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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continuous_barycentric_coord (i : ι) : continuous (b.coord i) | (b.coord i).continuous_of_finite_dimensional | lemma | continuous_barycentric_coord | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smooth_barycentric_coord (b : affine_basis ι 𝕜 E) (i : ι) : cont_diff 𝕜 ⊤ (b.coord i) | (⟨b.coord i, continuous_barycentric_coord b i⟩ : E →A[𝕜] 𝕜).cont_diff | lemma | smooth_barycentric_coord | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_basis",
"cont_diff",
"continuous_barycentric_coord"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_basis.interior_convex_hull {ι E : Type*} [finite ι] [normed_add_comm_group E]
[normed_space ℝ E] (b : affine_basis ι ℝ E) :
interior (convex_hull ℝ (range b)) = {x | ∀ i, 0 < b.coord i x} | begin
casesI subsingleton_or_nontrivial ι,
{ -- The zero-dimensional case.
have : range b = univ,
from affine_subspace.eq_univ_of_subsingleton_span_eq_top (subsingleton_range _) b.tot,
simp [this] },
{ -- The positive-dimensional case.
haveI : finite_dimensional ℝ E := b.finite_dimensional,
... | lemma | affine_basis.interior_convex_hull | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_basis",
"affine_subspace.eq_univ_of_subsingleton_span_eq_top",
"continuous_barycentric_coord",
"convex_hull",
"finite",
"finite_dimensional",
"interior",
"interior_Ici",
"interior_Inter",
"is_open_map.preimage_interior_eq_interior_preimage",
"is_open_map_barycentric_coord",
"normed_add... | Given a finite-dimensional normed real vector space, the interior of the convex hull of an
affine basis is the set of points whose barycentric coordinates are strictly positive with respect
to this basis.
TODO Restate this result for affine spaces (instead of vector spaces) once the definition of
convexity is generali... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.exists_between_affine_independent_span_eq_top {s u : set P} (hu : is_open u)
(hsu : s ⊆ u) (hne : s.nonempty) (h : affine_independent ℝ (coe : s → P)) :
∃ t : set P, s ⊆ t ∧ t ⊆ u ∧ affine_independent ℝ (coe : t → P) ∧ affine_span ℝ t = ⊤ | begin
obtain ⟨q, hq⟩ := hne,
obtain ⟨ε, ε0, hεu⟩ := metric.nhds_basis_closed_ball.mem_iff.1 (hu.mem_nhds $ hsu hq),
obtain ⟨t, ht₁, ht₂, ht₃⟩ := exists_subset_affine_independent_affine_span_eq_top h,
let f : P → P := λ y, line_map q y (ε / dist y q),
have hf : ∀ y, f y ∈ u,
{ refine λ y, hεu _,
simp onl... | lemma | is_open.exists_between_affine_independent_span_eq_top | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"abs_div",
"abs_of_nonneg",
"abs_of_pos",
"affine_independent",
"affine_span",
"affine_span_eq_affine_span_line_map_units",
"dist_eq_norm_vsub",
"dist_vadd_left",
"div_mul_comm",
"div_ne_zero",
"div_self_le_one",
"exists_subset_affine_independent_affine_span_eq_top",
"is_open",
"metric.mem... | Given a set `s` of affine-independent points belonging to an open set `u`, we may extend `s` to
an affine basis, all of whose elements belong to `u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.exists_subset_affine_independent_span_eq_top {u : set P} (hu : is_open u)
(hne : u.nonempty) :
∃ s ⊆ u, affine_independent ℝ (coe : s → P) ∧ affine_span ℝ s = ⊤ | begin
rcases hne with ⟨x, hx⟩,
rcases hu.exists_between_affine_independent_span_eq_top (singleton_subset_iff.mpr hx)
(singleton_nonempty _) (affine_independent_of_subsingleton _ _) with ⟨s, -, hsu, hs⟩,
exact ⟨s, hsu, hs⟩
end | lemma | is_open.exists_subset_affine_independent_span_eq_top | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_independent",
"affine_independent_of_subsingleton",
"affine_span",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.affine_span_eq_top {u : set P} (hu : is_open u) (hne : u.nonempty) :
affine_span ℝ u = ⊤ | let ⟨s, hsu, hs, hs'⟩ := hu.exists_subset_affine_independent_span_eq_top hne
in top_unique $ hs' ▸ affine_span_mono _ hsu | lemma | is_open.affine_span_eq_top | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_span",
"affine_span_mono",
"is_open",
"top_unique"
] | The affine span of a nonempty open set is `⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_span_eq_top_of_nonempty_interior {s : set V}
(hs : (interior $ convex_hull ℝ s).nonempty) :
affine_span ℝ s = ⊤ | top_unique $ is_open_interior.affine_span_eq_top hs ▸
(affine_span_mono _ interior_subset).trans_eq (affine_span_convex_hull _) | lemma | affine_span_eq_top_of_nonempty_interior | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_span",
"affine_span_convex_hull",
"affine_span_mono",
"convex_hull",
"interior",
"interior_subset",
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_basis.centroid_mem_interior_convex_hull {ι} [fintype ι] (b : affine_basis ι ℝ V) :
finset.univ.centroid ℝ b ∈ interior (convex_hull ℝ (range b)) | begin
haveI := b.nonempty,
simp only [b.interior_convex_hull, mem_set_of_eq, b.coord_apply_centroid (finset.mem_univ _),
inv_pos, nat.cast_pos, finset.card_pos, finset.univ_nonempty, forall_true_iff]
end | lemma | affine_basis.centroid_mem_interior_convex_hull | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_basis",
"convex_hull",
"finset.card_pos",
"finset.mem_univ",
"finset.univ_nonempty",
"fintype",
"forall_true_iff",
"interior",
"inv_pos",
"nat.cast_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_convex_hull_nonempty_iff_affine_span_eq_top [finite_dimensional ℝ V] {s : set V} :
(interior (convex_hull ℝ s)).nonempty ↔ affine_span ℝ s = ⊤ | begin
refine ⟨affine_span_eq_top_of_nonempty_interior, λ h, _⟩,
obtain ⟨t, hts, b, hb⟩ := affine_basis.exists_affine_subbasis h,
suffices : (interior (convex_hull ℝ (range b))).nonempty,
{ rw [hb, subtype.range_coe_subtype, set_of_mem_eq] at this,
refine this.mono _,
mono* },
lift t to finset V using ... | lemma | interior_convex_hull_nonempty_iff_affine_span_eq_top | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_basis.exists_affine_subbasis",
"affine_span",
"convex_hull",
"finite_dimensional",
"finset",
"interior",
"lift",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.interior_nonempty_iff_affine_span_eq_top [finite_dimensional ℝ V] {s : set V}
(hs : convex ℝ s) : (interior s).nonempty ↔ affine_span ℝ s = ⊤ | by rw [← interior_convex_hull_nonempty_iff_affine_span_eq_top, hs.convex_hull_eq] | lemma | convex.interior_nonempty_iff_affine_span_eq_top | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_span",
"convex",
"finite_dimensional",
"interior",
"interior_convex_hull_nonempty_iff_affine_span_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_isometry extends P →ᵃ[𝕜] P₂ | (norm_map : ∀ x : V, ‖linear x‖ = ‖x‖) | structure | affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | An `𝕜`-affine isometric embedding of one normed add-torsor over a normed `𝕜`-space into
another. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry : V →ₗᵢ[𝕜] V₂ | { norm_map' := f.norm_map,
.. f.linear } | def | affine_isometry.linear_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"linear_isometry"
] | The underlying linear map of an affine isometry is in fact a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_eq_linear_isometry : f.linear = f.linear_isometry.to_linear_map | by { ext, refl } | lemma | affine_isometry.linear_eq_linear_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_affine_map : ⇑f.to_affine_map = f | rfl | lemma | affine_isometry.coe_to_affine_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_map_injective : injective (to_affine_map : (P →ᵃⁱ[𝕜] P₂) → (P →ᵃ[𝕜] P₂)) | | ⟨f, _⟩ ⟨g, _⟩ rfl := rfl | lemma | affine_isometry.to_affine_map_injective | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @injective (P →ᵃⁱ[𝕜] P₂) (P → P₂) coe_fn | affine_map.coe_fn_injective.comp to_affine_map_injective | lemma | affine_isometry.coe_fn_injective | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : P →ᵃⁱ[𝕜] P₂} (h : ∀ x, f x = g x) : f = g | coe_fn_injective $ funext h | lemma | affine_isometry.ext | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry : V →ᵃⁱ[𝕜] V₂ | { norm_map := f.norm_map,
.. f.to_linear_map.to_affine_map } | def | linear_isometry.to_affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Reinterpret a linear isometry as an affine isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_affine_isometry : ⇑(f.to_affine_isometry : V →ᵃⁱ[𝕜] V₂) = f | rfl | lemma | linear_isometry.coe_to_affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_linear_isometry : f.to_affine_isometry.linear_isometry = f | by { ext, refl } | lemma | linear_isometry.to_affine_isometry_linear_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_to_affine_map :
f.to_affine_isometry.to_affine_map = f.to_linear_map.to_affine_map | rfl | lemma | linear_isometry.to_affine_isometry_to_affine_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_vadd (p : P) (v : V) : f (v +ᵥ p) = f.linear_isometry v +ᵥ f p | f.to_affine_map.map_vadd p v | lemma | affine_isometry.map_vadd | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_vsub (p1 p2 : P) : f.linear_isometry (p1 -ᵥ p2) = f p1 -ᵥ f p2 | f.to_affine_map.linear_map_vsub p1 p2 | lemma | affine_isometry.map_vsub | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_map (x y : P) : dist (f x) (f y) = dist x y | by rw [dist_eq_norm_vsub V₂, dist_eq_norm_vsub V, ← map_vsub, f.linear_isometry.norm_map] | lemma | affine_isometry.dist_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"dist_eq_norm_vsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_map (x y : P) : nndist (f x) (f y) = nndist x y | by simp [nndist_dist] | lemma | affine_isometry.nndist_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"nndist_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_map (x y : P) : edist (f x) (f y) = edist x y | by simp [edist_dist] | lemma | affine_isometry.edist_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"edist_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry : isometry f | f.edist_map | lemma | affine_isometry.isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective : injective f₁ | f₁.isometry.injective | lemma | affine_isometry.injective | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_iff {x y : P₁} : f₁ x = f₁ y ↔ x = y | f₁.injective.eq_iff | lemma | affine_isometry.map_eq_iff | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne {x y : P₁} (h : x ≠ y) : f₁ x ≠ f₁ y | f₁.injective.ne h | lemma | affine_isometry.map_ne | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz : lipschitz_with 1 f | f.isometry.lipschitz | lemma | affine_isometry.lipschitz | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz : antilipschitz_with 1 f | f.isometry.antilipschitz | lemma | affine_isometry.antilipschitz | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous f | f.isometry.continuous | lemma | affine_isometry.continuous | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image (s : set P) : emetric.diam (f '' s) = emetric.diam s | f.isometry.ediam_image s | lemma | affine_isometry.ediam_image | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_range : emetric.diam (range f) = emetric.diam (univ : set P) | f.isometry.ediam_range | lemma | affine_isometry.ediam_range | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_image (s : set P) : metric.diam (f '' s) = metric.diam s | f.isometry.diam_image s | lemma | affine_isometry.diam_image | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_range : metric.diam (range f) = metric.diam (univ : set P) | f.isometry.diam_range | lemma | affine_isometry.diam_range | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {α : Type*} [topological_space α] {g : α → P} :
continuous (f ∘ g) ↔ continuous g | f.isometry.comp_continuous_iff | lemma | affine_isometry.comp_continuous_iff | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : P →ᵃⁱ[𝕜] P | ⟨affine_map.id 𝕜 P, λ x, rfl⟩ | def | affine_isometry.id | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | The identity affine isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(id : P →ᵃⁱ[𝕜] P) = _root_.id | rfl | lemma | affine_isometry.coe_id | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : P) : (affine_isometry.id : P →ᵃⁱ[𝕜] P) x = x | rfl | lemma | affine_isometry.id_apply | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_isometry.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_to_affine_map : (id.to_affine_map : P →ᵃ[𝕜] P) = affine_map.id 𝕜 P | rfl | lemma | affine_isometry.id_to_affine_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : P →ᵃⁱ[𝕜] P₃ | ⟨g.to_affine_map.comp f.to_affine_map, λ x, (g.norm_map _).trans (f.norm_map _)⟩ | def | affine_isometry.comp | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Composition of affine isometries. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) :
⇑(g.comp f) = g ∘ f | rfl | lemma | affine_isometry.coe_comp | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp : (id : P₂ →ᵃⁱ[𝕜] P₂).comp f = f | ext $ λ x, rfl | lemma | affine_isometry.id_comp | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id : f.comp id = f | ext $ λ x, rfl | lemma | affine_isometry.comp_id | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | affine_isometry.comp_assoc | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : P →ᵃⁱ[𝕜] P) = _root_.id | rfl | lemma | affine_isometry.coe_one | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : P →ᵃⁱ[𝕜] P) : ⇑(f * g) = f ∘ g | rfl | lemma | affine_isometry.coe_mul | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtypeₐᵢ (s : affine_subspace 𝕜 P) [nonempty s] : s →ᵃⁱ[𝕜] P | { norm_map := s.direction.subtypeₗᵢ.norm_map,
.. s.subtype } | def | affine_subspace.subtypeₐᵢ | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_subspace"
] | `affine_subspace.subtype` as an `affine_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtypeₐᵢ_linear (s : affine_subspace 𝕜 P) [nonempty s] :
s.subtypeₐᵢ.linear = s.direction.subtype | rfl | lemma | affine_subspace.subtypeₐᵢ_linear | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtypeₐᵢ_linear_isometry (s : affine_subspace 𝕜 P) [nonempty s] :
s.subtypeₐᵢ.linear_isometry = s.direction.subtypeₗᵢ | rfl | lemma | affine_subspace.subtypeₐᵢ_linear_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subtypeₐᵢ (s : affine_subspace 𝕜 P) [nonempty s] : ⇑s.subtypeₐᵢ = s.subtype | rfl | lemma | affine_subspace.coe_subtypeₐᵢ | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtypeₐᵢ_to_affine_map (s : affine_subspace 𝕜 P) [nonempty s] :
s.subtypeₐᵢ.to_affine_map = s.subtype | rfl | lemma | affine_subspace.subtypeₐᵢ_to_affine_map | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_isometry_equiv extends P ≃ᵃ[𝕜] P₂ | (norm_map : ∀ x, ‖linear x‖ = ‖x‖) | structure | affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | A affine isometric equivalence between two normed vector spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv : V ≃ₗᵢ[𝕜] V₂ | { norm_map' := e.norm_map,
.. e.linear } | def | affine_isometry_equiv.linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"linear_isometry_equiv"
] | The underlying linear equiv of an affine isometry equiv is in fact a linear isometry equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_eq_linear_isometry : e.linear = e.linear_isometry_equiv.to_linear_equiv | by { ext, refl } | lemma | affine_isometry_equiv.linear_eq_linear_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (e : P ≃ᵃ[𝕜] P₂) (he : ∀ x, ‖e.linear x‖ = ‖x‖) :
⇑(mk e he) = e | rfl | lemma | affine_isometry_equiv.coe_mk | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_affine_equiv (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.to_affine_equiv = e | rfl | lemma | affine_isometry_equiv.coe_to_affine_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_equiv_injective : injective (to_affine_equiv : (P ≃ᵃⁱ[𝕜] P₂) → (P ≃ᵃ[𝕜] P₂)) | | ⟨e, _⟩ ⟨_, _⟩ rfl := rfl | lemma | affine_isometry_equiv.to_affine_equiv_injective | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {e e' : P ≃ᵃⁱ[𝕜] P₂} (h : ∀ x, e x = e' x) : e = e' | to_affine_equiv_injective $ affine_equiv.ext h | lemma | affine_isometry_equiv.ext | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry : P →ᵃⁱ[𝕜] P₂ | ⟨e.1.to_affine_map, e.2⟩ | def | affine_isometry_equiv.to_affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Reinterpret a `affine_isometry_equiv` as a `affine_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_affine_isometry : ⇑e.to_affine_isometry = e | rfl | lemma | affine_isometry_equiv.coe_to_affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ p' : P₁, e p' = e' (p' -ᵥ p) +ᵥ e p) :
P₁ ≃ᵃⁱ[𝕜] P₂ | { norm_map := e'.norm_map,
.. affine_equiv.mk' e e'.to_linear_equiv p h } | def | affine_isometry_equiv.mk' | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_equiv.mk'",
"mk'"
] | Construct an affine isometry equivalence by verifying the relation between the map and its
linear part at one base point. Namely, this function takes a map `e : P₁ → P₂`, a linear isometry
equivalence `e' : V₁ ≃ᵢₗ[k] V₂`, and a point `p` such that for any other point `p'` we have
`e p' = e' (p' -ᵥ p) +ᵥ e p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p h) : ⇑(mk' e e' p h) = e | rfl | lemma | affine_isometry_equiv.coe_mk' | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_equiv_mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p h) :
(mk' e e' p h).linear_isometry_equiv = e' | by { ext, refl } | lemma | affine_isometry_equiv.linear_isometry_equiv_mk' | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"linear_isometry_equiv",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv : V ≃ᵃⁱ[𝕜] V₂ | { norm_map := e.norm_map,
.. e.to_linear_equiv.to_affine_equiv } | def | linear_isometry_equiv.to_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Reinterpret a linear isometry equiv as an affine isometry equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_affine_isometry_equiv : ⇑(e.to_affine_isometry_equiv : V ≃ᵃⁱ[𝕜] V₂) = e | rfl | lemma | linear_isometry_equiv.coe_to_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv_linear_isometry_equiv :
e.to_affine_isometry_equiv.linear_isometry_equiv = e | by { ext, refl } | lemma | linear_isometry_equiv.to_affine_isometry_equiv_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv_to_affine_equiv :
e.to_affine_isometry_equiv.to_affine_equiv = e.to_linear_equiv.to_affine_equiv | rfl | lemma | linear_isometry_equiv.to_affine_isometry_equiv_to_affine_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_isometry_equiv_to_affine_isometry :
e.to_affine_isometry_equiv.to_affine_isometry = e.to_linear_isometry.to_affine_isometry | rfl | lemma | linear_isometry_equiv.to_affine_isometry_equiv_to_affine_isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry : isometry e | e.to_affine_isometry.isometry | lemma | affine_isometry_equiv.isometry | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_isometry_equiv : P ≃ᵢ P₂ | ⟨e.to_affine_equiv.to_equiv, e.isometry⟩ | def | affine_isometry_equiv.to_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Reinterpret a `affine_isometry_equiv` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_isometry_equiv : ⇑e.to_isometry_equiv = e | rfl | lemma | affine_isometry_equiv.coe_to_isometry_equiv | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_univ (e : P ≃ᵃⁱ[𝕜] P₂) : set.range e = set.univ | by { rw ← coe_to_isometry_equiv, exact isometry_equiv.range_eq_univ _, } | lemma | affine_isometry_equiv.range_eq_univ | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"isometry_equiv.range_eq_univ",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph : P ≃ₜ P₂ | e.to_isometry_equiv.to_homeomorph | def | affine_isometry_equiv.to_homeomorph | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Reinterpret a `affine_isometry_equiv` as an `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_homeomorph : ⇑e.to_homeomorph = e | rfl | lemma | affine_isometry_equiv.coe_to_homeomorph | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous e | e.isometry.continuous | lemma | affine_isometry_equiv.continuous | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at {x} : continuous_at e x | e.continuous.continuous_at | lemma | affine_isometry_equiv.continuous_at | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on {s} : continuous_on e s | e.continuous.continuous_on | lemma | affine_isometry_equiv.continuous_on | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at {s x} : continuous_within_at e s x | e.continuous.continuous_within_at | lemma | affine_isometry_equiv.continuous_within_at | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : P ≃ᵃⁱ[𝕜] P | ⟨affine_equiv.refl 𝕜 P, λ x, rfl⟩ | def | affine_isometry_equiv.refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Identity map as a `affine_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_refl : ⇑(refl 𝕜 P) = id | rfl | lemma | affine_isometry_equiv.coe_refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_equiv_refl : (refl 𝕜 P).to_affine_equiv = affine_equiv.refl 𝕜 P | rfl | lemma | affine_isometry_equiv.to_affine_equiv_refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"affine_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_isometry_equiv_refl : (refl 𝕜 P).to_isometry_equiv = isometry_equiv.refl P | rfl | lemma | affine_isometry_equiv.to_isometry_equiv_refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"isometry_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph_refl : (refl 𝕜 P).to_homeomorph = homeomorph.refl P | rfl | lemma | affine_isometry_equiv.to_homeomorph_refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [
"homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm : P₂ ≃ᵃⁱ[𝕜] P | { norm_map := e.linear_isometry_equiv.symm.norm_map,
.. e.to_affine_equiv.symm } | def | affine_isometry_equiv.symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | The inverse `affine_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (x : P₂) : e (e.symm x) = x | e.to_affine_equiv.apply_symm_apply x | lemma | affine_isometry_equiv.apply_symm_apply | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (x : P) : e.symm (e x) = x | e.to_affine_equiv.symm_apply_apply x | lemma | affine_isometry_equiv.symm_apply_apply | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm : e.symm.symm = e | ext $ λ x, rfl | lemma | affine_isometry_equiv.symm_symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_equiv_symm : e.to_affine_equiv.symm = e.symm.to_affine_equiv | rfl | lemma | affine_isometry_equiv.to_affine_equiv_symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_isometry_equiv_symm : e.to_isometry_equiv.symm = e.symm.to_isometry_equiv | rfl | lemma | affine_isometry_equiv.to_isometry_equiv_symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph_symm : e.to_homeomorph.symm = e.symm.to_homeomorph | rfl | lemma | affine_isometry_equiv.to_homeomorph_symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e' : P₂ ≃ᵃⁱ[𝕜] P₃) : P ≃ᵃⁱ[𝕜] P₃ | ⟨e.to_affine_equiv.trans e'.to_affine_equiv, λ x, (e'.norm_map _).trans (e.norm_map _)⟩ | def | affine_isometry_equiv.trans | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | Composition of `affine_isometry_equiv`s as a `affine_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_trans (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ | rfl | lemma | affine_isometry_equiv.coe_trans | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl : e.trans (refl 𝕜 P₂) = e | ext $ λ x, rfl | lemma | affine_isometry_equiv.trans_refl | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans : (refl 𝕜 P).trans e = e | ext $ λ x, rfl | lemma | affine_isometry_equiv.refl_trans | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm : e.trans e.symm = refl 𝕜 P | ext e.symm_apply_apply | lemma | affine_isometry_equiv.self_trans_symm | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self : e.symm.trans e = refl 𝕜 P₂ | ext e.apply_symm_apply | lemma | affine_isometry_equiv.symm_trans_self | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_trans (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) :
⇑(e₁.trans e₂).symm = e₁.symm ∘ e₂.symm | rfl | lemma | affine_isometry_equiv.coe_symm_trans | analysis.normed_space | src/analysis/normed_space/affine_isometry.lean | [
"analysis.normed_space.linear_isometry",
"analysis.normed.group.add_torsor",
"analysis.normed_space.basic",
"linear_algebra.affine_space.restrict",
"algebra.char_p.invertible"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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