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
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coe_smul_rightₗ (c : M →L[R] S) :
⇑(smul_rightₗ c : M₂ →ₗ[T] (M →L[R] M₂)) = c.smul_right | rfl | lemma | continuous_linear_map.coe_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars (f : M →L[A] M₂) : M →L[R] M₂ | ⟨(f : M →ₗ[A] M₂).restrict_scalars R, f.continuous⟩ | def | continuous_linear_map.restrict_scalars | 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"
] | [
"restrict_scalars"
] | If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous
`R`-linear map. We assume `linear_map.compatible_smul M M₂ R A` to match assumptions of
`linear_map.map_smul_of_tower`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars (f : M →L[A] M₂) :
(f.restrict_scalars R : M →ₗ[R] M₂) = (f : M →ₗ[A] M₂).restrict_scalars R | rfl | lemma | continuous_linear_map.coe_restrict_scalars | 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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars' (f : M →L[A] M₂) : ⇑(f.restrict_scalars R) = f | rfl | lemma | continuous_linear_map.coe_restrict_scalars' | 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 | |
restrict_scalars_zero : (0 : M →L[A] M₂).restrict_scalars R = 0 | rfl | lemma | continuous_linear_map.restrict_scalars_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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_add (f g : M →L[A] M₂) :
(f + g).restrict_scalars R = f.restrict_scalars R + g.restrict_scalars R | rfl | lemma | continuous_linear_map.restrict_scalars_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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_neg (f : M →L[A] M₂) :
(-f).restrict_scalars R = -f.restrict_scalars R | rfl | lemma | continuous_linear_map.restrict_scalars_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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_smul (c : S) (f : M →L[A] M₂) :
(c • f).restrict_scalars R = c • f.restrict_scalars R | rfl | lemma | continuous_linear_map.restrict_scalars_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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalarsₗ : (M →L[A] M₂) →ₗ[S] (M →L[R] M₂) | { to_fun := restrict_scalars R,
map_add' := restrict_scalars_add,
map_smul' := restrict_scalars_smul } | def | continuous_linear_map.restrict_scalarsₗ | 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"
] | [
"restrict_scalars"
] | `continuous_linear_map.restrict_scalars` as a `linear_map`. See also
`continuous_linear_map.restrict_scalarsL`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalarsₗ : ⇑(restrict_scalarsₗ A M M₂ R S) = restrict_scalars R | rfl | lemma | continuous_linear_map.coe_restrict_scalarsₗ | 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"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_map (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁₂] M₂ | { cont := e.continuous_to_fun,
..e.to_linear_equiv.to_linear_map } | def | continuous_linear_equiv.to_continuous_linear_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"
] | [
"cont"
] | A continuous linear equivalence induces a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_def_rev (e : M₁ ≃SL[σ₁₂] M₂) : e.to_continuous_linear_map = e | rfl | theorem | continuous_linear_equiv.coe_def_rev | 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_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : (e : M₁ →SL[σ₁₂] M₂) b = e b | rfl | theorem | continuous_linear_equiv.coe_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_to_linear_equiv (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.to_linear_equiv = f | rfl | lemma | continuous_linear_equiv.coe_to_linear_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe (e : M₁ ≃SL[σ₁₂] M₂) : ⇑(e : M₁ →SL[σ₁₂] M₂) = e | rfl | lemma | continuous_linear_equiv.coe_coe | 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"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_injective :
function.injective (to_linear_equiv : (M₁ ≃SL[σ₁₂] M₂) → (M₁ ≃ₛₗ[σ₁₂] M₂)) | | ⟨e, _, _⟩ ⟨e', _, _⟩ rfl := rfl | lemma | continuous_linear_equiv.to_linear_equiv_injective | 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 | |
ext {f g : M₁ ≃SL[σ₁₂] M₂} (h : (f : M₁ → M₂) = g) : f = g | to_linear_equiv_injective $ linear_equiv.ext $ congr_fun h | lemma | continuous_linear_equiv.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"
] | [
"linear_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : (M₁ ≃SL[σ₁₂] M₂) → (M₁ →SL[σ₁₂] M₂)) | λ e e' h, ext $ funext $ continuous_linear_map.ext_iff.1 h | lemma | continuous_linear_equiv.coe_injective | 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_inj {e e' : M₁ ≃SL[σ₁₂] M₂} : (e : M₁ →SL[σ₁₂] M₂) = e' ↔ e = e' | coe_injective.eq_iff | lemma | continuous_linear_equiv.coe_inj | 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_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) : M₁ ≃ₜ M₂ | { to_equiv := e.to_linear_equiv.to_equiv, ..e } | def | continuous_linear_equiv.to_homeomorph | 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"
] | [] | A continuous linear equivalence induces a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.to_homeomorph = e | rfl | lemma | continuous_linear_equiv.coe_to_homeomorph | 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 | |
image_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e '' closure s = closure (e '' s) | e.to_homeomorph.image_closure s | lemma | continuous_linear_equiv.image_closure | 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"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e ⁻¹' closure s = closure (e ⁻¹' s) | e.to_homeomorph.preimage_closure s | lemma | continuous_linear_equiv.preimage_closure | 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"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_image (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} :
is_closed (e '' s) ↔ is_closed s | e.to_homeomorph.is_closed_image | lemma | continuous_linear_equiv.is_closed_image | 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"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e (𝓝 x) = 𝓝 (e x) | e.to_homeomorph.map_nhds_eq x | lemma | continuous_linear_equiv.map_nhds_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (e : M₁ ≃SL[σ₁₂] M₂) : e (0 : M₁) = 0 | (e : M₁ →SL[σ₁₂] M₂).map_zero | lemma | continuous_linear_equiv.map_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 | |
map_add (e : M₁ ≃SL[σ₁₂] M₂) (x y : M₁) : e (x + y) = e x + e y | (e : M₁ →SL[σ₁₂] M₂).map_add x y | lemma | continuous_linear_equiv.map_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e (c • x) = σ₁₂ c • (e x) | (e : M₁ →SL[σ₁₂] M₂).map_smulₛₗ c x | lemma | continuous_linear_equiv.map_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul [module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M₁) :
e (c • x) = c • (e x) | (e : M₁ →L[R₁] M₂).map_smul c x | lemma | continuous_linear_equiv.map_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"
] | [
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ x = 0 | e.to_linear_equiv.map_eq_zero_iff | lemma | continuous_linear_equiv.map_eq_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (e : M₁ ≃SL[σ₁₂] M₂) : continuous (e : M₁ → M₂) | e.continuous_to_fun | lemma | continuous_linear_equiv.continuous | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} : continuous_on (e : M₁ → M₂) s | e.continuous.continuous_on | lemma | continuous_linear_equiv.continuous_on | 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_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : continuous_at (e : M₁ → M₂) x | e.continuous.continuous_at | lemma | continuous_linear_equiv.continuous_at | 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_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} {x : M₁} :
continuous_within_at (e : M₁ → M₂) s x | e.continuous.continuous_within_at | lemma | continuous_linear_equiv.continuous_within_at | 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_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_on_iff
{α : Type*} [topological_space α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} {s : set α} :
continuous_on (e ∘ f) s ↔ continuous_on f s | e.to_homeomorph.comp_continuous_on_iff _ _ | lemma | continuous_linear_equiv.comp_continuous_on_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"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff
{α : Type*} [topological_space α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} :
continuous (e ∘ f) ↔ continuous f | e.to_homeomorph.comp_continuous_iff | lemma | continuous_linear_equiv.comp_continuous_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"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext₁ [topological_space R₁] {f g : R₁ ≃L[R₁] M₁} (h : f 1 = g 1) : f = g | ext $ funext $ λ x, mul_one x ▸ by rw [← smul_eq_mul, map_smul, h, map_smul] | lemma | continuous_linear_equiv.ext₁ | 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"
] | [
"mul_one",
"smul_eq_mul",
"topological_space"
] | An extensionality lemma for `R ≃L[R] M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : M₁ ≃L[R₁] M₁ | { continuous_to_fun := continuous_id,
continuous_inv_fun := continuous_id,
.. linear_equiv.refl R₁ M₁ } | def | continuous_linear_equiv.refl | 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_id",
"linear_equiv.refl"
] | The identity map as a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_refl :
↑(continuous_linear_equiv.refl R₁ M₁) = continuous_linear_map.id R₁ M₁ | rfl | lemma | continuous_linear_equiv.coe_refl | 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_equiv.refl",
"continuous_linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_refl' : ⇑(continuous_linear_equiv.refl R₁ M₁) = id | rfl | lemma | continuous_linear_equiv.coe_refl' | 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_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ | { continuous_to_fun := e.continuous_inv_fun,
continuous_inv_fun := e.continuous_to_fun,
.. e.to_linear_equiv.symm } | def | continuous_linear_equiv.symm | 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 inverse of a continuous linear equivalence as a continuous linear equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_to_linear_equiv (e : M₁ ≃SL[σ₁₂] M₂) :
e.symm.to_linear_equiv = e.to_linear_equiv.symm | by { ext, refl } | lemma | continuous_linear_equiv.symm_to_linear_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) :
e.to_homeomorph.symm = e.symm.to_homeomorph | rfl | lemma | continuous_linear_equiv.symm_to_homeomorph | 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 | |
simps.apply (h : M₁ ≃SL[σ₁₂] M₂) : M₁ → M₂ | h | def | continuous_linear_equiv.simps.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"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁ | h.symm
initialize_simps_projections continuous_linear_equiv
(to_linear_equiv_to_fun → apply, to_linear_equiv_inv_fun → symm_apply) | def | continuous_linear_equiv.simps.symm_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"
] | [
"continuous_linear_equiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e.symm (𝓝 (e x)) = 𝓝 x | e.to_homeomorph.symm_map_nhds_eq x | lemma | continuous_linear_equiv.symm_map_nhds_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : M₁ ≃SL[σ₁₃] M₃ | { continuous_to_fun := e₂.continuous_to_fun.comp e₁.continuous_to_fun,
continuous_inv_fun := e₁.continuous_inv_fun.comp e₂.continuous_inv_fun,
.. e₁.to_linear_equiv.trans e₂.to_linear_equiv } | def | continuous_linear_equiv.trans | 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 composition of two continuous linear equivalences as a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_to_linear_equiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) :
(e₁.trans e₂).to_linear_equiv = e₁.to_linear_equiv.trans e₂.to_linear_equiv | by { ext, refl } | lemma | continuous_linear_equiv.trans_to_linear_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) :
(M₁ × M₃) ≃L[R₁] (M₂ × M₄) | { continuous_to_fun := e.continuous_to_fun.prod_map e'.continuous_to_fun,
continuous_inv_fun := e.continuous_inv_fun.prod_map e'.continuous_inv_fun,
.. e.to_linear_equiv.prod e'.to_linear_equiv } | def | continuous_linear_equiv.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"
] | Product of two continuous linear equivalences. The map comes from `equiv.prod_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_apply [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂)
(e' : M₃ ≃L[R₁] M₄) (x) :
e.prod e' x = (e x.1, e' x.2) | rfl | lemma | continuous_linear_equiv.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 | |
coe_prod [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂)
(e' : M₃ ≃L[R₁] M₄) :
(e.prod e' : (M₁ × M₃) →L[R₁] (M₂ × M₄)) = (e : M₁ →L[R₁] M₂).prod_map (e' : M₃ →L[R₁] M₄) | rfl | lemma | continuous_linear_equiv.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"
] | [
"module",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_symm [module R₁ M₂] [module R₁ M₃] [module R₁ M₄]
(e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) :
(e.prod e').symm = e.symm.prod e'.symm | rfl | lemma | continuous_linear_equiv.prod_symm | 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 | |
bijective (e : M₁ ≃SL[σ₁₂] M₂) : function.bijective e | e.to_linear_equiv.to_equiv.bijective | theorem | continuous_linear_equiv.bijective | 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 | |
injective (e : M₁ ≃SL[σ₁₂] M₂) : function.injective e | e.to_linear_equiv.to_equiv.injective | theorem | continuous_linear_equiv.injective | 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 | |
surjective (e : M₁ ≃SL[σ₁₂] M₂) : function.surjective e | e.to_linear_equiv.to_equiv.surjective | theorem | continuous_linear_equiv.surjective | 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 | |
trans_apply (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) (c : M₁) :
(e₁.trans e₂) c = e₂ (e₁ c) | rfl | theorem | continuous_linear_equiv.trans_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 | |
apply_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (c : M₂) : e (e.symm c) = c | e.1.right_inv c | theorem | continuous_linear_equiv.apply_symm_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 | |
symm_apply_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : e.symm (e b) = b | e.1.left_inv b | theorem | continuous_linear_equiv.symm_apply_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 | |
symm_trans_apply (e₁ : M₂ ≃SL[σ₂₁] M₁) (e₂ : M₃ ≃SL[σ₃₂] M₂) (c : M₁) :
(e₂.trans e₁).symm c = e₂.symm (e₁.symm c) | rfl | theorem | continuous_linear_equiv.symm_trans_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 | |
symm_image_image (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e.symm '' (e '' s) = s | e.to_linear_equiv.to_equiv.symm_image_image s | theorem | continuous_linear_equiv.symm_image_image | 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 | |
image_symm_image (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e '' (e.symm '' s) = s | e.symm.symm_image_image s | theorem | continuous_linear_equiv.image_symm_image | 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_coe (f : M₁ ≃SL[σ₁₂] M₂) (f' : M₂ ≃SL[σ₂₃] M₃) :
(f' : M₂ →SL[σ₂₃] M₃).comp (f : M₁ →SL[σ₁₂] M₂) = (f.trans f' : M₁ →SL[σ₁₃] M₃) | rfl | lemma | continuous_linear_equiv.comp_coe | 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_comp_coe_symm (e : M₁ ≃SL[σ₁₂] M₂) :
(e : M₁ →SL[σ₁₂] M₂).comp (e.symm : M₂ →SL[σ₂₁] M₁) = continuous_linear_map.id R₂ M₂ | continuous_linear_map.ext e.apply_symm_apply | theorem | continuous_linear_equiv.coe_comp_coe_symm | 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.ext",
"continuous_linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_comp_coe (e : M₁ ≃SL[σ₁₂] M₂) :
(e.symm : M₂ →SL[σ₂₁] M₁).comp (e : M₁ →SL[σ₁₂] M₂) = continuous_linear_map.id R₁ M₁ | continuous_linear_map.ext e.symm_apply_apply | theorem | continuous_linear_equiv.coe_symm_comp_coe | 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.ext",
"continuous_linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_self (e : M₁ ≃SL[σ₁₂] M₂) :
(e.symm : M₂ → M₁) ∘ (e : M₁ → M₂) = id | by{ ext x, exact symm_apply_apply e x } | lemma | continuous_linear_equiv.symm_comp_self | 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 | |
self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) :
(e : M₁ → M₂) ∘ (e.symm : M₂ → M₁) = id | by{ ext x, exact apply_symm_apply e x } | lemma | continuous_linear_equiv.self_comp_symm | 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 | |
symm_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e | by { ext x, refl } | theorem | continuous_linear_equiv.symm_symm | 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 | |
refl_symm :
(continuous_linear_equiv.refl R₁ M₁).symm = continuous_linear_equiv.refl R₁ M₁ | rfl | lemma | continuous_linear_equiv.refl_symm | 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_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : e.symm.symm x = e x | rfl | theorem | continuous_linear_equiv.symm_symm_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 | |
symm_apply_eq (e : M₁ ≃SL[σ₁₂] M₂) {x y} : e.symm x = y ↔ x = e y | e.to_linear_equiv.symm_apply_eq | lemma | continuous_linear_equiv.symm_apply_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) {x y} : y = e.symm x ↔ e y = x | e.to_linear_equiv.eq_symm_apply | lemma | continuous_linear_equiv.eq_symm_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 | |
image_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e '' s = e.symm ⁻¹' s | e.to_linear_equiv.to_equiv.image_eq_preimage s | lemma | continuous_linear_equiv.image_eq_preimage | 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 | |
image_symm_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e.symm '' s = e ⁻¹' s | by rw [e.symm.image_eq_preimage, e.symm_symm] | lemma | continuous_linear_equiv.image_symm_eq_preimage | 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 | |
symm_preimage_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) :
e.symm ⁻¹' (e ⁻¹' s) = s | e.to_linear_equiv.to_equiv.symm_preimage_preimage s | lemma | continuous_linear_equiv.symm_preimage_preimage | 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 | |
preimage_symm_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) :
e ⁻¹' (e.symm ⁻¹' s) = s | e.symm.symm_preimage_preimage s | lemma | continuous_linear_equiv.preimage_symm_preimage | 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 | |
uniform_embedding {E₁ E₂ : Type*} [uniform_space E₁] [uniform_space E₂]
[add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]
[uniform_add_group E₁] [uniform_add_group E₂]
(e : E₁ ≃SL[σ₁₂] E₂) :
uniform_embedding e | e.to_linear_equiv.to_equiv.uniform_embedding
e.to_continuous_linear_map.uniform_continuous
e.symm.to_continuous_linear_map.uniform_continuous | lemma | continuous_linear_equiv.uniform_embedding | 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",
"module",
"uniform_add_group",
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_equiv.uniform_embedding {E₁ E₂ : Type*} [uniform_space E₁]
[uniform_space E₂] [add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]
[uniform_add_group E₁] [uniform_add_group E₂]
(e : E₁ ≃ₛₗ[σ₁₂] E₂) (h₁ : continuous e) (h₂ : continuous e.symm) :
uniform_embedding e | continuous_linear_equiv.uniform_embedding
({ continuous_to_fun := h₁,
continuous_inv_fun := h₂,
.. e } : E₁ ≃SL[σ₁₂] E₂) | lemma | linear_equiv.uniform_embedding | 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",
"continuous",
"continuous_linear_equiv.uniform_embedding",
"module",
"uniform_add_group",
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_inverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : function.left_inverse f₂ f₁)
(h₂ : function.right_inverse f₂ f₁) :
M₁ ≃SL[σ₁₂] M₂ | { to_fun := f₁,
continuous_to_fun := f₁.continuous,
inv_fun := f₂,
continuous_inv_fun := f₂.continuous,
left_inv := h₁,
right_inv := h₂,
.. f₁ } | def | continuous_linear_equiv.equiv_of_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"
] | [
"inv_fun"
] | Create a `continuous_linear_equiv` from two `continuous_linear_map`s that are
inverse of each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_inverse_apply (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂ x) :
equiv_of_inverse f₁ f₂ h₁ h₂ x = f₁ x | rfl | lemma | continuous_linear_equiv.equiv_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_equiv_of_inverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂) :
(equiv_of_inverse f₁ f₂ h₁ h₂).symm = equiv_of_inverse f₂ f₁ h₂ h₁ | rfl | lemma | continuous_linear_equiv.symm_equiv_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
automorphism_group : group (M₁ ≃L[R₁] M₁) | { mul := λ f g, g.trans f,
one := continuous_linear_equiv.refl R₁ M₁,
inv := λ f, f.symm,
mul_assoc := λ f g h, by {ext, refl},
mul_one := λ f, by {ext, refl},
one_mul := λ f, by {ext, refl},
mul_left_inv := λ f, by {ext, exact f.left_inv x} } | instance | continuous_linear_equiv.automorphism_group | 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_equiv.refl",
"group",
"mul_assoc",
"mul_left_inv",
"mul_one",
"one_mul"
] | The continuous linear equivalences from `M` to itself form a group under composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift : ulift M₁ ≃L[R₁] M₁ | { map_add' := λ x y, rfl,
map_smul' := λ c x, rfl,
continuous_to_fun := continuous_ulift_down,
continuous_inv_fun := continuous_ulift_up,
.. equiv.ulift } | def | continuous_linear_equiv.ulift | 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_ulift_down",
"continuous_ulift_up",
"equiv.ulift"
] | The continuous linear equivalence between `ulift M₁` and `M₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow_congr_equiv (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) :
(M₁ →SL[σ₁₄] M₄) ≃ (M₂ →SL[σ₂₃] M₃) | { to_fun := λ f, (e₄₃ : M₄ →SL[σ₄₃] M₃).comp (f.comp (e₁₂.symm : M₂ →SL[σ₂₁] M₁)),
inv_fun := λ f, (e₄₃.symm : M₃ →SL[σ₃₄] M₄).comp (f.comp (e₁₂ : M₁ →SL[σ₁₂] M₂)),
left_inv := λ f, continuous_linear_map.ext $ λ x,
by simp only [continuous_linear_map.comp_apply, symm_apply_apply, coe_coe],
right_inv := λ f, c... | def | continuous_linear_equiv.arrow_congr_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"
] | [
"coe_coe",
"continuous_linear_map.comp_apply",
"continuous_linear_map.ext",
"inv_fun"
] | A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of
continuous linear maps. See also `continuous_linear_equiv.arrow_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_prod (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) :
(M × M₃) ≃L[R] M₂ × M₄ | { continuous_to_fun := (e.continuous_to_fun.comp continuous_fst).prod_mk
((e'.continuous_to_fun.comp continuous_snd).add $ f.continuous.comp continuous_fst),
continuous_inv_fun := (e.continuous_inv_fun.comp continuous_fst).prod_mk
(e'.continuous_inv_fun.comp $ continuous_snd.sub $ f.continuous.comp $
e.... | def | continuous_linear_equiv.skew_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"
] | [
"continuous_fst",
"continuous_snd"
] | Equivalence given by a block lower diagonal matrix. `e` and `e'` are diagonal square blocks,
and `f` is a rectangular block below the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_prod_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :
e.skew_prod e' f x = (e x.1, e' x.2 + f x.1) | rfl | lemma | continuous_linear_equiv.skew_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_prod_symm_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :
(e.skew_prod e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) | rfl | lemma | continuous_linear_equiv.skew_prod_symm_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 | |
map_sub (e : M ≃SL[σ₁₂] M₂) (x y : M) : e (x - y) = e x - e y | (e : M →SL[σ₁₂] M₂).map_sub x y | lemma | continuous_linear_equiv.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 | |
map_neg (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x | (e : M →SL[σ₁₂] M₂).map_neg x | lemma | continuous_linear_equiv.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 | |
of_unit (f : (M →L[R] M)ˣ) : (M ≃L[R] M) | { to_linear_equiv :=
{ to_fun := f.val,
map_add' := by simp,
map_smul' := by simp,
inv_fun := f.inv,
left_inv := λ x, show (f.inv * f.val) x = x, by {rw f.inv_val, simp},
right_inv := λ x, show (f.val * f.inv) x = x, by {rw f.val_inv, simp}, },
continuous_to_fun := f.val.continuous,
co... | def | continuous_linear_equiv.of_unit | 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"
] | An invertible continuous linear map `f` determines a continuous equivalence from `M` to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_unit (f : (M ≃L[R] M)) : (M →L[R] M)ˣ | { val := f,
inv := f.symm,
val_inv := by {ext, simp},
inv_val := by {ext, simp} } | def | continuous_linear_equiv.to_unit | 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"
] | [] | A continuous equivalence from `M` to itself determines an invertible continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_equiv : (M →L[R] M)ˣ ≃* (M ≃L[R] M) | { to_fun := of_unit,
inv_fun := to_unit,
left_inv := λ f, by {ext, refl},
right_inv := λ f, by {ext, refl},
map_mul' := λ x y, by {ext, refl} } | def | continuous_linear_equiv.units_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"
] | The units of the algebra of continuous `R`-linear endomorphisms of `M` is multiplicatively
equivalent to the type of continuous linear equivalences between `M` and itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_equiv_apply (f : (M →L[R] M)ˣ) (x : M) :
units_equiv R M f x = f x | rfl | lemma | continuous_linear_equiv.units_equiv_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 | |
units_equiv_aut : Rˣ ≃ (R ≃L[R] R) | { to_fun := λ u, equiv_of_inverse
(continuous_linear_map.smul_right (1 : R →L[R] R) ↑u)
(continuous_linear_map.smul_right (1 : R →L[R] R) ↑u⁻¹)
(λ x, by simp) (λ x, by simp),
inv_fun := λ e, ⟨e 1, e.symm 1,
by rw [← smul_eq_mul, ← map_smul, smul_eq_mul, mul_one, symm_apply_apply],
by rw [← smul_eq... | def | continuous_linear_equiv.units_equiv_aut | 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.smul_right",
"inv_fun",
"mul_one",
"smul_eq_mul",
"units.ext"
] | Continuous linear equivalences `R ≃L[R] R` are enumerated by `Rˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_equiv_aut_apply (u : Rˣ) (x : R) : units_equiv_aut R u x = x * u | rfl | lemma | continuous_linear_equiv.units_equiv_aut_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 | |
units_equiv_aut_apply_symm (u : Rˣ) (x : R) :
(units_equiv_aut R u).symm x = x * ↑u⁻¹ | rfl | lemma | continuous_linear_equiv.units_equiv_aut_apply_symm | 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 | |
units_equiv_aut_symm_apply (e : R ≃L[R] R) :
↑((units_equiv_aut R).symm e) = e 1 | rfl | lemma | continuous_linear_equiv.units_equiv_aut_symm_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 | |
equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : function.right_inverse f₂ f₁) :
M ≃L[R] M₂ × ker f₁ | equiv_of_inverse (f₁.prod (f₁.proj_ker_of_right_inverse f₂ h)) (f₂.coprod (ker f₁).subtypeL)
(λ x, by simp)
(λ ⟨x, y⟩, by simp [h x]) | def | continuous_linear_equiv.equiv_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"
] | [] | A pair of continuous linear maps such that `f₁ ∘ f₂ = id` generates a continuous
linear equivalence `e` between `M` and `M₂ × f₁.ker` such that `(e x).2 = x` for `x ∈ f₁.ker`,
`(e x).1 = f₁ x`, and `(e (f₂ y)).2 = 0`. The map is given by `e x = (f₁ x, x - f₂ (f₁ x))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
(h : function.right_inverse f₂ f₁) (x : M) :
(equiv_of_right_inverse f₁ f₂ h x).1 = f₁ x | rfl | lemma | continuous_linear_equiv.fst_equiv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
(h : function.right_inverse f₂ f₁) (x : M) :
((equiv_of_right_inverse f₁ f₂ h x).2 : M) = x - f₂ (f₁ x) | rfl | lemma | continuous_linear_equiv.snd_equiv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_right_inverse_symm_apply (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
(h : function.right_inverse f₂ f₁) (y : M₂ × ker f₁) :
(equiv_of_right_inverse f₁ f₂ h).symm y = f₂ y.1 + y.2 | rfl | lemma | continuous_linear_equiv.equiv_of_right_inverse_symm_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 |
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