem_bot R).1 x.mem⟩ | instance | submodule.unique_bot | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule",
"subtype.ext",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_iff (p : submodule R M) : p = ⊥ ↔ ∀ x ∈ p, x = (0 : M) | ⟨ λ h, h.symm ▸ λ x hx, (mem_bot R).mp hx,
λ h, eq_bot_iff.mpr (λ x hx, (mem_bot R).mpr (h x hx)) ⟩ | lemma | submodule.eq_bot_iff | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"eq_bot_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_ext (x y : (⊥ : submodule R M)) : x = y | begin
rcases x with ⟨x, xm⟩, rcases y with ⟨y, ym⟩, congr,
rw (submodule.eq_bot_iff _).mp rfl x xm,
rw (submodule.eq_bot_iff _).mp rfl y ym,
end | lemma | submodule.bot_ext | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule",
"submodule.eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot_iff (p : submodule R M) : p ≠ ⊥ ↔ ∃ x ∈ p, x ≠ (0 : M) | by { haveI := classical.prop_decidable, simp_rw [ne.def, p.eq_bot_iff, not_forall] } | lemma | submodule.ne_bot_iff | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"not_forall",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonzero_mem_of_bot_lt {p : submodule R M} (bot_lt : ⊥ < p) : ∃ a : p, a ≠ 0 | let ⟨b, hb₁, hb₂⟩ := p.ne_bot_iff.mp bot_lt.ne' in ⟨⟨b, hb₁⟩, hb₂ ∘ (congr_arg coe)⟩ | lemma | submodule.nonzero_mem_of_bot_lt | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_ne_zero_of_ne_bot {p : submodule R M} (h : p ≠ ⊥) : ∃ b : M, b ∈ p ∧ b ≠ 0 | let ⟨b, hb₁, hb₂⟩ := p.ne_bot_iff.mp h in ⟨b, hb₁, hb₂⟩ | lemma | submodule.exists_mem_ne_zero_of_ne_bot | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_equiv_punit : (⊥ : submodule R M) ≃ₗ[R] punit | { to_fun := λ x, punit.star,
inv_fun := λ x, 0,
map_add' := by { intros, ext, },
map_smul' := by { intros, ext, },
left_inv := by { intro x, ext, },
right_inv := by { intro x, ext, }, } | def | submodule.bot_equiv_punit | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"inv_fun",
"submodule"
] | The bottom submodule is linearly equivalent to punit as an `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_bot_of_subsingleton (p : submodule R M) [subsingleton p] : p = ⊥ | begin
rw eq_bot_iff,
intros v hv,
exact congr_arg coe (subsingleton.elim (⟨v, hv⟩ : p) 0)
end | lemma | submodule.eq_bot_of_subsingleton | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"eq_bot_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_coe : ((⊤ : submodule R M) : set M) = set.univ | rfl | lemma | submodule.top_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_to_add_submonoid : (⊤ : submodule R M).to_add_submonoid = ⊤ | rfl | lemma | submodule.top_to_add_submonoid | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top {x : M} : x ∈ (⊤ : submodule R M) | trivial | lemma | submodule.mem_top | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_top : restrict_scalars S (⊤ : submodule R M) = ⊤ | rfl | lemma | submodule.restrict_scalars_top | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"restrict_scalars",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_eq_top_iff {p : submodule R M} :
restrict_scalars S p = ⊤ ↔ p = ⊤ | by simp [set_like.ext_iff] | lemma | submodule.restrict_scalars_eq_top_iff | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"restrict_scalars",
"set_like.ext_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff' {p : submodule R M} : p = ⊤ ↔ ∀ x, x ∈ p | eq_top_iff.trans ⟨λ h x, h trivial, λ h x _, h x⟩ | lemma | submodule.eq_top_iff' | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_equiv : (⊤ : submodule R M) ≃ₗ[R] M | { to_fun := λ x, x,
inv_fun := λ x, ⟨x, by simp⟩,
map_add' := by { intros, refl, },
map_smul' := by { intros, refl, },
left_inv := by { intro x, ext, refl, },
right_inv := by { intro x, refl, }, } | def | submodule.top_equiv | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"inv_fun",
"submodule"
] | The top submodule is linearly equivalent to the module.
This is the module version of `add_submonoid.top_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inf_le' {S : set (submodule R M)} {p} : p ∈ S → Inf S ≤ p | set.bInter_subset_of_mem | lemma | submodule.Inf_le' | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"set.bInter_subset_of_mem",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_Inf' {S : set (submodule R M)} {p} : (∀q ∈ S, p ≤ q) → p ≤ Inf S | set.subset_Inter₂ | lemma | submodule.le_Inf' | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"set.subset_Inter₂",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe : ↑(p ⊓ q) = (p ∩ q : set M) | rfl | theorem | submodule.inf_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {p q : submodule R M} {x : M} :
x ∈ p ⊓ q ↔ x ∈ p ∧ x ∈ q | iff.rfl | theorem | submodule.mem_inf | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_coe (P : set (submodule R M)) : (↑(Inf P) : set M) = ⋂ p ∈ P, ↑p | rfl | theorem | submodule.Inf_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_inf_coe {ι} (s : finset ι) (p : ι → submodule R M) :
(↑(s.inf p) : set M) = ⋂ i ∈ s, ↑(p i) | begin
letI := classical.dec_eq ι,
refine s.induction_on _ (λ i s hi ih, _),
{ simp },
{ rw [finset.inf_insert, inf_coe, ih],
simp },
end | theorem | submodule.finset_inf_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"classical.dec_eq",
"finset",
"finset.inf_insert",
"ih",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_coe {ι} (p : ι → submodule R M) :
(↑⨅ i, p i : set M) = ⋂ i, ↑(p i) | by rw [infi, Inf_coe]; ext a; simp; exact
⟨λ h i, h _ i rfl, λ h i x e, e ▸ h _⟩ | theorem | submodule.infi_coe | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"infi",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (submodule R M)} {x : M} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | set.mem_Inter₂ | lemma | submodule.mem_Inf | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"set.mem_Inter₂",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι} (p : ι → submodule R M) {x} :
x ∈ (⨅ i, p i) ↔ ∀ i, x ∈ p i | by rw [← set_like.mem_coe, infi_coe, set.mem_Inter]; refl | theorem | submodule.mem_infi | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"set.mem_Inter",
"set_like.mem_coe",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_finset_inf {ι} {s : finset ι} {p : ι → submodule R M} {x : M} :
x ∈ s.inf p ↔ ∀ i ∈ s, x ∈ p i | by simp only [← set_like.mem_coe, finset_inf_coe, set.mem_Inter] | theorem | submodule.mem_finset_inf | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"finset",
"set.mem_Inter",
"set_like.mem_coe",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_left {S T : submodule R M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T | show S ≤ S ⊔ T, from le_sup_left | lemma | submodule.mem_sup_left | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"le_sup_left",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_right {S T : submodule R M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T | show T ≤ S ⊔ T, from le_sup_right | lemma | submodule.mem_sup_right | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"le_sup_right",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_sup {S T : submodule R M} {s t : M} (hs : s ∈ S) (ht : t ∈ T) : s + t ∈ S ⊔ T | add_mem (mem_sup_left hs) (mem_sup_right ht) | lemma | submodule.add_mem_sup | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_sup {R' M' : Type*} [ring R'] [add_comm_group M'] [module R' M']
{S T : submodule R' M'} {s t : M'} (hs : s ∈ S) (ht : t ∈ T) :
s - t ∈ S ⊔ T | begin
rw sub_eq_add_neg,
exact add_mem_sup hs (neg_mem ht),
end | lemma | submodule.sub_mem_sup | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_comm_group",
"module",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_supr_of_mem {ι : Sort*} {b : M} {p : ι → submodule R M} (i : ι) (h : b ∈ p i) :
b ∈ (⨆i, p i) | have p i ≤ (⨆i, p i) := le_supr p i,
@this b h | lemma | submodule.mem_supr_of_mem | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"le_supr",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mem_supr {ι : Type*} [fintype ι] {f : ι → M} {p : ι → submodule R M}
(h : ∀ i, f i ∈ p i) :
∑ i, f i ∈ ⨆ i, p i | sum_mem $ λ i hi, mem_supr_of_mem i (h i) | lemma | submodule.sum_mem_supr | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"fintype",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mem_bsupr {ι : Type*} {s : finset ι} {f : ι → M} {p : ι → submodule R M}
(h : ∀ i ∈ s, f i ∈ p i) :
∑ i in s, f i ∈ ⨆ i ∈ s, p i | sum_mem $ λ i hi, mem_supr_of_mem i $ mem_supr_of_mem hi (h i hi) | lemma | submodule.sum_mem_bsupr | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"finset",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup_of_mem {S : set (submodule R M)} {s : submodule R M}
(hs : s ∈ S) : ∀ {x : M}, x ∈ s → x ∈ Sup S | show s ≤ Sup S, from le_Sup hs | lemma | submodule.mem_Sup_of_mem | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"le_Sup",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_def {p p' : submodule R M} :
disjoint p p' ↔ ∀ x ∈ p, x ∈ p' → x = (0:M) | disjoint_iff_inf_le.trans $ show (∀ x, x ∈ p ∧ x ∈ p' → x ∈ ({0} : set M)) ↔ _, by simp | theorem | submodule.disjoint_def | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"disjoint",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_def' {p p' : submodule R M} :
disjoint p p' ↔ ∀ (x ∈ p) (y ∈ p'), x = y → x = (0:M) | disjoint_def.trans ⟨λ h x hx y hy hxy, h x hx $ hxy.symm ▸ hy,
λ h x hx hx', h _ hx x hx' rfl⟩ | theorem | submodule.disjoint_def' | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"disjoint",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_coe_mem_of_disjoint (hpq : disjoint p q) {a : p} (ha : (a : M) ∈ q) :
a = 0 | by exact_mod_cast disjoint_def.mp hpq a (coe_mem a) ha | lemma | submodule.eq_zero_of_coe_mem_of_disjoint | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_submonoid.to_nat_submodule : add_submonoid M ≃o submodule ℕ M | { to_fun := λ S,
{ smul_mem' := λ r s hs, show r • s ∈ S, from nsmul_mem hs _, ..S },
inv_fun := submodule.to_add_submonoid,
left_inv := λ ⟨S, _, _⟩, rfl,
right_inv := λ ⟨S, _, _, _⟩, rfl,
map_rel_iff' := λ a b, iff.rfl } | def | add_submonoid.to_nat_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_submonoid",
"inv_fun",
"submodule"
] | An additive submonoid is equivalent to a ℕ-submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_submonoid.to_nat_submodule_symm :
⇑(add_submonoid.to_nat_submodule.symm : _ ≃o add_submonoid M) = submodule.to_add_submonoid | rfl | lemma | add_submonoid.to_nat_submodule_symm | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_submonoid.coe_to_nat_submodule (S : add_submonoid M) :
(S.to_nat_submodule : set M) = S | rfl | lemma | add_submonoid.coe_to_nat_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_submonoid.to_nat_submodule_to_add_submonoid (S : add_submonoid M) :
S.to_nat_submodule.to_add_submonoid = S | add_submonoid.to_nat_submodule.symm_apply_apply S | lemma | add_submonoid.to_nat_submodule_to_add_submonoid | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.to_add_submonoid_to_nat_submodule (S : submodule ℕ M) :
S.to_add_submonoid.to_nat_submodule = S | add_submonoid.to_nat_submodule.apply_symm_apply S | lemma | submodule.to_add_submonoid_to_nat_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_subgroup.to_int_submodule : add_subgroup M ≃o submodule ℤ M | { to_fun := λ S,
{ smul_mem' := λ r s hs, S.zsmul_mem hs _, ..S},
inv_fun := submodule.to_add_subgroup,
left_inv := λ ⟨S, _, _, _⟩, rfl,
right_inv := λ ⟨S, _, _, _⟩, rfl,
map_rel_iff' := λ a b, iff.rfl } | def | add_subgroup.to_int_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_subgroup",
"inv_fun",
"submodule",
"submodule.to_add_subgroup"
] | An additive subgroup is equivalent to a ℤ-submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_subgroup.to_int_submodule_symm :
⇑(add_subgroup.to_int_submodule.symm : _ ≃o add_subgroup M) = submodule.to_add_subgroup | rfl | lemma | add_subgroup.to_int_submodule_symm | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_subgroup",
"submodule.to_add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_subgroup.coe_to_int_submodule (S : add_subgroup M) :
(S.to_int_submodule : set M) = S | rfl | lemma | add_subgroup.coe_to_int_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_subgroup.to_int_submodule_to_add_subgroup (S : add_subgroup M) :
S.to_int_submodule.to_add_subgroup = S | add_subgroup.to_int_submodule.symm_apply_apply S | lemma | add_subgroup.to_int_submodule_to_add_subgroup | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.to_add_subgroup_to_int_submodule (S : submodule ℤ M) :
S.to_add_subgroup.to_int_submodule = S | add_subgroup.to_int_submodule.apply_symm_apply S | lemma | submodule.to_add_subgroup_to_int_submodule | algebra.module.submodule | src/algebra/module/submodule/lattice.lean | [
"algebra.module.submodule.basic",
"algebra.punit_instances"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pointwise_neg : has_neg (submodule R M) | { neg := λ p,
{ carrier := -(p : set M),
smul_mem' := λ r m hm, set.mem_neg.2 $ smul_neg r m ▸ p.smul_mem r $ set.mem_neg.1 hm,
..(- p.to_add_submonoid) } } | def | submodule.has_pointwise_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"smul_neg",
"submodule"
] | The submodule with every element negated. Note if `R` is a ring and not just a semiring, this
is a no-op, as shown by `submodule.neg_eq_self`.
Recall that When `R` is the semiring corresponding to the nonnegative elements of `R'`,
`submodule R' M` is the type of cones of `M`. This instance reflects such cones about `0... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_set_neg (S : submodule R M) : ↑(-S) = -(S : set M) | rfl | lemma | submodule.coe_set_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_to_add_submonoid (S : submodule R M) :
(-S).to_add_submonoid = -S.to_add_submonoid | rfl | lemma | submodule.neg_to_add_submonoid | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_neg {g : M} {S : submodule R M} : g ∈ -S ↔ -g ∈ S | iff.rfl | lemma | submodule.mem_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_involutive_pointwise_neg : has_involutive_neg (submodule R M) | { neg := has_neg.neg,
neg_neg := λ S, set_like.coe_injective $ neg_neg _ } | def | submodule.has_involutive_pointwise_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"has_involutive_neg",
"set_like.coe_injective",
"submodule"
] | `submodule.has_pointwise_neg` is involutive.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_le_neg (S T : submodule R M) : -S ≤ -T ↔ S ≤ T | set_like.coe_subset_coe.symm.trans set.neg_subset_neg | lemma | submodule.neg_le_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_le (S T : submodule R M) : -S ≤ T ↔ S ≤ -T | set_like.coe_subset_coe.symm.trans set.neg_subset | lemma | submodule.neg_le | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_order_iso : submodule R M ≃o submodule R M | { to_equiv := equiv.neg _,
map_rel_iff' := neg_le_neg } | def | submodule.neg_order_iso | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | `submodule.has_pointwise_neg` as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_neg (s : set M) : span R (-s) = -(span R s) | begin
apply le_antisymm,
{ rw [span_le, coe_set_neg, ←set.neg_subset, neg_neg],
exact subset_span },
{ rw [neg_le, span_le, coe_set_neg, ←set.neg_subset],
exact subset_span }
end | lemma | submodule.closure_neg | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_inf (S T : submodule R M) : -(S ⊓ T) = (-S) ⊓ (-T) | set_like.coe_injective set.inter_neg | lemma | submodule.neg_inf | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_sup (S T : submodule R M) : -(S ⊔ T) = (-S) ⊔ (-T) | (neg_order_iso : submodule R M ≃o submodule R M).map_sup S T | lemma | submodule.neg_sup | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_bot : -(⊥ : submodule R M) = ⊥ | set_like.coe_injective $ (set.neg_singleton 0).trans $ congr_arg _ neg_zero | lemma | submodule.neg_bot | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_top : -(⊤ : submodule R M) = ⊤ | set_like.coe_injective $ set.neg_univ | lemma | submodule.neg_top | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_infi {ι : Sort*} (S : ι → submodule R M) : -(⨅ i, S i) = ⨅ i, -S i | (neg_order_iso : submodule R M ≃o submodule R M).map_infi _ | lemma | submodule.neg_infi | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"map_infi",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_supr {ι : Sort*} (S : ι → submodule R M) : -(⨆ i, S i) = ⨆ i, -(S i) | (neg_order_iso : submodule R M ≃o submodule R M).map_supr _ | lemma | submodule.neg_supr | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"map_supr",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_self [ring R] [add_comm_group M] [module R M] (p : submodule R M) : -p = p | ext $ λ _, p.neg_mem_iff | lemma | submodule.neg_eq_self | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"add_comm_group",
"module",
"neg_eq_self",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_add_comm_monoid : add_comm_monoid (submodule R M) | { add := (⊔),
add_assoc := λ _ _ _, sup_assoc,
zero := ⊥,
zero_add := λ _, bot_sup_eq,
add_zero := λ _, sup_bot_eq,
add_comm := λ _ _, sup_comm } | instance | submodule.pointwise_add_comm_monoid | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"add_comm_monoid",
"bot_sup_eq",
"submodule",
"sup_assoc",
"sup_bot_eq",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_sup (p q : submodule R M) : p + q = p ⊔ q | rfl | lemma | submodule.add_eq_sup | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"add_eq_sup",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_bot : (0 : submodule R M) = ⊥ | rfl | lemma | submodule.zero_eq_bot | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_distrib_mul_action : distrib_mul_action α (submodule R M) | { smul := λ a S, S.map (distrib_mul_action.to_linear_map R M a : M →ₗ[R] M),
one_smul := λ S,
(congr_arg (λ f : module.End R M, S.map f) (linear_map.ext $ by exact one_smul α)).trans
S.map_id,
mul_smul := λ a₁ a₂ S,
(congr_arg (λ f : module.End R M, S.map f) (linear_map.ext $ by exact mul_smul _ _)).t... | def | submodule.pointwise_distrib_mul_action | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"distrib_mul_action",
"distrib_mul_action.to_linear_map",
"linear_map.ext",
"module.End",
"one_smul",
"smul_add",
"smul_zero",
"submodule"
] | The action on a submodule corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pointwise_smul (a : α) (S : submodule R M) : ↑(a • S) = a • (S : set M) | rfl | lemma | submodule.coe_pointwise_smul | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_add_submonoid (a : α) (S : submodule R M) :
(a • S).to_add_submonoid = a • S.to_add_submonoid | rfl | lemma | submodule.pointwise_smul_to_add_submonoid | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_add_subgroup {R M : Type*}
[ring R] [add_comm_group M] [distrib_mul_action α M] [module R M] [smul_comm_class α R M]
(a : α) (S : submodule R M) :
(a • S).to_add_subgroup = a • S.to_add_subgroup | rfl | lemma | submodule.pointwise_smul_to_add_subgroup | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"add_comm_group",
"distrib_mul_action",
"module",
"ring",
"smul_comm_class",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul (m : M) (a : α) (S : submodule R M) : m ∈ S → a • m ∈ a • S | (set.smul_mem_smul_set : _ → _ ∈ a • (S : set M)) | lemma | submodule.smul_mem_pointwise_smul | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"set.smul_mem_smul_set",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_bot' (a : α) : a • (⊥ : submodule R M) = ⊥ | map_bot _ | lemma | submodule.smul_bot' | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | See also `submodule.smul_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_sup' (a : α) (S T : submodule R M) : a • (S ⊔ T) = a • S ⊔ a • T | map_sup _ _ _ | lemma | submodule.smul_sup' | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"submodule"
] | See also `submodule.smul_sup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_span (a : α) (s : set M) : a • span R s = span R (a • s) | map_span _ _ | lemma | submodule.smul_span | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_smul (a : α) (s : set M) : span R (a • s) = a • span R s | eq.symm (span_image _).symm | lemma | submodule.span_smul | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_central_scalar [distrib_mul_action αᵐᵒᵖ M] [smul_comm_class αᵐᵒᵖ R M]
[is_central_scalar α M] :
is_central_scalar α (submodule R M) | ⟨λ a S, congr_arg (λ f : module.End R M, S.map f) $ linear_map.ext $ by exact op_smul_eq_smul _⟩ | instance | submodule.pointwise_central_scalar | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"distrib_mul_action",
"is_central_scalar",
"linear_map.ext",
"module.End",
"smul_comm_class",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_self_of_tower {α : Type*}
[semiring α] [module α R] [module α M] [smul_comm_class α R M] [is_scalar_tower α R M]
(a : α) (S : submodule R M) : a • S ≤ S | begin
rintro y ⟨x, hx, rfl⟩,
exact smul_of_tower_mem _ a hx,
end | lemma | submodule.smul_le_self_of_tower | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"is_scalar_tower",
"module",
"semiring",
"smul_comm_class",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_mul_action_with_zero : mul_action_with_zero α (submodule R M) | { zero_smul := λ S,
(congr_arg (λ f : M →ₗ[R] M, S.map f) (linear_map.ext $ by exact zero_smul α)).trans S.map_zero,
.. submodule.pointwise_distrib_mul_action } | def | submodule.pointwise_mul_action_with_zero | algebra.module.submodule | src/algebra/module/submodule/pointwise.lean | [
"group_theory.subgroup.pointwise",
"linear_algebra.span"
] | [
"linear_map.ext",
"mul_action_with_zero",
"submodule",
"submodule.pointwise_distrib_mul_action",
"zero_smul"
] | The action on a submodule corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale.
This is a stronger version of `submodule.pointwise_distrib_mul_action`. Note that `add_smul` does
not hold so this cannot be stated as a `module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_algebra : Type (max u₁ u₂) | G →₀ k | def | monoid_algebra | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [] | The monoid algebra over a semiring `k` generated by the monoid `G`.
It is the type of finite formal `k`-linear combinations of terms of `G`,
endowed with the convolution product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_nc (f : k →+ R) (g : G → R) : monoid_algebra k G →+ R | lift_add_hom (λ x : G, (add_monoid_hom.mul_right (g x)).comp f) | def | monoid_algebra.lift_nc | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"add_monoid_hom.mul_right",
"monoid_algebra"
] | A non-commutative version of `monoid_algebra.lift`: given a additive homomorphism `f : k →+ R`
and a homomorphism `g : G → R`, returns the additive homomorphism from
`monoid_algebra k G` such that `lift_nc f g (single a b) = f b * g a`. If `f` is a ring homomorphism
and the range of either `f` or `g` is in center of `R... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_nc_single (f : k →+ R) (g : G → R) (a : G) (b : k) :
lift_nc f g (single a b) = f b * g a | lift_add_hom_apply_single _ _ _ | lemma | monoid_algebra.lift_nc_single | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def {f g : monoid_algebra k G} :
f * g = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ * a₂) (b₁ * b₂)) | rfl | lemma | monoid_algebra.mul_def | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nc_mul {g_hom : Type*} [mul_hom_class g_hom G R] (f : k →+* R) (g : g_hom)
(a b : monoid_algebra k G) (h_comm : ∀ {x y}, y ∈ a.support → commute (f (b x)) (g y)) :
lift_nc (f : k →+ R) g (a * b) = lift_nc (f : k →+ R) g a * lift_nc (f : k →+ R) g b | begin
conv_rhs { rw [← sum_single a, ← sum_single b] },
simp_rw [mul_def, (lift_nc _ g).map_finsupp_sum, lift_nc_single, finsupp.sum_mul,
finsupp.mul_sum],
refine finset.sum_congr rfl (λ y hy, finset.sum_congr rfl (λ x hx, _)),
simp [mul_assoc, (h_comm hy).left_comm]
end | lemma | monoid_algebra.lift_nc_mul | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"commute",
"finsupp.mul_sum",
"finsupp.sum_mul",
"monoid_algebra",
"mul_assoc",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : monoid_algebra k G) = single 1 1 | rfl | lemma | monoid_algebra.one_def | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nc_one {g_hom : Type*} [one_hom_class g_hom G R] (f : k →+* R) (g : g_hom) :
lift_nc (f : k →+ R) g 1 = 1 | by simp [one_def] | lemma | monoid_algebra.lift_nc_one | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_def (n : ℕ) : (n : monoid_algebra k G) = single 1 n | rfl | lemma | monoid_algebra.nat_cast_def | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nc_ring_hom (f : k →+* R) (g : G →* R) (h_comm : ∀ x y, commute (f x) (g y)) :
monoid_algebra k G →+* R | { to_fun := lift_nc (f : k →+ R) g,
map_one' := lift_nc_one _ _,
map_mul' := λ a b, lift_nc_mul _ _ _ _ $ λ _ _ _, h_comm _ _,
..(lift_nc (f : k →+ R) g)} | def | monoid_algebra.lift_nc_ring_hom | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"commute",
"monoid_algebra"
] | `lift_nc` as a `ring_hom`, for when `f x` and `g y` commute | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_cast_def [ring k] [mul_one_class G] (z : ℤ) : (z : monoid_algebra k G) = single 1 z | rfl | lemma | monoid_algebra.int_cast_def | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"monoid_algebra",
"mul_one_class",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_distrib_mul_action_self [group G] [semiring k] :
distrib_mul_action G (monoid_algebra k G) | finsupp.comap_distrib_mul_action | def | monoid_algebra.comap_distrib_mul_action_self | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"distrib_mul_action",
"finsupp.comap_distrib_mul_action",
"group",
"monoid_algebra",
"semiring"
] | This is not an instance as it conflicts with `monoid_algebra.distrib_mul_action` when `G = kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_apply [decidable_eq G] [has_mul G] (f g : monoid_algebra k G) (x : G) :
(f * g) x = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, if a₁ * a₂ = x then b₁ * b₂ else 0) | begin
rw [mul_def],
simp only [finsupp.sum_apply, single_apply],
end | lemma | monoid_algebra.mul_apply | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"finsupp.sum_apply",
"monoid_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply_antidiagonal [has_mul G] (f g : monoid_algebra k G) (x : G) (s : finset (G × G))
(hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x) :
(f * g) x = ∑ p in s, (f p.1 * g p.2) | by classical; exact
let F : G × G → k := λ p, if p.1 * p.2 = x then f p.1 * g p.2 else 0 in
calc (f * g) x = (∑ a₁ in f.support, ∑ a₂ in g.support, F (a₁, a₂)) :
mul_apply f g x
... = ∑ p in f.support ×ˢ g.support, F p : finset.sum_product.symm
... = ∑ p in (f.support ×ˢ g.support).filter (λ p : G × G, p.1 * p.2 = x)... | lemma | monoid_algebra.mul_apply_antidiagonal | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"filter",
"finset",
"monoid_algebra",
"mul_zero",
"not_and",
"not_not",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_mul_single [has_mul G] {a₁ a₂ : G} {b₁ b₂ : k} :
(single a₁ b₁ : monoid_algebra k G) * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂) | (sum_single_index (by simp only [zero_mul, single_zero, sum_zero])).trans
(sum_single_index (by rw [mul_zero, single_zero])) | lemma | monoid_algebra.single_mul_single | algebra.monoid_algebra | src/algebra/monoid_algebra/basic.lean | [
"algebra.algebra.equiv",
"algebra.big_operators.finsupp",
"algebra.hom.non_unital_alg",
"algebra.module.big_operators",
"linear_algebra.finsupp"
] | [
"monoid_algebra",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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