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 |
|---|---|---|---|---|---|---|---|---|---|---|
mem_to_add_submonoid (p : submodule R M) (x : M) : x ∈ p.to_add_submonoid ↔ x ∈ p | iff.rfl | theorem | submodule.mem_to_add_submonoid | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk {S : set M} {x : M} (h₁ h₂ h₃) : x ∈ (⟨S, h₁, h₂, h₃⟩ : submodule R M) ↔ x ∈ S | iff.rfl | lemma | submodule.mem_mk | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_set_mk (S : set M) (h₁ h₂ h₃) :
((⟨S, h₁, h₂, h₃⟩ : submodule R M) : set M) = S | rfl | lemma | submodule.coe_set_mk | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_mk {S S' : set M} (h₁ h₂ h₃ h₁' h₂' h₃') :
(⟨S, h₁, h₂, h₃⟩ : submodule R M) ≤ (⟨S', h₁', h₂', h₃'⟩ : submodule R M) ↔ S ⊆ S' | iff.rfl | lemma | submodule.mk_le_mk | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q | set_like.ext h | theorem | submodule.ext | 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"
] | [
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (p : submodule R M) (s : set M) (hs : s = ↑p) : submodule R M | { carrier := s,
zero_mem' := hs.symm ▸ p.zero_mem',
add_mem' := λ _ _, hs.symm ▸ p.add_mem',
smul_mem' := hs.symm ▸ p.smul_mem' } | def | submodule.copy | 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"
] | [
"submodule"
] | Copy of a submodule with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : submodule R M) (s : set M) (hs : s = ↑S) :
(S.copy s hs : set M) = s | rfl | lemma | submodule.coe_copy | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (S : submodule R M) (s : set M) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | submodule.copy_eq | 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"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_injective :
injective (to_add_submonoid : submodule R M → add_submonoid M) | λ p q h, set_like.ext'_iff.2 (show _, from set_like.ext'_iff.1 h) | theorem | submodule.to_add_submonoid_injective | 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_submonoid",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_eq : p.to_add_submonoid = q.to_add_submonoid ↔ p = q | to_add_submonoid_injective.eq_iff | theorem | submodule.to_add_submonoid_eq | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_strict_mono :
strict_mono (to_add_submonoid : submodule R M → add_submonoid M) | λ _ _, id | lemma | submodule.to_add_submonoid_strict_mono | 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_submonoid",
"strict_mono",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_le : p.to_add_submonoid ≤ q.to_add_submonoid ↔ p ≤ q | iff.rfl | lemma | submodule.to_add_submonoid_le | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_mono : monotone (to_add_submonoid : submodule R M → add_submonoid M) | to_add_submonoid_strict_mono.monotone | lemma | submodule.to_add_submonoid_mono | 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_submonoid",
"monotone",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_submonoid (p : submodule R M) :
(p.to_add_submonoid : set M) = p | rfl | theorem | submodule.coe_to_add_submonoid | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sub_mul_action_injective :
injective (to_sub_mul_action : submodule R M → sub_mul_action R M) | λ p q h, set_like.ext'_iff.2 (show _, from set_like.ext'_iff.1 h) | theorem | submodule.to_sub_mul_action_injective | 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"
] | [
"sub_mul_action",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sub_mul_action_eq : p.to_sub_mul_action = q.to_sub_mul_action ↔ p = q | to_sub_mul_action_injective.eq_iff | theorem | submodule.to_sub_mul_action_eq | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sub_mul_action_strict_mono :
strict_mono (to_sub_mul_action : submodule R M → sub_mul_action R M) | λ _ _, id | lemma | submodule.to_sub_mul_action_strict_mono | 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"
] | [
"strict_mono",
"sub_mul_action",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sub_mul_action_mono : monotone (to_sub_mul_action : submodule R M → sub_mul_action R M) | to_sub_mul_action_strict_mono.monotone | lemma | submodule.to_sub_mul_action_mono | 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"
] | [
"monotone",
"sub_mul_action",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_sub_mul_action (p : submodule R M) :
(p.to_sub_mul_action : set M) = p | rfl | theorem | submodule.coe_to_sub_mul_action | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_module : module R S' | subtype.coe_injective.module R (add_submonoid_class.subtype S') (set_like.coe_smul S') | instance | smul_mem_class.to_module | 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"
] | [
"module",
"set_like.coe_smul"
] | A submodule of a `module` is a `module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype : S' →ₗ[R] M | ⟨coe, λ _ _, rfl, λ _ _, rfl⟩ | def | smul_mem_class.subtype | 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"
] | [] | The natural `R`-linear map from a submodule of an `R`-module `M` to `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_subtype : (smul_mem_class.subtype S' : S' → M) = coe | rfl | theorem | smul_mem_class.coe_subtype | 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"
] | [
"smul_mem_class.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier : x ∈ p.carrier ↔ x ∈ (p : set M) | iff.rfl | lemma | submodule.mem_carrier | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem : (0 : M) ∈ p | zero_mem _ | lemma | submodule.zero_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem (h₁ : x ∈ p) (h₂ : y ∈ p) : x + y ∈ p | add_mem h₁ h₂ | lemma | submodule.add_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem (r : R) (h : x ∈ p) : r • x ∈ p | p.smul_mem' r h | lemma | submodule.smul_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_of_tower_mem [has_smul S R] [has_smul S M] [is_scalar_tower S R M]
(r : S) (h : x ∈ p) : r • x ∈ p | p.to_sub_mul_action.smul_of_tower_mem r h | lemma | submodule.smul_of_tower_mem | 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"
] | [
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mem {t : finset ι} {f : ι → M} : (∀c∈t, f c ∈ p) → (∑ i in t, f i) ∈ p | sum_mem | lemma | submodule.sum_mem | 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"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_smul_mem {t : finset ι} {f : ι → M} (r : ι → R)
(hyp : ∀ c ∈ t, f c ∈ p) : (∑ i in t, r i • f i) ∈ p | sum_mem (λ i hi, smul_mem _ _ (hyp i hi)) | lemma | submodule.sum_smul_mem | 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"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_iff' [group G] [mul_action G M] [has_smul G R] [is_scalar_tower G R M]
(g : G) : g • x ∈ p ↔ x ∈ p | p.to_sub_mul_action.smul_mem_iff' g | lemma | submodule.smul_mem_iff' | 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"
] | [
"group",
"has_smul",
"is_scalar_tower",
"mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower' {S' : Type*}
[has_smul S R] [has_smul S M] [has_smul S' R] [has_smul S' M] [has_smul S S']
[is_scalar_tower S' R M] [is_scalar_tower S S' M] [is_scalar_tower S R M] :
is_scalar_tower S S' p | p.to_sub_mul_action.is_scalar_tower' | instance | submodule.is_scalar_tower' | 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"
] | [
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty : (p : set M).nonempty | ⟨0, p.zero_mem⟩ | lemma | submodule.nonempty | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero {x} (h : x ∈ p) : (⟨x, h⟩ : p) = 0 ↔ x = 0 | subtype.ext_iff_val | lemma | submodule.mk_eq_zero | 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"
] | [
"subtype.ext_iff_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero {x : p} : (x : M) = 0 ↔ x = 0 | (set_like.coe_eq_coe : (x : M) = (0 : p) ↔ x = 0) | lemma | submodule.coe_eq_zero | 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"
] | [
"set_like.coe_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : p) : (↑(x + y) : M) = ↑x + ↑y | rfl | lemma | submodule.coe_add | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : p) : M) = 0 | rfl | lemma | submodule.coe_zero | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : R) (x : p) : ((r • x : p) : M) = r • ↑x | rfl | lemma | submodule.coe_smul | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_of_tower [has_smul S R] [has_smul S M] [is_scalar_tower S R M]
(r : S) (x : p) : ((r • x : p) : M) = r • ↑x | rfl | lemma | submodule.coe_smul_of_tower | 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"
] | [
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (x : M) (hx : x ∈ p) : ((⟨x, hx⟩ : p) : M) = x | rfl | lemma | submodule.coe_mk | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mem (x : p) : (x : M) ∈ p | x.2 | lemma | submodule.coe_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' [semiring S] [has_smul S R] [module S M] [is_scalar_tower S R M] : module S p | by refine {smul := (•), ..p.to_sub_mul_action.mul_action', ..};
{ intros, apply set_coe.ext, simp [smul_add, add_smul, mul_smul] } | instance | submodule.module' | 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_smul",
"has_smul",
"is_scalar_tower",
"module",
"semiring",
"set_coe.ext",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors [no_zero_smul_divisors R M] : no_zero_smul_divisors R p | ⟨λ c x h,
have c = 0 ∨ (x : M) = 0,
from eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg coe h),
this.imp_right (@subtype.ext_iff _ _ x 0).mpr⟩ | instance | submodule.no_zero_smul_divisors | 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"
] | [
"no_zero_smul_divisors",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype : p →ₗ[R] M | by refine {to_fun := coe, ..}; simp [coe_smul] | def | submodule.subtype | 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"
] | [] | Embedding of a submodule `p` to the ambient space `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype_apply (x : p) : p.subtype x = x | rfl | theorem | submodule.subtype_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subtype : ((submodule.subtype p) : p → M) = coe | rfl | lemma | submodule.coe_subtype | 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"
] | [
"submodule.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_subtype : injective p.subtype | subtype.coe_injective | lemma | submodule.injective_subtype | 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"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sum (x : ι → p) (s : finset ι) : ↑(∑ i in s, x i) = ∑ i in s, (x i : M) | map_sum p.subtype _ _ | lemma | submodule.coe_sum | 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"
] | [
"finset"
] | Note the `add_submonoid` version of this lemma is called `add_submonoid.coe_finset_sum`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_comm_class [has_vadd M β] [has_vadd α β] [vadd_comm_class M α β] :
vadd_comm_class p α β | ⟨λ a, (vadd_comm (a : M) : _)⟩ | instance | submodule.vadd_comm_class | 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"
] | [
"has_vadd",
"vadd_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_def [has_vadd M α] (g : p) (m : α) : g +ᵥ m = (g : M) +ᵥ m | rfl | lemma | submodule.vadd_def | 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"
] | [
"has_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (V : submodule R M) : submodule S M | { carrier := V,
zero_mem' := V.zero_mem,
smul_mem' := λ c m h, V.smul_of_tower_mem c h,
add_mem' := λ x y hx hy, V.add_mem hx hy } | def | submodule.restrict_scalars | 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"
] | [
"restrict_scalars",
"submodule"
] | `V.restrict_scalars S` is the `S`-submodule of the `S`-module given by restriction of scalars,
corresponding to `V`, an `R`-submodule of the original `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars (V : submodule R M) : (V.restrict_scalars S : set M) = V | rfl | lemma | submodule.coe_restrict_scalars | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_mem (V : submodule R M) (m : M) : m ∈ V.restrict_scalars S ↔ m ∈ V | iff.refl _ | lemma | submodule.restrict_scalars_mem | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_self (V : submodule R M) : V.restrict_scalars R = V | set_like.coe_injective rfl | lemma | submodule.restrict_scalars_self | 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"
] | [
"set_like.coe_injective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_injective :
function.injective (restrict_scalars S : submodule R M → submodule S M) | λ V₁ V₂ h, ext $ set.ext_iff.1 (set_like.ext'_iff.1 h : _) | lemma | submodule.restrict_scalars_injective | 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"
] | [
"restrict_scalars",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_inj {V₁ V₂ : submodule R M} :
restrict_scalars S V₁ = restrict_scalars S V₂ ↔ V₁ = V₂ | (restrict_scalars_injective S _ _).eq_iff | lemma | submodule.restrict_scalars_inj | 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"
] | [
"restrict_scalars",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.orig_module (p : submodule R M) :
module R (p.restrict_scalars S) | (by apply_instance : module R p) | instance | submodule.restrict_scalars.orig_module | 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"
] | [
"module",
"submodule"
] | Even though `p.restrict_scalars S` has type `submodule S M`, it is still an `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars_embedding : submodule R M ↪o submodule S M | { to_fun := restrict_scalars S,
inj' := restrict_scalars_injective S R M,
map_rel_iff' := λ p q, by simp [set_like.le_def] } | def | submodule.restrict_scalars_embedding | 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"
] | [
"restrict_scalars",
"set_like.le_def",
"submodule"
] | `restrict_scalars S` is an embedding of the lattice of `R`-submodules into
the lattice of `S`-submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars_equiv (p : submodule R M) : p.restrict_scalars S ≃ₗ[R] p | { to_fun := id, inv_fun := id, map_smul' := λ c x, rfl, .. add_equiv.refl p } | def | submodule.restrict_scalars_equiv | 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"
] | [
"inv_fun",
"submodule"
] | Turning `p : submodule R M` into an `S`-submodule gives the same module structure
as turning it into a type and adding a module structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_mem (hx : x ∈ p) : -x ∈ p | neg_mem hx | lemma | submodule.neg_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup : add_subgroup M | { neg_mem' := λ _, p.neg_mem , .. p.to_add_submonoid } | def | submodule.to_add_subgroup | 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_subgroup"
] | Reinterpret a submodule as an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_add_subgroup : (p.to_add_subgroup : set M) = p | rfl | lemma | submodule.coe_to_add_subgroup | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_add_subgroup : x ∈ p.to_add_subgroup ↔ x ∈ p | iff.rfl | lemma | submodule.mem_to_add_subgroup | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_injective : injective (to_add_subgroup : submodule R M → add_subgroup M) | | p q h := set_like.ext (set_like.ext_iff.1 h : _) | theorem | submodule.to_add_subgroup_injective | 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_subgroup",
"set_like.ext",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_eq : p.to_add_subgroup = p'.to_add_subgroup ↔ p = p' | to_add_subgroup_injective.eq_iff | theorem | submodule.to_add_subgroup_eq | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_strict_mono :
strict_mono (to_add_subgroup : submodule R M → add_subgroup M) | λ _ _, id | lemma | submodule.to_add_subgroup_strict_mono | 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_subgroup",
"strict_mono",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_le : p.to_add_subgroup ≤ p'.to_add_subgroup ↔ p ≤ p' | iff.rfl | lemma | submodule.to_add_subgroup_le | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_subgroup_mono : monotone (to_add_subgroup : submodule R M → add_subgroup M) | to_add_subgroup_strict_mono.monotone | lemma | submodule.to_add_subgroup_mono | 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_subgroup",
"monotone",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem : x ∈ p → y ∈ p → x - y ∈ p | sub_mem | lemma | submodule.sub_mem | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mem_iff : -x ∈ p ↔ x ∈ p | neg_mem_iff | lemma | submodule.neg_mem_iff | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_iff_left : y ∈ p → (x + y ∈ p ↔ x ∈ p) | add_mem_cancel_right | lemma | submodule.add_mem_iff_left | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_iff_right : x ∈ p → (x + y ∈ p ↔ y ∈ p) | add_mem_cancel_left | lemma | submodule.add_mem_iff_right | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : p) : ((-x : p) : M) = -x | add_subgroup_class.coe_neg _ | lemma | submodule.coe_neg | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (x y : p) : (↑(x - y) : M) = ↑x - ↑y | add_subgroup_class.coe_sub _ _ | lemma | submodule.coe_sub | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_iff_left (hy : y ∈ p) : (x - y) ∈ p ↔ x ∈ p | by rw [sub_eq_add_neg, p.add_mem_iff_left (p.neg_mem hy)] | lemma | submodule.sub_mem_iff_left | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_iff_right (hx : x ∈ p) : (x - y) ∈ p ↔ y ∈ p | by rw [sub_eq_add_neg, p.add_mem_iff_right hx, p.neg_mem_iff] | lemma | submodule.sub_mem_iff_right | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_of_ortho {x : M} {N : submodule R M}
(ortho : ∀ (c : R) (y ∈ N), c • x + y = (0 : M) → c = 0) :
x ∉ N | by { intro hx, simpa using ortho (-1) x hx } | lemma | submodule.not_mem_of_ortho | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_ortho {x : M} {N : submodule R M}
(ortho : ∀ (c : R) (y ∈ N), c • x + y = (0 : M) → c = 0) :
x ≠ 0 | mt (λ h, show x ∈ N, from h.symm ▸ N.zero_mem) (not_mem_of_ortho ortho) | lemma | submodule.ne_zero_of_ortho | 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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_add_comm_monoid
{M} [ordered_add_comm_monoid M] [module R M] (S : submodule R M) :
ordered_add_comm_monoid S | subtype.coe_injective.ordered_add_comm_monoid coe rfl (λ _ _, rfl) (λ _ _, rfl) | instance | submodule.to_ordered_add_comm_monoid | 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"
] | [
"module",
"ordered_add_comm_monoid",
"submodule"
] | A submodule of an `ordered_add_comm_monoid` is an `ordered_add_comm_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_add_comm_monoid
{M} [linear_ordered_add_comm_monoid M] [module R M] (S : submodule R M) :
linear_ordered_add_comm_monoid S | subtype.coe_injective.linear_ordered_add_comm_monoid coe rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | submodule.to_linear_ordered_add_comm_monoid | 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"
] | [
"linear_ordered_add_comm_monoid",
"module",
"submodule"
] | A submodule of a `linear_ordered_add_comm_monoid` is a `linear_ordered_add_comm_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_cancel_add_comm_monoid
{M} [ordered_cancel_add_comm_monoid M] [module R M] (S : submodule R M) :
ordered_cancel_add_comm_monoid S | subtype.coe_injective.ordered_cancel_add_comm_monoid coe rfl (λ _ _, rfl) (λ _ _, rfl) | instance | submodule.to_ordered_cancel_add_comm_monoid | 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"
] | [
"module",
"ordered_cancel_add_comm_monoid",
"submodule"
] | A submodule of an `ordered_cancel_add_comm_monoid` is an `ordered_cancel_add_comm_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_cancel_add_comm_monoid
{M} [linear_ordered_cancel_add_comm_monoid M] [module R M] (S : submodule R M) :
linear_ordered_cancel_add_comm_monoid S | subtype.coe_injective.linear_ordered_cancel_add_comm_monoid coe rfl (λ _ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) | instance | submodule.to_linear_ordered_cancel_add_comm_monoid | 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"
] | [
"linear_ordered_cancel_add_comm_monoid",
"module",
"submodule"
] | A submodule of a `linear_ordered_cancel_add_comm_monoid` is a
`linear_ordered_cancel_add_comm_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_add_comm_group
{M} [ordered_add_comm_group M] [module R M] (S : submodule R M) :
ordered_add_comm_group S | subtype.coe_injective.ordered_add_comm_group coe
rfl (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | submodule.to_ordered_add_comm_group | 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"
] | [
"module",
"ordered_add_comm_group",
"submodule"
] | A submodule of an `ordered_add_comm_group` is an `ordered_add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_add_comm_group
{M} [linear_ordered_add_comm_group M] [module R M] (S : submodule R M) :
linear_ordered_add_comm_group S | subtype.coe_injective.linear_ordered_add_comm_group coe
rfl (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | submodule.to_linear_ordered_add_comm_group | 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"
] | [
"linear_ordered_add_comm_group",
"module",
"submodule"
] | A submodule of a `linear_ordered_add_comm_group` is a
`linear_ordered_add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mem_iff (s0 : s ≠ 0) : s • x ∈ p ↔ x ∈ p | p.to_sub_mul_action.smul_mem_iff s0 | theorem | submodule.smul_mem_iff | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subspace (R : Type u) (M : Type v)
[division_ring R] [add_comm_group M] [module R M] | submodule R M | abbreviation | subspace | 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_group",
"division_ring",
"module",
"submodule"
] | Subspace of a vector space. Defined to equal `submodule`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂ (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) (q : submodule R N) : submodule R P | ⨆ s : p, q.map $ f s | def | submodule.map₂ | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule"
] | Map a pair of submodules under a bilinear map.
This is the submodule version of `set.image2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_mem_map₂ (f : M →ₗ[R] N →ₗ[R] P) {m : M} {n : N}
{p : submodule R M} {q : submodule R N} (hm : m ∈ p) (hn : n ∈ q) : f m n ∈ map₂ f p q | (le_supr _ ⟨m, hm⟩ : _ ≤ map₂ f p q) ⟨n, hn, rfl⟩ | theorem | submodule.apply_mem_map₂ | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"le_supr",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_le {f : M →ₗ[R] N →ₗ[R] P}
{p : submodule R M} {q : submodule R N} {r : submodule R P} :
map₂ f p q ≤ r ↔ ∀ (m ∈ p) (n ∈ q), f m n ∈ r | ⟨λ H m hm n hn, H $ apply_mem_map₂ _ hm hn,
λ H, supr_le $ λ ⟨m, hm⟩, map_le_iff_le_comap.2 $ λ n hn, H m hm n hn⟩ | theorem | submodule.map₂_le | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : set M) (t : set N) :
map₂ f (span R s) (span R t) = span R (set.image2 (λ m n, f m n) s t) | begin
apply le_antisymm,
{ rw map₂_le, intros a ha b hb,
apply span_induction ha,
work_on_goal 1 { intros, apply span_induction hb,
work_on_goal 1 { intros, exact subset_span ⟨_, _, ‹_›, ‹_›, rfl⟩ } },
all_goals {
intros,
simp only [linear_map.map_zero, linear_map.zero_apply, zero_mem,... | theorem | submodule.map₂_span_span | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"linear_map.add_apply",
"linear_map.map_add",
"linear_map.map_smul",
"linear_map.map_zero",
"linear_map.smul_apply",
"linear_map.zero_apply",
"set.image2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_bot_right (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) : map₂ f p ⊥ = ⊥ | eq_bot_iff.2 $ map₂_le.2 $ λ m hm n hn,
by { rw [submodule.mem_bot] at hn ⊢, rw [hn, linear_map.map_zero] } | theorem | submodule.map₂_bot_right | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"linear_map.map_zero",
"submodule",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_bot_left (f : M →ₗ[R] N →ₗ[R] P) (q : submodule R N) : map₂ f ⊥ q = ⊥ | eq_bot_iff.2 $ map₂_le.2 $ λ m hm n hn,
by { rw [submodule.mem_bot] at hm ⊢, rw [hm, linear_map.map_zero₂] } | theorem | submodule.map₂_bot_left | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"linear_map.map_zero₂",
"submodule",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_le_map₂ {f : M →ₗ[R] N →ₗ[R] P}
{p₁ p₂ : submodule R M} {q₁ q₂ : submodule R N} (hp : p₁ ≤ p₂) (hq : q₁ ≤ q₂) :
map₂ f p₁ q₁ ≤ map₂ f p₂ q₂ | map₂_le.2 $ λ m hm n hn, apply_mem_map₂ _ (hp hm) (hq hn) | theorem | submodule.map₂_le_map₂ | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_le_map₂_left {f : M →ₗ[R] N →ₗ[R] P}
{p₁ p₂ : submodule R M} {q : submodule R N} (h : p₁ ≤ p₂) : map₂ f p₁ q ≤ map₂ f p₂ q | map₂_le_map₂ h (le_refl q) | theorem | submodule.map₂_le_map₂_left | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_le_map₂_right {f : M →ₗ[R] N →ₗ[R] P}
{p : submodule R M} {q₁ q₂ : submodule R N} (h : q₁ ≤ q₂): map₂ f p q₁ ≤ map₂ f p q₂ | map₂_le_map₂ (le_refl p) h | theorem | submodule.map₂_le_map₂_right | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_sup_right (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) (q₁ q₂ : submodule R N) :
map₂ f p (q₁ ⊔ q₂) = map₂ f p q₁ ⊔ map₂ f p q₂ | le_antisymm (map₂_le.2 $ λ m hm np hnp, let ⟨n, hn, p, hp, hnp⟩ := mem_sup.1 hnp in
mem_sup.2 ⟨_, apply_mem_map₂ _ hm hn, _, apply_mem_map₂ _ hm hp, hnp ▸ (map_add _ _ _).symm⟩)
(sup_le (map₂_le_map₂_right le_sup_left) (map₂_le_map₂_right le_sup_right)) | theorem | submodule.map₂_sup_right | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"le_sup_left",
"le_sup_right",
"submodule",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_sup_left (f : M →ₗ[R] N →ₗ[R] P) (p₁ p₂ : submodule R M) (q : submodule R N) :
map₂ f (p₁ ⊔ p₂) q = map₂ f p₁ q ⊔ map₂ f p₂ q | le_antisymm (map₂_le.2 $ λ mn hmn p hp, let ⟨m, hm, n, hn, hmn⟩ := mem_sup.1 hmn in
mem_sup.2 ⟨_, apply_mem_map₂ _ hm hp, _, apply_mem_map₂ _ hn hp,
hmn ▸ (linear_map.map_add₂ _ _ _ _).symm⟩)
(sup_le (map₂_le_map₂_left le_sup_left) (map₂_le_map₂_left le_sup_right)) | theorem | submodule.map₂_sup_left | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"le_sup_left",
"le_sup_right",
"linear_map.map_add₂",
"submodule",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image2_subset_map₂ (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) (q : submodule R N) :
set.image2 (λ m n, f m n) (↑p : set M) (↑q : set N) ⊆ (↑(map₂ f p q) : set P) | by { rintros _ ⟨i, j, hi, hj, rfl⟩, exact apply_mem_map₂ _ hi hj } | lemma | submodule.image2_subset_map₂ | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_eq_span_image2 (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) (q : submodule R N) :
map₂ f p q = span R (set.image2 (λ m n, f m n) (p : set M) (q : set N)) | by rw [← map₂_span_span, span_eq, span_eq] | lemma | submodule.map₂_eq_span_image2 | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_flip (f : M →ₗ[R] N →ₗ[R] P) (p : submodule R M) (q : submodule R N) :
map₂ f.flip q p = map₂ f p q | by { rw [map₂_eq_span_image2, map₂_eq_span_image2, set.image2_swap], refl } | lemma | submodule.map₂_flip | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2_swap",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_supr_left (f : M →ₗ[R] N →ₗ[R] P) (s : ι → submodule R M) (t : submodule R N) :
map₂ f (⨆ i, s i) t = ⨆ i, map₂ f (s i) t | begin
suffices :
map₂ f (⨆ i, span R (s i : set M)) (span R t) = (⨆ i, map₂ f (span R (s i)) (span R t)),
{ simpa only [span_eq] using this },
simp_rw [map₂_span_span, ← span_Union, map₂_span_span, set.image2_Union_left],
end | lemma | submodule.map₂_supr_left | algebra.module.submodule | src/algebra/module/submodule/bilinear.lean | [
"linear_algebra.span",
"linear_algebra.bilinear_map"
] | [
"set.image2_Union_left",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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