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 |
|---|---|---|---|---|---|---|---|---|---|---|
apply_module : module (M₁ →L[R₁] M₁) M₁ | module.comp_hom _ to_linear_map_ring_hom | instance | continuous_linear_map.apply_module | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module",
"module.comp_hom"
] | The tautological action by `M₁ →L[R₁] M₁` on `M`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a | rfl | lemma | continuous_linear_map.smul_def | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul : has_faithful_smul (M₁ →L[R₁] M₁) M₁ | ⟨λ _ _, continuous_linear_map.ext⟩ | instance | continuous_linear_map.apply_has_faithful_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_faithful_smul"
] | `continuous_linear_map.apply_module` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_smul_comm_class : smul_comm_class R₁ (M₁ →L[R₁] M₁) M₁ | { smul_comm := λ r e m, (e.map_smul r m).symm } | instance | continuous_linear_map.apply_smul_comm_class | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_smul_comm_class' : smul_comm_class (M₁ →L[R₁] M₁) R₁ M₁ | { smul_comm := continuous_linear_map.map_smul } | instance | continuous_linear_map.apply_smul_comm_class' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.map_smul",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) :
M₁ →L[R₁] (M₂ × M₃) | ⟨(f₁ : M₁ →ₗ[R₁] M₂).prod f₂, f₁.2.prod_mk f₂.2⟩ | def | continuous_linear_map.prod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | The cartesian product of two bounded linear maps, as a bounded linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂)
(f₂ : M₁ →L[R₁] M₃) :
(f₁.prod f₂ : M₁ →ₗ[R₁] M₂ × M₃) = linear_map.prod f₁ f₂ | rfl | lemma | continuous_linear_map.coe_prod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.prod",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_apply [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂)
(f₂ : M₁ →L[R₁] M₃) (x : M₁) :
f₁.prod f₂ x = (f₁ x, f₂ x) | rfl | lemma | continuous_linear_map.prod_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl [module R₁ M₂] : M₁ →L[R₁] M₁ × M₂ | (id R₁ M₁).prod 0 | def | continuous_linear_map.inl | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | The left injection into a product is a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr [module R₁ M₂] : M₂ →L[R₁] M₁ × M₂ | (0 : M₂ →L[R₁] M₁).prod (id R₁ M₂) | def | continuous_linear_map.inr | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | The right injection into a product is a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_apply [module R₁ M₂] (x : M₁) : inl R₁ M₁ M₂ x = (x, 0) | rfl | lemma | continuous_linear_map.inl_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_apply [module R₁ M₂] (x : M₂) : inr R₁ M₁ M₂ x = (0, x) | rfl | lemma | continuous_linear_map.inr_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inl [module R₁ M₂] :
(inl R₁ M₁ M₂ : M₁ →ₗ[R₁] M₁ × M₂) = linear_map.inl R₁ M₁ M₂ | rfl | lemma | continuous_linear_map.coe_inl | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.inl",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inr [module R₁ M₂] :
(inr R₁ M₁ M₂ : M₂ →ₗ[R₁] M₁ × M₂) = linear_map.inr R₁ M₁ M₂ | rfl | lemma | continuous_linear_map.coe_inr | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.inr",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_ker [t1_space M₂] [continuous_semilinear_map_class F σ₁₂ M₁ M₂]
(f : F) : is_closed (ker f : set M₁) | continuous_iff_is_closed.1 (map_continuous f) _ is_closed_singleton | lemma | continuous_linear_map.is_closed_ker | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_semilinear_map_class",
"is_closed",
"is_closed_singleton",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']
[module R₁ M'] [t1_space M₂] [continuous_semilinear_map_class F σ₁₂ M' M₂]
(f : F) : is_complete (ker f : set M') | (is_closed_ker f).is_complete | lemma | continuous_linear_map.is_complete_ker | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"complete_space",
"continuous_semilinear_map_class",
"is_complete",
"module",
"t1_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']
[module R₁ M'] [t1_space M₂] [continuous_semilinear_map_class F σ₁₂ M' M₂]
(f : F) : complete_space (ker f) | (is_closed_ker f).complete_space_coe | instance | continuous_linear_map.complete_space_ker | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"complete_space",
"continuous_semilinear_map_class",
"module",
"t1_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
ker (f.prod g) = ker f ⊓ ker g | linear_map.ker_prod f g | lemma | continuous_linear_map.ker_prod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.ker_prod",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
M₁ →SL[σ₁₂] p | { cont := f.continuous.subtype_mk _,
to_linear_map := (f : M₁ →ₛₗ[σ₁₂] M₂).cod_restrict p h} | def | continuous_linear_map.cod_restrict | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"submodule"
] | Restrict codomain of a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
(f.cod_restrict p h : M₁ →ₛₗ[σ₁₂] p) = (f : M₁ →ₛₗ[σ₁₂] M₂).cod_restrict p h | rfl | lemma | continuous_linear_map.coe_cod_restrict | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_cod_restrict_apply (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p)
(x) :
(f.cod_restrict p h x : M₂) = f x | rfl | lemma | continuous_linear_map.coe_cod_restrict_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :
ker (f.cod_restrict p h) = ker f | (f : M₁ →ₛₗ[σ₁₂] M₂).ker_cod_restrict p h | lemma | continuous_linear_map.ker_cod_restrict | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.subtypeL (p : submodule R₁ M₁) : p →L[R₁] M₁ | { cont := continuous_subtype_val,
to_linear_map := p.subtype } | def | submodule.subtypeL | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous_subtype_val",
"submodule"
] | `submodule.subtype` as a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.submodule.coe_subtypeL (p : submodule R₁ M₁) :
(p.subtypeL : p →ₗ[R₁] M₁) = p.subtype | rfl | lemma | submodule.coe_subtypeL | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.coe_subtypeL' (p : submodule R₁ M₁) :
⇑p.subtypeL = p.subtype | rfl | lemma | submodule.coe_subtypeL' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.subtypeL_apply (p : submodule R₁ M₁) (x : p) :
p.subtypeL x = x | rfl | lemma | submodule.subtypeL_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.range_subtypeL (p : submodule R₁ M₁) :
range p.subtypeL = p | submodule.range_subtype _ | lemma | submodule.range_subtypeL | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule",
"submodule.range_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.submodule.ker_subtypeL (p : submodule R₁ M₁) :
ker p.subtypeL = ⊥ | submodule.ker_subtype _ | lemma | submodule.ker_subtypeL | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule",
"submodule.ker_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst [module R₁ M₂] : M₁ × M₂ →L[R₁] M₁ | { cont := continuous_fst, to_linear_map := linear_map.fst R₁ M₁ M₂ } | def | continuous_linear_map.fst | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous_fst",
"linear_map.fst",
"module"
] | `prod.fst` as a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd [module R₁ M₂] : M₁ × M₂ →L[R₁] M₂ | { cont := continuous_snd, to_linear_map := linear_map.snd R₁ M₁ M₂ } | def | continuous_linear_map.snd | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous_snd",
"linear_map.snd",
"module"
] | `prod.snd` as a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fst [module R₁ M₂] : ↑(fst R₁ M₁ M₂) = linear_map.fst R₁ M₁ M₂ | rfl | lemma | continuous_linear_map.coe_fst | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.fst",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst' [module R₁ M₂] : ⇑(fst R₁ M₁ M₂) = prod.fst | rfl | lemma | continuous_linear_map.coe_fst' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_snd [module R₁ M₂] : ↑(snd R₁ M₁ M₂) = linear_map.snd R₁ M₁ M₂ | rfl | lemma | continuous_linear_map.coe_snd | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.snd",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_snd' [module R₁ M₂] : ⇑(snd R₁ M₁ M₂) = prod.snd | rfl | lemma | continuous_linear_map.coe_snd' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod_snd [module R₁ M₂] : (fst R₁ M₁ M₂).prod (snd R₁ M₁ M₂) = id R₁ (M₁ × M₂) | ext $ λ ⟨x, y⟩, rfl | lemma | continuous_linear_map.fst_prod_snd | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
(fst R₁ M₂ M₃).comp (f.prod g) = f | ext $ λ x, rfl | lemma | continuous_linear_map.fst_comp_prod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
(snd R₁ M₂ M₃).comp (f.prod g) = g | ext $ λ x, rfl | lemma | continuous_linear_map.snd_comp_prod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
(M₁ × M₃) →L[R₁] (M₂ × M₄) | (f₁.comp (fst R₁ M₁ M₃)).prod (f₂.comp (snd R₁ M₁ M₃)) | def | continuous_linear_map.prod_map | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module",
"prod_map"
] | `prod.map` of two continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_map [module R₁ M₂] [module R₁ M₃] [module R₁ M₄]
(f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
↑(f₁.prod_map f₂) = ((f₁ : M₁ →ₗ[R₁] M₂).prod_map (f₂ : M₃ →ₗ[R₁] M₄)) | rfl | lemma | continuous_linear_map.coe_prod_map | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_map' [module R₁ M₂] [module R₁ M₃] [module R₁ M₄]
(f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
⇑(f₁.prod_map f₂) = prod.map f₁ f₂ | rfl | lemma | continuous_linear_map.coe_prod_map' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃] (f₁ : M₁ →L[R₁] M₃)
(f₂ : M₂ →L[R₁] M₃) :
(M₁ × M₂) →L[R₁] M₃ | ⟨linear_map.coprod f₁ f₂, (f₁.cont.comp continuous_fst).add (f₂.cont.comp continuous_snd)⟩ | def | continuous_linear_map.coprod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_fst",
"continuous_snd",
"has_continuous_add",
"module"
] | The continuous linear map given by `(x, y) ↦ f₁ x + f₂ y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃]
(f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) :
(f₁.coprod f₂ : (M₁ × M₂) →ₗ[R₁] M₃) = linear_map.coprod f₁ f₂ | rfl | lemma | continuous_linear_map.coe_coprod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add",
"linear_map.coprod",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_apply [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃]
(f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x) :
f₁.coprod f₂ x = f₁ x.1 + f₂ x.2 | rfl | lemma | continuous_linear_map.coprod_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃] (f₁ : M₁ →L[R₁] M₃)
(f₂ : M₂ →L[R₁] M₃) :
range (f₁.coprod f₂) = range f₁ ⊔ range f₂ | linear_map.range_coprod _ _ | lemma | continuous_linear_map.range_coprod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add",
"linear_map.range_coprod",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_fst_add_comp_snd [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃]
(f : M₁ →L[R₁] M₃) (g : M₂ →L[R₁] M₃) :
f.comp (continuous_linear_map.fst R₁ M₁ M₂) +
g.comp (continuous_linear_map.snd R₁ M₁ M₂) =
f.coprod g | rfl | lemma | continuous_linear_map.comp_fst_add_comp_snd | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"has_continuous_add",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_inl_inr [has_continuous_add M₁] [has_continuous_add M'₁] :
(continuous_linear_map.inl R₁ M₁ M'₁).coprod (continuous_linear_map.inr R₁ M₁ M'₁) =
continuous_linear_map.id R₁ (M₁ × M'₁) | by { apply coe_injective, apply linear_map.coprod_inl_inr } | lemma | continuous_linear_map.coprod_inl_inr | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.id",
"continuous_linear_map.inl",
"continuous_linear_map.inr",
"has_continuous_add",
"linear_map.coprod_inl_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right (c : M₁ →L[R] S) (f : M₂) : M₁ →L[R] M₂ | { cont := c.2.smul continuous_const,
..c.to_linear_map.smul_right f } | def | continuous_linear_map.smul_right | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous_const"
] | The linear map `λ x, c x • f`. Associates to a scalar-valued linear map and an element of
`M₂` the `M₂`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M₂`).
See also `continuous_linear_map.smul_rightₗ` and `continuous_linear_map.smul_rightL`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_right_apply {c : M₁ →L[R] S} {f : M₂} {x : M₁} :
(smul_right c f : M₁ → M₂) x = c x • f | rfl | lemma | continuous_linear_map.smul_right_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right_one_one (c : R₁ →L[R₁] M₂) : smul_right (1 : R₁ →L[R₁] R₁) (c 1) = c | by ext; simp [← continuous_linear_map.map_smul_of_tower] | lemma | continuous_linear_map.smul_right_one_one | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.map_smul_of_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right_one_eq_iff {f f' : M₂} :
smul_right (1 : R₁ →L[R₁] R₁) f = smul_right (1 : R₁ →L[R₁] R₁) f' ↔ f = f' | by simp only [ext_ring_iff, smul_right_apply, one_apply, one_smul] | lemma | continuous_linear_map.smul_right_one_eq_iff | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right_comp [has_continuous_mul R₁] {x : M₂} {c : R₁} :
(smul_right (1 : R₁ →L[R₁] R₁) x).comp (smul_right (1 : R₁ →L[R₁] R₁) c) =
smul_right (1 : R₁ →L[R₁] R₁) (c • x) | by { ext, simp [mul_smul] } | lemma | continuous_linear_map.smul_right_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_singleton (x : M₁) : R₁ →L[R₁] M₁ | { to_linear_map := linear_map.to_span_singleton R₁ M₁ x,
cont := continuous_id.smul continuous_const } | def | continuous_linear_map.to_span_singleton | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous_const",
"linear_map.to_span_singleton"
] | Given an element `x` of a topological space `M` over a semiring `R`, the natural continuous
linear map from `R` to `M` by taking multiples of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_span_singleton_apply (x : M₁) (r : R₁) : to_span_singleton R₁ x r = r • x | rfl | lemma | continuous_linear_map.to_span_singleton_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_singleton_add [has_continuous_add M₁] (x y : M₁) :
to_span_singleton R₁ (x + y) = to_span_singleton R₁ x + to_span_singleton R₁ y | by { ext1, simp [to_span_singleton_apply], } | lemma | continuous_linear_map.to_span_singleton_add | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_singleton_smul' {α} [monoid α] [distrib_mul_action α M₁]
[has_continuous_const_smul α M₁]
[smul_comm_class R₁ α M₁] (c : α) (x : M₁) :
to_span_singleton R₁ (c • x) = c • to_span_singleton R₁ x | by { ext1, rw [to_span_singleton_apply, smul_apply, to_span_singleton_apply, smul_comm], } | lemma | continuous_linear_map.to_span_singleton_smul' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"distrib_mul_action",
"has_continuous_const_smul",
"monoid",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_singleton_smul (R) {M₁} [comm_semiring R] [add_comm_monoid M₁] [module R M₁]
[topological_space R] [topological_space M₁] [has_continuous_smul R M₁] (c : R) (x : M₁) :
to_span_singleton R (c • x) = c • to_span_singleton R x | to_span_singleton_smul' R c x | lemma | continuous_linear_map.to_span_singleton_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"comm_semiring",
"has_continuous_smul",
"module",
"topological_space"
] | A special case of `to_span_singleton_smul'` for when `R` is commutative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi (f : Πi, M →L[R] φ i) : M →L[R] (Πi, φ i) | ⟨linear_map.pi (λ i, f i), continuous_pi (λ i, (f i).continuous)⟩ | def | continuous_linear_map.pi | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous",
"continuous_pi"
] | `pi` construction for continuous linear functions. From a family of continuous linear functions
it produces a continuous linear function into a family of topological modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pi' (f : Π i, M →L[R] φ i) : ⇑(pi f) = λ c i, f i c | rfl | lemma | continuous_linear_map.coe_pi' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pi (f : Π i, M →L[R] φ i) :
(pi f : M →ₗ[R] Π i, φ i) = linear_map.pi (λ i, f i) | rfl | lemma | continuous_linear_map.coe_pi | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_apply (f : Πi, M →L[R] φ i) (c : M) (i : ι) :
pi f c i = f i c | rfl | lemma | continuous_linear_map.pi_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_eq_zero (f : Πi, M →L[R] φ i) : pi f = 0 ↔ (∀i, f i = 0) | by { simp only [ext_iff, pi_apply, function.funext_iff], exact forall_swap } | lemma | continuous_linear_map.pi_eq_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"forall_swap",
"function.funext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_zero : pi (λi, 0 : Πi, M →L[R] φ i) = 0 | ext $ λ _, rfl | lemma | continuous_linear_map.pi_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_comp (f : Πi, M →L[R] φ i) (g : M₂ →L[R] M) : (pi f).comp g = pi (λi, (f i).comp g) | rfl | lemma | continuous_linear_map.pi_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj (i : ι) : (Πi, φ i) →L[R] φ i | ⟨linear_map.proj i, continuous_apply _⟩ | def | continuous_linear_map.proj | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_apply"
] | The projections from a family of topological modules are continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_apply (i : ι) (b : Πi, φ i) : (proj i : (Πi, φ i) →L[R] φ i) b = b i | rfl | lemma | continuous_linear_map.proj_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_pi (f : Πi, M₂ →L[R] φ i) (i : ι) : (proj i).comp (pi f) = f i | ext $ assume c, rfl | lemma | continuous_linear_map.proj_pi | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_ker_proj : (⨅i, ker (proj i : (Πi, φ i) →L[R] φ i) :
submodule R (Πi, φ i)) = ⊥ | linear_map.infi_ker_proj | lemma | continuous_linear_map.infi_ker_proj | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.infi_ker_proj",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_ker_proj_equiv {I J : set ι} [decidable_pred (λi, i ∈ I)]
(hd : disjoint I J) (hu : set.univ ⊆ I ∪ J) :
(⨅i ∈ J, ker (proj i : (Πi, φ i) →L[R] φ i) :
submodule R (Πi, φ i)) ≃L[R] (Πi:I, φ i) | { to_linear_equiv := linear_map.infi_ker_proj_equiv R φ hd hu,
continuous_to_fun := continuous_pi (λ i, begin
have := @continuous_subtype_coe _ _
(λ x, x ∈ (⨅i ∈ J, ker (proj i : (Πi, φ i) →L[R] φ i) : submodule R (Πi, φ i))),
have := continuous.comp (by exact continuous_apply i) this,
exact this
... | def | continuous_linear_map.infi_ker_proj_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous.comp",
"continuous.subtype_mk",
"continuous_apply",
"continuous_pi",
"continuous_subtype_coe",
"disjoint",
"linear_map.infi_ker_proj_equiv",
"submodule"
] | If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections
of `φ` is linearly equivalent to the product over `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = - (f x) | map_neg _ _ | lemma | continuous_linear_map.map_neg | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (f : M →SL[σ₁₂] M₂) (x y : M) : f (x - y) = f x - f y | map_sub _ _ _ | lemma | continuous_linear_map.map_sub | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply' (f g : M →SL[σ₁₂] M₂) (x : M) : ((f : M →ₛₗ[σ₁₂] M₂) - g) x = f x - g x | rfl | lemma | continuous_linear_map.sub_apply' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃}
(h : ker f ⊔ ker g = ⊤) :
range (f.prod g) = (range f).prod (range g) | linear_map.range_prod_eq h | lemma | continuous_linear_map.range_prod_eq | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.range_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_prod_ker_le_ker_coprod [has_continuous_add M₃]
(f : M →L[R] M₃) (g : M₂ →L[R] M₃) :
(linear_map.ker f).prod (linear_map.ker g) ≤ linear_map.ker (f.coprod g) | linear_map.ker_prod_ker_le_ker_coprod f.to_linear_map g.to_linear_map | lemma | continuous_linear_map.ker_prod_ker_le_ker_coprod | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add",
"linear_map.ker",
"linear_map.ker_prod_ker_le_ker_coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_coprod_of_disjoint_range [has_continuous_add M₃]
(f : M →L[R] M₃) (g : M₂ →L[R] M₃) (hd : disjoint (range f) (range g)) :
linear_map.ker (f.coprod g) = (linear_map.ker f).prod (linear_map.ker g) | linear_map.ker_coprod_of_disjoint_range f.to_linear_map g.to_linear_map hd | lemma | continuous_linear_map.ker_coprod_of_disjoint_range | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"disjoint",
"has_continuous_add",
"linear_map.ker",
"linear_map.ker_coprod_of_disjoint_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : M →SL[σ₁₂] M₂) (x : M) : (-f) x = - (f x) | rfl | lemma | continuous_linear_map.neg_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (f : M →SL[σ₁₂] M₂) : (↑(-f) : M →ₛₗ[σ₁₂] M₂) = -f | rfl | lemma | continuous_linear_map.coe_neg | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg' (f : M →SL[σ₁₂] M₂) : ⇑(-f) = -f | rfl | lemma | continuous_linear_map.coe_neg' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x | rfl | lemma | continuous_linear_map.sub_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (f g : M →SL[σ₁₂] M₂) : (↑(f - g) : M →ₛₗ[σ₁₂] M₂) = f - g | rfl | lemma | continuous_linear_map.coe_sub | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g | rfl | lemma | continuous_linear_map.coe_sub' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_neg [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₂]
[topological_add_group M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
g.comp (-f) = -g.comp f | by { ext, simp } | lemma | continuous_linear_map.comp_neg | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"ring_hom_comp_triple",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₃]
(g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
(-g).comp f = -g.comp f | by { ext, simp } | lemma | continuous_linear_map.neg_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"ring_hom_comp_triple",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_sub [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₂]
[topological_add_group M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂) :
g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂ | by { ext, simp } | lemma | continuous_linear_map.comp_sub | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"ring_hom_comp_triple",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₃]
(g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
(g₁ - g₂).comp f = g₁.comp f - g₂.comp f | by { ext, simp } | lemma | continuous_linear_map.sub_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"ring_hom_comp_triple",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right_one_pow [topological_space R] [topological_ring R] (c : R) (n : ℕ) :
(smul_right (1 : R →L[R] R) c)^n = smul_right (1 : R →L[R] R) (c^n) | begin
induction n with n ihn,
{ ext, simp },
{ rw [pow_succ, ihn, mul_def, smul_right_comp, smul_eq_mul, pow_succ'] }
end | lemma | continuous_linear_map.smul_right_one_pow | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"pow_succ",
"pow_succ'",
"smul_eq_mul",
"topological_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_ker_of_right_inverse [topological_add_group M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M)
(h : function.right_inverse f₂ f₁) :
M →L[R] (linear_map.ker f₁) | (id R M - f₂.comp f₁).cod_restrict (linear_map.ker f₁) $ λ x, by simp [h (f₁ x)] | def | continuous_linear_map.proj_ker_of_right_inverse | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.ker",
"topological_add_group"
] | Given a right inverse `f₂ : M₂ →L[R] M` to `f₁ : M →L[R] M₂`,
`proj_ker_of_right_inverse f₁ f₂ h` is the projection `M →L[R] f₁.ker` along `f₂.range`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_proj_ker_of_right_inverse_apply [topological_add_group M]
(f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁) (x : M) :
(f₁.proj_ker_of_right_inverse f₂ h x : M) = x - f₂ (f₁ x) | rfl | lemma | continuous_linear_map.coe_proj_ker_of_right_inverse_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_ker_of_right_inverse_apply_idem [topological_add_group M]
(f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁)
(x : linear_map.ker f₁) : f₁.proj_ker_of_right_inverse f₂ h x = x | subtype.ext_iff_val.2 $ by simp | lemma | continuous_linear_map.proj_ker_of_right_inverse_apply_idem | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.ker",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_ker_of_right_inverse_comp_inv [topological_add_group M]
(f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁) (y : M₂) :
f₁.proj_ker_of_right_inverse f₂ h (f₂ y) = 0 | subtype.ext_iff_val.2 $ by simp [h y] | lemma | continuous_linear_map.proj_ker_of_right_inverse_comp_inv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_of_ne_zero [topological_space R] [division_ring R]
[has_continuous_sub R] [add_comm_group M] [topological_space M] [has_continuous_add M]
[module R M] [has_continuous_smul R M] (f : M →L[R] R) (hf : f ≠ 0) : is_open_map f | let ⟨x, hx⟩ := exists_ne_zero hf in is_open_map.of_sections $ λ y,
⟨λ a, y + (a - f y) • (f x)⁻¹ • x, continuous.continuous_at $ by continuity,
by simp, λ a, by simp [hx]⟩ | lemma | continuous_linear_map.is_open_map_of_ne_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_group",
"continuity",
"continuous.continuous_at",
"division_ring",
"has_continuous_add",
"has_continuous_smul",
"has_continuous_sub",
"is_open_map",
"is_open_map.of_sections",
"module",
"topological_space"
] | A nonzero continuous linear functional is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_comp (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
(c • h).comp f = c • (h.comp f) | rfl | lemma | continuous_linear_map.smul_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_smul [linear_map.compatible_smul N₂ N₃ S R]
(hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂) :
hₗ.comp (c • fₗ) = c • (hₗ.comp fₗ) | by { ext x, exact hₗ.map_smul_of_tower c (fₗ x) } | lemma | continuous_linear_map.comp_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.compatible_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_smulₛₗ [smul_comm_class R₂ R₂ M₂] [smul_comm_class R₃ R₃ M₃]
[has_continuous_const_smul R₂ M₂] [has_continuous_const_smul R₃ M₃]
(h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) :
h.comp (c • f) = (σ₂₃ c) • (h.comp f) | by { ext x, simp only [coe_smul', coe_comp', function.comp_app, pi.smul_apply,
continuous_linear_map.map_smulₛₗ] } | lemma | continuous_linear_map.comp_smulₛₗ | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.map_smulₛₗ",
"has_continuous_const_smul",
"pi.smul_apply",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ (M →L[R] N₂ × N₃) | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ⟨(fst _ _ _).comp f, (snd _ _ _).comp f⟩,
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl } | def | continuous_linear_map.prod_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"inv_fun"
] | `continuous_linear_map.prod` as an `equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_ext_iff {f g : M × N₂ →L[R] N₃} :
f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) | by { simp only [← coe_inj, linear_map.prod_ext_iff], refl } | lemma | continuous_linear_map.prod_ext_iff | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.prod_ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ext {f g : M × N₂ →L[R] N₃} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))
(hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g | prod_ext_iff.2 ⟨hl, hr⟩ | lemma | continuous_linear_map.prod_ext | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prodₗ : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ₗ[S] (M →L[R] N₂ × N₃) | { map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
.. prod_equiv } | def | continuous_linear_map.prodₗ | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | `continuous_linear_map.prod` as a `linear_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lm : (M →L[R] N₃) →ₗ[S] (M →ₗ[R] N₃) | { to_fun := coe,
map_add' := λ f g, coe_add f g,
map_smul' := λ c f, coe_smul c f } | def | continuous_linear_map.coe_lm | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | The coercion from `M →L[R] M₂` to `M →ₗ[R] M₂`, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lmₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] (M →ₛₗ[σ₁₃] M₃) | { to_fun := coe,
map_add' := λ f g, coe_add f g,
map_smul' := λ c f, coe_smul c f } | def | continuous_linear_map.coe_lmₛₗ | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | The coercion from `M →SL[σ] M₂` to `M →ₛₗ[σ] M₂`, as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_rightₗ (c : M →L[R] S) : M₂ →ₗ[T] (M →L[R] M₂) | { to_fun := c.smul_right,
map_add' := λ x y, by { ext e, apply smul_add },
map_smul' := λ a x, by { ext e, dsimp, apply smul_comm } } | def | continuous_linear_map.smul_rightₗ | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"smul_add"
] | Given `c : E →L[𝕜] 𝕜`, `c.smul_rightₗ` is the linear map from `F` to `E →L[𝕜] F`
sending `f` to `λ e, c e • f`. See also `continuous_linear_map.smul_rightL`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.