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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|---|---|---|---|---|---|---|---|---|---|---|
norm_map (x : E) : ‖f x‖ = ‖x‖ | semilinear_isometry_class.norm_map f x | lemma | linear_isometry.norm_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_map (x : E) : ‖f x‖₊ = ‖x‖₊ | nnreal.eq $ norm_map f x | lemma | linear_isometry.nnnorm_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry : isometry f | add_monoid_hom_class.isometry_of_norm _ (norm_map _) | lemma | linear_isometry.isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_image_iff [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) {s : set E} :
is_complete (f '' s) ↔ is_complete s | is_complete_image_iff (semilinear_isometry_class.isometry f).uniform_inducing | lemma | linear_isometry.is_complete_image_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"is_complete",
"is_complete_image_iff",
"semilinear_isometry_class",
"semilinear_isometry_class.isometry",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_map_iff [ring_hom_surjective σ₁₂] {p : submodule R E} :
is_complete (p.map f.to_linear_map : set E₂) ↔ is_complete (p : set E) | f.is_complete_image_iff | lemma | linear_isometry.is_complete_map_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"is_complete",
"ring_hom_surjective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_map_iff' [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) [ring_hom_surjective σ₁₂]
{p : submodule R E} : is_complete (p.map f : set E₂) ↔ is_complete (p : set E) | is_complete_image_iff f | lemma | linear_isometry.is_complete_map_iff' | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"is_complete",
"is_complete_image_iff",
"ring_hom_surjective",
"semilinear_isometry_class",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_map [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) [ring_hom_surjective σ₁₂]
(p : submodule R E) [complete_space p] : complete_space (p.map f) | ((is_complete_map_iff' f).2 $ complete_space_coe_iff_is_complete.1 ‹_›).complete_space_coe | instance | linear_isometry.complete_space_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"complete_space",
"ring_hom_surjective",
"semilinear_isometry_class",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_map' [ring_hom_surjective σ₁₂] (p : submodule R E) [complete_space p] :
complete_space (p.map f.to_linear_map) | (f.is_complete_map_iff.2 $ complete_space_coe_iff_is_complete.1 ‹_›).complete_space_coe | instance | linear_isometry.complete_space_map' | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"complete_space",
"ring_hom_surjective",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_map (x y : E) : dist (f x) (f y) = dist x y | f.isometry.dist_eq x y | lemma | linear_isometry.dist_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_map (x y : E) : edist (f x) (f y) = edist x y | f.isometry.edist_eq x y | lemma | linear_isometry.edist_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective : injective f₁ | isometry.injective (linear_isometry.isometry f₁) | lemma | linear_isometry.injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry.injective",
"linear_isometry.isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_iff {x y : F} : f₁ x = f₁ y ↔ x = y | f₁.injective.eq_iff | lemma | linear_isometry.map_eq_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne {x y : F} (h : x ≠ y) : f₁ x ≠ f₁ y | f₁.injective.ne h | lemma | linear_isometry.map_ne | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ball (x : E) (r : ℝ) :
f ⁻¹' (metric.ball (f x) r) = metric.ball x r | f.isometry.preimage_ball x r | lemma | linear_isometry.preimage_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_sphere (x : E) (r : ℝ) :
f ⁻¹' (metric.sphere (f x) r) = metric.sphere x r | f.isometry.preimage_sphere x r | lemma | linear_isometry.preimage_sphere | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (x : E) (r : ℝ) :
f ⁻¹' (metric.closed_ball (f x) r) = metric.closed_ball x r | f.isometry.preimage_closed_ball x r | lemma | linear_isometry.preimage_closed_ball | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image (s : set E) : emetric.diam (f '' s) = emetric.diam s | f.isometry.ediam_image s | lemma | linear_isometry.ediam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_range : emetric.diam (range f) = emetric.diam (univ : set E) | f.isometry.ediam_range | lemma | linear_isometry.ediam_range | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image (s : set E) : metric.diam (f '' s) = metric.diam s | isometry.diam_image (linear_isometry.isometry f) s | lemma | linear_isometry.diam_image | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry.diam_image",
"linear_isometry.isometry",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_range : metric.diam (range f) = metric.diam (univ : set E) | isometry.diam_range (linear_isometry.isometry f) | lemma | linear_isometry.diam_range | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry.diam_range",
"linear_isometry.isometry",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_map : E →SL[σ₁₂] E₂ | ⟨f.to_linear_map, f.continuous⟩ | def | linear_isometry.to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Interpret a linear isometry as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_map_injective :
function.injective (to_continuous_linear_map : _ → E →SL[σ₁₂] E₂) | λ x y h, coe_injective (congr_arg _ h : ⇑x.to_continuous_linear_map = _) | lemma | linear_isometry.to_continuous_linear_map_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_map_inj {f g : E →ₛₗᵢ[σ₁₂] E₂} :
f.to_continuous_linear_map = g.to_continuous_linear_map ↔ f = g | to_continuous_linear_map_injective.eq_iff | lemma | linear_isometry.to_continuous_linear_map_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_map : ⇑f.to_continuous_linear_map = f | rfl | lemma | linear_isometry.coe_to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {α : Type*} [topological_space α] {g : α → E} :
continuous (f ∘ g) ↔ continuous g | f.isometry.comp_continuous_iff | lemma | linear_isometry.comp_continuous_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : E →ₗᵢ[R] E | ⟨linear_map.id, λ x, rfl⟩ | def | linear_isometry.id | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | The identity linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ((id : E →ₗᵢ[R] E) : E → E) = _root_.id | rfl | lemma | linear_isometry.coe_id | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : E) : (id : E →ₗᵢ[R] E) x = x | rfl | lemma | linear_isometry.id_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_to_linear_map : (id.to_linear_map : E →ₗ[R] E) = linear_map.id | rfl | lemma | linear_isometry.id_to_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_to_continuous_linear_map :
id.to_continuous_linear_map = continuous_linear_map.id R E | rfl | lemma | linear_isometry.id_to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous_linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂) : E →ₛₗᵢ[σ₁₃] E₃ | ⟨g.to_linear_map.comp f.to_linear_map, λ x, (norm_map g _).trans (norm_map f _)⟩ | def | linear_isometry.comp | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Composition of linear isometries. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (f : E →ₛₗᵢ[σ₁₂] E₂) :
⇑(g.comp f) = g ∘ f | rfl | lemma | linear_isometry.coe_comp | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp : (id : E₂ →ₗᵢ[R₂] E₂).comp f = f | ext $ λ x, rfl | lemma | linear_isometry.id_comp | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : E₃ →ₛₗᵢ[σ₃₄] E₄) (g : E₂ →ₛₗᵢ[σ₂₃] E₃) (h : E →ₛₗᵢ[σ₁₂] E₂) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | linear_isometry.comp_assoc | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : E →ₗᵢ[R] E) : E → E) = _root_.id | rfl | lemma | linear_isometry.coe_one | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : E →ₗᵢ[R] E) : ⇑(f * g) = f ∘ g | rfl | lemma | linear_isometry.coe_mul | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : E →ₗᵢ[R] E) = id | rfl | lemma | linear_isometry.one_def | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (f g : E →ₗᵢ[R] E) : (f * g : E →ₗᵢ[R] E) = f.comp g | rfl | lemma | linear_isometry.mul_def | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.to_linear_isometry (f : E →ₛₗ[σ₁₂] E₂) (hf : isometry f) : E →ₛₗᵢ[σ₁₂] E₂ | { norm_map' := by { simp_rw [←dist_zero_right, ←f.map_zero], exact λ x, hf.dist_eq x _ },
.. f } | def | linear_map.to_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry"
] | Construct a `linear_isometry` from a `linear_map` satisfying `isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtypeₗᵢ : p →ₗᵢ[R'] E | ⟨p.subtype, λ x, rfl⟩ | def | submodule.subtypeₗᵢ | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | `submodule.subtype` as a `linear_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_subtypeₗᵢ : ⇑p.subtypeₗᵢ = p.subtype | rfl | lemma | submodule.coe_subtypeₗᵢ | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtypeₗᵢ_to_linear_map : p.subtypeₗᵢ.to_linear_map = p.subtype | rfl | lemma | submodule.subtypeₗᵢ_to_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtypeₗᵢ_to_continuous_linear_map :
p.subtypeₗᵢ.to_continuous_linear_map = p.subtypeL | rfl | lemma | submodule.subtypeₗᵢ_to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_equiv (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁]
[ring_hom_inv_pair σ₂₁ σ₁₂] (E E₂ : Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] extends E ≃ₛₗ[σ₁₂] E₂ | (norm_map' : ∀ x, ‖to_linear_equiv x‖ = ‖x‖) | structure | linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"ring_hom_inv_pair",
"seminormed_add_comm_group"
] | A semilinear isometric equivalence between two normed vector spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semilinear_isometry_equiv_class (𝓕 : Type*) {R R₂ : out_param Type*}
[semiring R] [semiring R₂] (σ₁₂ : out_param $ R →+* R₂) {σ₂₁ : out_param $ R₂ →+* R}
[ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] (E E₂ : out_param Type*)
[seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [mod... | (norm_map : ∀ (f : 𝓕) (x : E), ‖f x‖ = ‖x‖) | class | semilinear_isometry_equiv_class | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"ring_hom_inv_pair",
"semilinear_equiv_class",
"seminormed_add_comm_group",
"semiring"
] | `semilinear_isometry_equiv_class F σ E E₂` asserts `F` is a type of bundled `σ`-semilinear
isometric equivs `E → E₂`.
See also `linear_isometry_equiv_class F R E E₂` for the case where `σ` is the identity map on `R`.
A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S`
is semilinea... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv_class (𝓕 : Type*) (R E E₂ : out_param Type*) [semiring R]
[seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [module R E₂] | semilinear_isometry_equiv_class 𝓕 (ring_hom.id R) E E₂ | abbreviation | linear_isometry_equiv_class | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"ring_hom.id",
"semilinear_isometry_equiv_class",
"seminormed_add_comm_group",
"semiring"
] | `linear_isometry_equiv_class F R E E₂` asserts `F` is a type of bundled `R`-linear isometries
`M → M₂`.
This is an abbreviation for `semilinear_isometry_equiv_class F (ring_hom.id R) E E₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_equiv_injective : injective (to_linear_equiv : (E ≃ₛₗᵢ[σ₁₂] E₂) → (E ≃ₛₗ[σ₁₂] E₂)) | | ⟨e, _⟩ ⟨_, _⟩ rfl := rfl | lemma | linear_isometry_equiv.to_linear_equiv_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} :
f.to_linear_equiv = g.to_linear_equiv ↔ f = g | to_linear_equiv_injective.eq_iff | lemma | linear_isometry_equiv.to_linear_equiv_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (E ≃ₛₗᵢ[σ₁₂] E₂) (E → E₂) coe_fn | fun_like.coe_injective | lemma | linear_isometry_equiv.coe_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (e : E ≃ₛₗ[σ₁₂] E₂) (he : ∀ x, ‖e x‖ = ‖x‖) :
⇑(mk e he) = e | rfl | lemma | linear_isometry_equiv.coe_mk | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_equiv (e : E ≃ₛₗᵢ[σ₁₂] E₂) : ⇑e.to_linear_equiv = e | rfl | lemma | linear_isometry_equiv.coe_to_linear_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {e e' : E ≃ₛₗᵢ[σ₁₂] E₂} (h : ∀ x, e x = e' x) : e = e' | to_linear_equiv_injective $ linear_equiv.ext h | lemma | linear_isometry_equiv.ext | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg {f : E ≃ₛₗᵢ[σ₁₂] E₂} : Π {x x' : E}, x = x' → f x = f x' | | _ _ rfl := rfl | lemma | linear_isometry_equiv.congr_arg | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : E ≃ₛₗᵢ[σ₁₂] E₂} (h : f = g) (x : E) : f x = g x | h ▸ rfl | lemma | linear_isometry_equiv.congr_fun | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bounds (e : E ≃ₛₗ[σ₁₂] E₂) (h₁ : ∀ x, ‖e x‖ ≤ ‖x‖) (h₂ : ∀ y, ‖e.symm y‖ ≤ ‖y‖) :
E ≃ₛₗᵢ[σ₁₂] E₂ | ⟨e, λ x, le_antisymm (h₁ x) $ by simpa only [e.symm_apply_apply] using h₂ (e x)⟩ | def | linear_isometry_equiv.of_bounds | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Construct a `linear_isometry_equiv` from a `linear_equiv` and two inequalities:
`∀ x, ‖e x‖ ≤ ‖x‖` and `∀ y, ‖e.symm y‖ ≤ ‖y‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_map (x : E) : ‖e x‖ = ‖x‖ | e.norm_map' x | lemma | linear_isometry_equiv.norm_map | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_isometry : E →ₛₗᵢ[σ₁₂] E₂ | ⟨e.1, e.2⟩ | def | linear_isometry_equiv.to_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Reinterpret a `linear_isometry_equiv` as a `linear_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_isometry_injective :
function.injective (to_linear_isometry : _ → E →ₛₗᵢ[σ₁₂] E₂) | λ x y h, coe_injective (congr_arg _ h : ⇑x.to_linear_isometry = _) | lemma | linear_isometry_equiv.to_linear_isometry_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_isometry_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} :
f.to_linear_isometry = g.to_linear_isometry ↔ f = g | to_linear_isometry_injective.eq_iff | lemma | linear_isometry_equiv.to_linear_isometry_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_isometry : ⇑e.to_linear_isometry = e | rfl | lemma | linear_isometry_equiv.coe_to_linear_isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry : isometry e | e.to_linear_isometry.isometry | lemma | linear_isometry_equiv.isometry | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_isometry_equiv : E ≃ᵢ E₂ | ⟨e.to_linear_equiv.to_equiv, e.isometry⟩ | def | linear_isometry_equiv.to_isometry_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Reinterpret a `linear_isometry_equiv` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_isometry_equiv_injective :
function.injective (to_isometry_equiv : (E ≃ₛₗᵢ[σ₁₂] E₂) → E ≃ᵢ E₂) | λ x y h, coe_injective (congr_arg _ h : ⇑x.to_isometry_equiv = _) | lemma | linear_isometry_equiv.to_isometry_equiv_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_isometry_equiv_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} :
f.to_isometry_equiv = g.to_isometry_equiv ↔ f = g | to_isometry_equiv_injective.eq_iff | lemma | linear_isometry_equiv.to_isometry_equiv_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_univ (e : E ≃ₛₗᵢ[σ₁₂] E₂) : set.range e = set.univ | by { rw ← coe_to_isometry_equiv, exact isometry_equiv.range_eq_univ _, } | lemma | linear_isometry_equiv.range_eq_univ | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"isometry_equiv.range_eq_univ",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph : E ≃ₜ E₂ | e.to_isometry_equiv.to_homeomorph | def | linear_isometry_equiv.to_homeomorph | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Reinterpret a `linear_isometry_equiv` as an `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_homeomorph_injective :
function.injective (to_homeomorph : (E ≃ₛₗᵢ[σ₁₂] E₂) → E ≃ₜ E₂) | λ x y h, coe_injective (congr_arg _ h : ⇑x.to_homeomorph = _) | lemma | linear_isometry_equiv.to_homeomorph_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} :
f.to_homeomorph = g.to_homeomorph ↔ f = g | to_homeomorph_injective.eq_iff | lemma | linear_isometry_equiv.to_homeomorph_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_equiv : E ≃SL[σ₁₂] E₂ | { .. e.to_linear_isometry.to_continuous_linear_map,
.. e.to_homeomorph } | def | linear_isometry_equiv.to_continuous_linear_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Interpret a `linear_isometry_equiv` as a continuous linear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_equiv_injective :
function.injective (to_continuous_linear_equiv : _ → E ≃SL[σ₁₂] E₂) | λ x y h, coe_injective (congr_arg _ h : ⇑x.to_continuous_linear_equiv = _) | lemma | linear_isometry_equiv.to_continuous_linear_equiv_injective | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_equiv_inj {f g : E ≃ₛₗᵢ[σ₁₂] E₂} :
f.to_continuous_linear_equiv = g.to_continuous_linear_equiv ↔ f = g | to_continuous_linear_equiv_injective.eq_iff | lemma | linear_isometry_equiv.to_continuous_linear_equiv_inj | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_equiv : ⇑e.to_continuous_linear_equiv = e | rfl | lemma | linear_isometry_equiv.coe_to_continuous_linear_equiv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : E ≃ₗᵢ[R] E | ⟨linear_equiv.refl R E, λ x, rfl⟩ | def | linear_isometry_equiv.refl | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Identity map as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift : ulift E ≃ₗᵢ[R] E | { norm_map' := λ x, rfl,
.. continuous_linear_equiv.ulift } | def | linear_isometry_equiv.ulift | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"continuous_linear_equiv.ulift"
] | Linear isometry equiv between a space and its lift to another universe. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_refl : ⇑(refl R E) = id | rfl | lemma | linear_isometry_equiv.coe_refl | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm : E₂ ≃ₛₗᵢ[σ₂₁] E | ⟨e.to_linear_equiv.symm,
λ x, (e.norm_map _).symm.trans $ congr_arg norm $ e.to_linear_equiv.apply_symm_apply x⟩ | def | linear_isometry_equiv.symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | The inverse `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (x : E₂) : e (e.symm x) = x | e.to_linear_equiv.apply_symm_apply x | lemma | linear_isometry_equiv.apply_symm_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (x : E) : e.symm (e x) = x | e.to_linear_equiv.symm_apply_apply x | lemma | linear_isometry_equiv.symm_apply_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff {x : E} : e x = 0 ↔ x = 0 | e.to_linear_equiv.map_eq_zero_iff | lemma | linear_isometry_equiv.map_eq_zero_iff | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_symm : e.to_linear_equiv.symm = e.symm.to_linear_equiv | rfl | lemma | linear_isometry_equiv.to_linear_equiv_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁]
[ring_hom_inv_pair σ₂₁ σ₁₂] (E E₂ : Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] (h : E ≃ₛₗᵢ[σ₁₂] E₂) : E → E₂ | h | def | linear_isometry_equiv.simps.apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"module",
"ring_hom_inv_pair",
"seminormed_add_comm_group"
] | 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 (σ₁₂ : R →+* R₂) {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁]
[ring_hom_inv_pair σ₂₁ σ₁₂] (E E₂ : Type*) [seminormed_add_comm_group E]
[seminormed_add_comm_group E₂]
[module R E] [module R₂ E₂] (h : E ≃ₛₗᵢ[σ₁₂] E₂) : E₂ → E | h.symm
initialize_simps_projections linear_isometry_equiv
(to_linear_equiv_to_fun → apply, to_linear_equiv_inv_fun → symm_apply) | def | linear_isometry_equiv.simps.symm_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [
"linear_isometry_equiv",
"module",
"ring_hom_inv_pair",
"seminormed_add_comm_group"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (e' : E₂ ≃ₛₗᵢ[σ₂₃] E₃) : E ≃ₛₗᵢ[σ₁₃] E₃ | ⟨e.to_linear_equiv.trans e'.to_linear_equiv, λ x, (e'.norm_map _).trans (e.norm_map _)⟩ | def | linear_isometry_equiv.trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | Composition of `linear_isometry_equiv`s as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_trans (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ | rfl | lemma | linear_isometry_equiv.coe_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_apply (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) (c : E) :
(e₁.trans e₂ : E ≃ₛₗᵢ[σ₁₃] E₃) c = e₂ (e₁ c) | rfl | lemma | linear_isometry_equiv.trans_apply | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_trans (e' : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :
(e.trans e').to_linear_equiv = e.to_linear_equiv.trans e'.to_linear_equiv | rfl | lemma | linear_isometry_equiv.to_linear_equiv_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl : e.trans (refl R₂ E₂) = e | ext $ λ x, rfl | lemma | linear_isometry_equiv.trans_refl | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans : (refl R E).trans e = e | ext $ λ x, rfl | lemma | linear_isometry_equiv.refl_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm : e.trans e.symm = refl R E | ext e.symm_apply_apply | lemma | linear_isometry_equiv.self_trans_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self : e.symm.trans e = refl R₂ E₂ | ext e.apply_symm_apply | lemma | linear_isometry_equiv.symm_trans_self | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_self : e.symm ∘ e = id | funext e.symm_apply_apply | lemma | linear_isometry_equiv.symm_comp_self | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_comp_symm : e ∘ e.symm = id | e.symm.symm_comp_self | lemma | linear_isometry_equiv.self_comp_symm | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :
(e₁.trans e₂).symm = e₂.symm.trans e₁.symm | rfl | lemma | linear_isometry_equiv.symm_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_trans (e₁ : E ≃ₛₗᵢ[σ₁₂] E₂) (e₂ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) :
⇑(e₁.trans e₂).symm = e₁.symm ∘ e₂.symm | rfl | lemma | linear_isometry_equiv.coe_symm_trans | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc (eEE₂ : E ≃ₛₗᵢ[σ₁₂] E₂) (eE₂E₃ : E₂ ≃ₛₗᵢ[σ₂₃] E₃) (eE₃E₄ : E₃ ≃ₛₗᵢ[σ₃₄] E₄) :
eEE₂.trans (eE₂E₃.trans eE₃E₄) = (eEE₂.trans eE₂E₃).trans eE₃E₄ | rfl | lemma | linear_isometry_equiv.trans_assoc | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : E ≃ₗᵢ[R] E) = id | rfl | lemma | linear_isometry_equiv.coe_one | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (e e' : E ≃ₗᵢ[R] E) : ⇑(e * e') = e ∘ e' | rfl | lemma | linear_isometry_equiv.coe_mul | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (e : E ≃ₗᵢ[R] E) : ⇑(e⁻¹) = e.symm | rfl | lemma | linear_isometry_equiv.coe_inv | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : E ≃ₗᵢ[R] E) = refl _ _ | rfl | lemma | linear_isometry_equiv.one_def | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (e e' : E ≃ₗᵢ[R] E) : (e * e' : E ≃ₗᵢ[R] E) = e'.trans e | rfl | lemma | linear_isometry_equiv.mul_def | analysis.normed_space | src/analysis/normed_space/linear_isometry.lean | [
"analysis.normed.group.basic",
"topology.algebra.module.basic",
"linear_algebra.basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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