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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|---|---|---|---|---|---|---|---|---|---|---|
trans_assoc (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄) :
ePP₂.trans (eP₂G.trans eGG') = (ePP₂.trans eP₂G).trans eGG' | rfl | lemma | affine_isometry_equiv.trans_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) = id | rfl | lemma | affine_isometry_equiv.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 (e e' : P ≃ᵃⁱ[𝕜] P) : ⇑(e * e') = e ∘ e' | rfl | lemma | affine_isometry_equiv.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 | |
coe_inv (e : P ≃ᵃⁱ[𝕜] P) : ⇑(e⁻¹) = e.symm | rfl | lemma | affine_isometry_equiv.coe_inv | 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) : e (v +ᵥ p) = e.linear_isometry_equiv v +ᵥ e p | e.to_affine_isometry.map_vadd p v | lemma | affine_isometry_equiv.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) : e.linear_isometry_equiv (p1 -ᵥ p2) = e p1 -ᵥ e p2 | e.to_affine_isometry.map_vsub p1 p2 | lemma | affine_isometry_equiv.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 (e x) (e y) = dist x y | e.to_affine_isometry.dist_map x y | lemma | affine_isometry_equiv.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_map (x y : P) : edist (e x) (e y) = edist x y | e.to_affine_isometry.edist_map x y | lemma | affine_isometry_equiv.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective : bijective e | e.1.bijective | lemma | affine_isometry_equiv.bijective | 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 | |
injective : injective e | e.1.injective | lemma | affine_isometry_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 | |
surjective : surjective e | e.1.surjective | lemma | affine_isometry_equiv.surjective | 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 : P} : e x = e y ↔ x = y | e.injective.eq_iff | lemma | affine_isometry_equiv.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 : P} (h : x ≠ y) : e x ≠ e y | e.injective.ne h | lemma | affine_isometry_equiv.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 e | e.isometry.lipschitz | lemma | affine_isometry_equiv.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 e | e.isometry.antilipschitz | lemma | affine_isometry_equiv.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 | |
ediam_image (s : set P) : emetric.diam (e '' s) = emetric.diam s | e.isometry.ediam_image s | lemma | affine_isometry_equiv.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 | |
diam_image (s : set P) : metric.diam (e '' s) = metric.diam s | e.isometry.diam_image s | lemma | affine_isometry_equiv.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 | |
comp_continuous_on_iff {f : α → P} {s : set α} :
continuous_on (e ∘ f) s ↔ continuous_on f s | e.isometry.comp_continuous_on_iff | lemma | affine_isometry_equiv.comp_continuous_on_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_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {f : α → P} :
continuous (e ∘ f) ↔ continuous f | e.isometry.comp_continuous_iff | lemma | affine_isometry_equiv.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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_const (p : P) : V ≃ᵃⁱ[𝕜] P | { norm_map := λ x, rfl,
.. affine_equiv.vadd_const 𝕜 p } | def | affine_isometry_equiv.vadd_const | 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.vadd_const"
] | The map `v ↦ v +ᵥ p` as an affine isometric equivalence between `V` and `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_vadd_const (p : P) : ⇑(vadd_const 𝕜 p) = λ v, v +ᵥ p | rfl | lemma | affine_isometry_equiv.coe_vadd_const | 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_vadd_const_symm (p : P) : ⇑(vadd_const 𝕜 p).symm = λ p', p' -ᵥ p | rfl | lemma | affine_isometry_equiv.coe_vadd_const_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 | |
vadd_const_to_affine_equiv (p : P) :
(vadd_const 𝕜 p).to_affine_equiv = affine_equiv.vadd_const 𝕜 p | rfl | lemma | affine_isometry_equiv.vadd_const_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"
] | [
"affine_equiv.vadd_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_vsub (p : P) : P ≃ᵃⁱ[𝕜] V | { norm_map := norm_neg,
.. affine_equiv.const_vsub 𝕜 p } | def | affine_isometry_equiv.const_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"
] | [
"affine_equiv.const_vsub"
] | `p' ↦ p -ᵥ p'` as an affine isometric equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const_vsub (p : P) : ⇑(const_vsub 𝕜 p) = (-ᵥ) p | rfl | lemma | affine_isometry_equiv.coe_const_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 | |
symm_const_vsub (p : P) :
(const_vsub 𝕜 p).symm
= (linear_isometry_equiv.neg 𝕜).to_affine_isometry_equiv.trans (vadd_const 𝕜 p) | by { ext, refl } | lemma | affine_isometry_equiv.symm_const_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"
] | [
"linear_isometry_equiv.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_vadd (v : V) : P ≃ᵃⁱ[𝕜] P | { norm_map := λ x, rfl,
.. affine_equiv.const_vadd 𝕜 P v } | def | affine_isometry_equiv.const_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"
] | [
"affine_equiv.const_vadd"
] | Translation by `v` (that is, the map `p ↦ v +ᵥ p`) as an affine isometric automorphism of `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const_vadd (v : V) : ⇑(const_vadd 𝕜 P v : P ≃ᵃⁱ[𝕜] P) = (+ᵥ) v | rfl | lemma | affine_isometry_equiv.coe_const_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 | |
const_vadd_zero : const_vadd 𝕜 P (0:V) = refl 𝕜 P | ext $ zero_vadd V | lemma | affine_isometry_equiv.const_vadd_zero | 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 | |
vadd_vsub {f : P → P₂} (hf : isometry f) {p : P} {g : V → V₂}
(hg : ∀ v, g v = f (v +ᵥ p) -ᵥ f p) : isometry g | begin
convert (vadd_const 𝕜 (f p)).symm.isometry.comp (hf.comp (vadd_const 𝕜 p).isometry),
exact funext hg
end | lemma | affine_isometry_equiv.vadd_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"
] | [
"isometry",
"vadd_vsub"
] | The map `g` from `V` to `V₂` corresponding to a map `f` from `P` to `P₂`, at a base point `p`,
is an isometry if `f` is one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection (x : P) : P ≃ᵃⁱ[𝕜] P | (const_vsub 𝕜 x).trans (vadd_const 𝕜 x) | def | affine_isometry_equiv.point_reflection | 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"
] | [] | Point reflection in `x` as an affine isometric automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_apply (x y : P) : (point_reflection 𝕜 x) y = x -ᵥ y +ᵥ x | rfl | lemma | affine_isometry_equiv.point_reflection_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 | |
point_reflection_to_affine_equiv (x : P) :
(point_reflection 𝕜 x).to_affine_equiv = affine_equiv.point_reflection 𝕜 x | rfl | lemma | affine_isometry_equiv.point_reflection_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"
] | [
"affine_equiv.point_reflection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_self (x : P) : point_reflection 𝕜 x x = x | affine_equiv.point_reflection_self 𝕜 x | lemma | affine_isometry_equiv.point_reflection_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"
] | [
"affine_equiv.point_reflection_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_involutive (x : P) : function.involutive (point_reflection 𝕜 x) | equiv.point_reflection_involutive x | lemma | affine_isometry_equiv.point_reflection_involutive | 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"
] | [
"equiv.point_reflection_involutive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_symm (x : P) : (point_reflection 𝕜 x).symm = point_reflection 𝕜 x | to_affine_equiv_injective $ affine_equiv.point_reflection_symm 𝕜 x | lemma | affine_isometry_equiv.point_reflection_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"
] | [
"affine_equiv.point_reflection_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_point_reflection_fixed (x y : P) :
dist (point_reflection 𝕜 x y) x = dist y x | by rw [← (point_reflection 𝕜 x).dist_map y x, point_reflection_self] | lemma | affine_isometry_equiv.dist_point_reflection_fixed | 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_point_reflection_self' (x y : P) :
dist (point_reflection 𝕜 x y) y = ‖bit0 (x -ᵥ y)‖ | by rw [point_reflection_apply, dist_eq_norm_vsub V, vadd_vsub_assoc, bit0] | lemma | affine_isometry_equiv.dist_point_reflection_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"
] | [
"dist_eq_norm_vsub",
"vadd_vsub_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_point_reflection_self (x y : P) :
dist (point_reflection 𝕜 x y) y = ‖(2:𝕜)‖ * dist x y | by rw [dist_point_reflection_self', ← two_smul' 𝕜 (x -ᵥ y), norm_smul, ← dist_eq_norm_vsub V] | lemma | affine_isometry_equiv.dist_point_reflection_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"
] | [
"dist_eq_norm_vsub",
"norm_smul",
"two_smul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_fixed_iff [invertible (2:𝕜)] {x y : P} :
point_reflection 𝕜 x y = y ↔ y = x | affine_equiv.point_reflection_fixed_iff_of_module 𝕜 | lemma | affine_isometry_equiv.point_reflection_fixed_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"
] | [
"affine_equiv.point_reflection_fixed_iff_of_module",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_point_reflection_self_real (x y : P) :
dist (point_reflection ℝ x y) y = 2 * dist x y | by { rw [dist_point_reflection_self, real.norm_two] } | lemma | affine_isometry_equiv.dist_point_reflection_self_real | 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"
] | [
"real.norm_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_midpoint_left (x y : P) :
point_reflection ℝ (midpoint ℝ x y) x = y | affine_equiv.point_reflection_midpoint_left x y | lemma | affine_isometry_equiv.point_reflection_midpoint_left | 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.point_reflection_midpoint_left",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_reflection_midpoint_right (x y : P) :
point_reflection ℝ (midpoint ℝ x y) y = x | affine_equiv.point_reflection_midpoint_right x y | lemma | affine_isometry_equiv.point_reflection_midpoint_right | 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.point_reflection_midpoint_right",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_map.continuous_linear_iff {f : P →ᵃ[𝕜] P₂} :
continuous f.linear ↔ continuous f | begin
inhabit P,
have : (f.linear : V → V₂) =
(affine_isometry_equiv.vadd_const 𝕜 $ f default).to_homeomorph.symm ∘ f ∘
(affine_isometry_equiv.vadd_const 𝕜 default).to_homeomorph,
{ ext v, simp },
rw this,
simp only [homeomorph.comp_continuous_iff, homeomorph.comp_continuous_iff'],
end | lemma | affine_map.continuous_linear_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"
] | [
"affine_isometry_equiv.vadd_const",
"continuous",
"homeomorph.comp_continuous_iff",
"homeomorph.comp_continuous_iff'"
] | If `f` is an affine map, then its linear part is continuous iff `f` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_map.is_open_map_linear_iff {f : P →ᵃ[𝕜] P₂} :
is_open_map f.linear ↔ is_open_map f | begin
inhabit P,
have : (f.linear : V → V₂) =
(affine_isometry_equiv.vadd_const 𝕜 $ f default).to_homeomorph.symm ∘ f ∘
(affine_isometry_equiv.vadd_const 𝕜 default).to_homeomorph,
{ ext v, simp },
rw this,
simp only [homeomorph.comp_is_open_map_iff, homeomorph.comp_is_open_map_iff'],
end | lemma | affine_map.is_open_map_linear_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"
] | [
"affine_isometry_equiv.vadd_const",
"homeomorph.comp_is_open_map_iff",
"homeomorph.comp_is_open_map_iff'",
"is_open_map"
] | If `f` is an affine map, then its linear part is an open map iff `f` is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_map_of_injective (E: affine_subspace 𝕜 P₁) [nonempty E]
(φ : P₁ →ᵃ[𝕜] P₂) (hφ : function.injective φ) : E ≃ᵃ[𝕜] E.map φ | { linear :=
(E.direction.equiv_map_of_injective φ.linear (φ.linear_injective_iff.mpr hφ)).trans
(linear_equiv.of_eq _ _ (affine_subspace.map_direction _ _).symm),
map_vadd' := λ p v, subtype.ext $ φ.map_vadd p v,
.. equiv.set.image _ (E : set P₁) hφ } | def | affine_subspace.equiv_map_of_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"
] | [
"affine_subspace",
"affine_subspace.map_direction",
"equiv.set.image",
"linear_equiv.of_eq",
"subtype.ext"
] | An affine subspace is isomorphic to its image under an injective affine map.
This is the affine version of `submodule.equiv_map_of_injective`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_equiv_map
(φ : P₁ →ᵃⁱ[𝕜] P₂) (E : affine_subspace 𝕜 P₁) [nonempty E] : E ≃ᵃⁱ[𝕜] E.map φ.to_affine_map | ⟨E.equiv_map_of_injective φ.to_affine_map φ.injective, (λ _, φ.norm_map _)⟩ | def | affine_subspace.isometry_equiv_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"
] | Restricts an affine isometry to an affine isometry equivalence between a nonempty affine
subspace `E` and its image.
This is an isometry version of `affine_subspace.equiv_map`, having a stronger premise and a stronger
conclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_equiv_map.apply_symm_apply
{E : affine_subspace 𝕜 P₁} [nonempty E]
{φ : P₁ →ᵃⁱ[𝕜] P₂} (x : E.map φ.to_affine_map) :
φ ((E.isometry_equiv_map φ).symm x) = x | congr_arg coe $ (E.isometry_equiv_map φ).apply_symm_apply _ | lemma | affine_subspace.isometry_equiv_map.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"
] | [
"affine_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_equiv_map.coe_apply
(φ : P₁ →ᵃⁱ[𝕜] P₂) (E : affine_subspace 𝕜 P₁) [nonempty E] (g: E) :
↑(E.isometry_equiv_map φ g) = φ g | rfl | lemma | affine_subspace.isometry_equiv_map.coe_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_subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_equiv_map.to_affine_map_eq
(φ : P₁ →ᵃⁱ[𝕜] P₂) (E : affine_subspace 𝕜 P₁) [nonempty E] :
(E.isometry_equiv_map φ).to_affine_map = E.equiv_map_of_injective φ.to_affine_map φ.injective | rfl | lemma | affine_subspace.isometry_equiv_map.to_affine_map_eq | 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 | |
norm_le_norm_one (φ : character_space 𝕜 A) :
‖to_normed_dual (φ : weak_dual 𝕜 A)‖ ≤ ‖(1 : A)‖ | continuous_linear_map.op_norm_le_bound _ (norm_nonneg (1 : A)) $
λ a, mul_comm (‖a‖) (‖(1 : A)‖) ▸ spectrum.norm_le_norm_mul_of_mem (apply_mem_spectrum φ a) | lemma | weak_dual.character_space.norm_le_norm_one | analysis.normed_space | src/analysis/normed_space/algebra.lean | [
"topology.algebra.module.character_space",
"analysis.normed_space.weak_dual",
"analysis.normed_space.spectrum"
] | [
"continuous_linear_map.op_norm_le_bound",
"mul_comm",
"spectrum.norm_le_norm_mul_of_mem",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_closed_ball_ball : mul_action (closed_ball (0 : 𝕜) 1) (ball (0 : E) r) | { smul := λ c x, ⟨(c : 𝕜) • x, mem_ball_zero_iff.2 $
by simpa only [norm_smul, one_mul]
using mul_lt_mul' (mem_closed_ball_zero_iff.1 c.2) (mem_ball_zero_iff.1 x.2)
(norm_nonneg _) one_pos⟩,
one_smul := λ x, subtype.ext $ one_smul 𝕜 _,
mul_smul := λ c₁ c₂ x, subtype.ext $ mul_smul _ _ _ } | instance | mul_action_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"mul_action",
"mul_lt_mul'",
"norm_smul",
"one_mul",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_closed_ball_ball :
has_continuous_smul (closed_ball (0 : 𝕜) 1) (ball (0 : E) r) | ⟨(continuous_subtype_val.fst'.smul continuous_subtype_val.snd').subtype_mk _⟩ | instance | has_continuous_smul_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_closed_ball_closed_ball :
mul_action (closed_ball (0 : 𝕜) 1) (closed_ball (0 : E) r) | { smul := λ c x, ⟨(c : 𝕜) • x, mem_closed_ball_zero_iff.2 $
by simpa only [norm_smul, one_mul]
using mul_le_mul (mem_closed_ball_zero_iff.1 c.2) (mem_closed_ball_zero_iff.1 x.2)
(norm_nonneg _) zero_le_one⟩,
one_smul := λ x, subtype.ext $ one_smul 𝕜 _,
mul_smul := λ c₁ c₂ x, subtype.ext $ mul_sm... | instance | mul_action_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"mul_action",
"mul_le_mul",
"norm_smul",
"one_mul",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_closed_ball_closed_ball :
has_continuous_smul (closed_ball (0 : 𝕜) 1) (closed_ball (0 : E) r) | ⟨(continuous_subtype_val.fst'.smul continuous_subtype_val.snd').subtype_mk _⟩ | instance | has_continuous_smul_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_sphere_ball : mul_action (sphere (0 : 𝕜) 1) (ball (0 : E) r) | { smul := λ c x, inclusion sphere_subset_closed_ball c • x,
one_smul := λ x, subtype.ext $ one_smul _ _,
mul_smul := λ c₁ c₂ x, subtype.ext $ mul_smul _ _ _ } | instance | mul_action_sphere_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"mul_action",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_sphere_ball :
has_continuous_smul (sphere (0 : 𝕜) 1) (ball (0 : E) r) | ⟨(continuous_subtype_val.fst'.smul continuous_subtype_val.snd').subtype_mk _⟩ | instance | has_continuous_smul_sphere_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_sphere_closed_ball : mul_action (sphere (0 : 𝕜) 1) (closed_ball (0 : E) r) | { smul := λ c x, inclusion sphere_subset_closed_ball c • x,
one_smul := λ x, subtype.ext $ one_smul _ _,
mul_smul := λ c₁ c₂ x, subtype.ext $ mul_smul _ _ _ } | instance | mul_action_sphere_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"mul_action",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_sphere_closed_ball :
has_continuous_smul (sphere (0 : 𝕜) 1) (closed_ball (0 : E) r) | ⟨(continuous_subtype_val.fst'.smul continuous_subtype_val.snd').subtype_mk _⟩ | instance | has_continuous_smul_sphere_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_sphere_sphere : mul_action (sphere (0 : 𝕜) 1) (sphere (0 : E) r) | { smul := λ c x, ⟨(c : 𝕜) • x, mem_sphere_zero_iff_norm.2 $
by rw [norm_smul, mem_sphere_zero_iff_norm.1 c.coe_prop, mem_sphere_zero_iff_norm.1 x.coe_prop,
one_mul]⟩,
one_smul := λ x, subtype.ext $ one_smul _ _,
mul_smul := λ c₁ c₂ x, subtype.ext $ mul_smul _ _ _ } | instance | mul_action_sphere_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"mul_action",
"norm_smul",
"one_mul",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_sphere_sphere :
has_continuous_smul (sphere (0 : 𝕜) 1) (sphere (0 : E) r) | ⟨(continuous_subtype_val.fst'.smul continuous_subtype_val.snd').subtype_mk _⟩ | instance | has_continuous_smul_sphere_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_closed_ball_closed_ball_closed_ball :
is_scalar_tower (closed_ball (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_closed_ball_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_closed_ball_closed_ball_ball :
is_scalar_tower (closed_ball (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_closed_ball_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_closed_ball_closed_ball :
is_scalar_tower (sphere (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_sphere_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_closed_ball_ball :
is_scalar_tower (sphere (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_sphere_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_sphere_closed_ball :
is_scalar_tower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_sphere_sphere_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_sphere_ball :
is_scalar_tower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_sphere_sphere_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_sphere_sphere :
is_scalar_tower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (sphere (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | is_scalar_tower_sphere_sphere_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_sphere_ball_ball :
is_scalar_tower (sphere (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : 𝕜')⟩ | instance | is_scalar_tower_sphere_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_closed_ball_ball_ball :
is_scalar_tower (closed_ball (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) | ⟨λ a b c, subtype.ext $ smul_assoc (a : 𝕜) (b : 𝕜') (c : 𝕜')⟩ | instance | is_scalar_tower_closed_ball_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"is_scalar_tower",
"smul_assoc",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_closed_ball_closed_ball_closed_ball :
smul_comm_class (closed_ball (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_closed_ball_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_closed_ball_closed_ball_ball :
smul_comm_class (closed_ball (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_closed_ball_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_closed_ball_closed_ball :
smul_comm_class (sphere (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_sphere_closed_ball_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_closed_ball_ball :
smul_comm_class (sphere (0 : 𝕜) 1) (closed_ball (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_sphere_closed_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_ball_ball [normed_algebra 𝕜 𝕜'] :
smul_comm_class (sphere (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : 𝕜')⟩ | instance | smul_comm_class_sphere_ball_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"normed_algebra",
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_sphere_closed_ball :
smul_comm_class (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (closed_ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_sphere_sphere_closed_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_sphere_ball :
smul_comm_class (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (ball (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_sphere_sphere_ball | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_sphere_sphere_sphere :
smul_comm_class (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (sphere (0 : E) r) | ⟨λ a b c, subtype.ext $ smul_comm (a : 𝕜) (b : 𝕜') (c : E)⟩ | instance | smul_comm_class_sphere_sphere_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0:E) r) : x ≠ - x | λ h, ne_zero_of_mem_sphere hr x ((self_eq_neg 𝕜 _).mp (by { conv_lhs {rw h}, simp })) | lemma | ne_neg_of_mem_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"self_eq_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_neg_of_mem_unit_sphere (x : sphere (0:E) 1) : x ≠ - x | ne_neg_of_mem_sphere 𝕜 one_ne_zero x | lemma | ne_neg_of_mem_unit_sphere | analysis.normed_space | src/analysis/normed_space/ball_action.lean | [
"analysis.normed.field.unit_ball",
"analysis.normed_space.basic"
] | [
"ne_neg_of_mem_sphere",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonlinear_right_inverse | (to_fun : F → E)
(nnnorm : ℝ≥0)
(bound' : ∀ y, ‖to_fun y‖ ≤ nnnorm * ‖y‖)
(right_inv' : ∀ y, f (to_fun y) = y) | structure | continuous_linear_map.nonlinear_right_inverse | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"bound'"
] | A (possibly nonlinear) right inverse to a continuous linear map, which doesn't have to be
linear itself but which satisfies a bound `‖inverse x‖ ≤ C * ‖x‖`. A surjective continuous linear
map doesn't always have a continuous linear right inverse, but it always has a nonlinear inverse
in this sense, by Banach's open map... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonlinear_right_inverse.right_inv {f : E →L[𝕜] F} (fsymm : nonlinear_right_inverse f)
(y : F) : f (fsymm y) = y | fsymm.right_inv' y | lemma | continuous_linear_map.nonlinear_right_inverse.right_inv | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonlinear_right_inverse.bound {f : E →L[𝕜] F} (fsymm : nonlinear_right_inverse f) (y : F) :
‖fsymm y‖ ≤ fsymm.nnnorm * ‖y‖ | fsymm.bound' y | lemma | continuous_linear_map.nonlinear_right_inverse.bound | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.to_nonlinear_right_inverse (f : E ≃L[𝕜] F) :
continuous_linear_map.nonlinear_right_inverse (f : E →L[𝕜] F) | { to_fun := f.inv_fun,
nnnorm := ‖(f.symm : F →L[𝕜] E)‖₊,
bound' := λ y, continuous_linear_map.le_op_norm (f.symm : F →L[𝕜] E) _,
right_inv' := f.apply_symm_apply } | def | continuous_linear_equiv.to_nonlinear_right_inverse | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"bound'",
"continuous_linear_map.le_op_norm",
"continuous_linear_map.nonlinear_right_inverse"
] | Given a continuous linear equivalence, the inverse is in particular an instance of
`nonlinear_right_inverse` (which turns out to be linear). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_approx_preimage_norm_le (surj : surjective f) :
∃C ≥ 0, ∀y, ∃x, dist (f x) y ≤ 1/2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ | begin
have A : (⋃n:ℕ, closure (f '' (ball 0 n))) = univ,
{ refine subset.antisymm (subset_univ _) (λy hy, _),
rcases surj y with ⟨x, hx⟩,
rcases exists_nat_gt (‖x‖) with ⟨n, hn⟩,
refine mem_Union.2 ⟨n, subset_closure _⟩,
refine (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩,
rwa [mem_ball, dist_eq_norm, sub_z... | lemma | continuous_linear_map.exists_approx_preimage_norm_le | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"closure",
"dist_comm",
"div_nonneg",
"div_pos",
"exists_nat_gt",
"half_pos",
"interior",
"inv_mul_cancel",
"inv_nonneg",
"is_closed_closure",
"mem_interior_iff_mem_nhds",
"metric.mem_nhds_iff",
"mul_le_mul",
"mul_le_mul_of_nonneg_left",
"nonempty_interior_of_Union_of_closed",
"norm_eq... | First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_preimage_norm_le (surj : surjective f) :
∃C > 0, ∀y, ∃x, f x = y ∧ ‖x‖ ≤ C * ‖y‖ | begin
obtain ⟨C, C0, hC⟩ := exists_approx_preimage_norm_le f surj,
/- Second step of the proof: starting from `y`, we want an exact preimage of `y`. Let `g y` be
the approximate preimage of `y` given by the first step, and `h y = y - f(g y)` the part that
has no preimage yet. We will iterate this process, takin... | theorem | continuous_linear_map.exists_preimage_norm_le | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"dist_comm",
"finset.range",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"norm_tsum_le_tsum_norm",
"one_div",
"one_mul",
"pow_succ",
"pow_zero",
"ring",
"squeeze_zero",
"summable",
"summable.mul_right",
"summable_geometric_of_lt_1",
"summable_geometric_two",
"summable_of_nonneg_of_le",
... | The Banach open mapping theorem: if a bounded linear map between Banach spaces is onto, then
any point has a preimage with controlled norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map (surj : surjective f) : is_open_map f | begin
assume s hs,
rcases exists_preimage_norm_le f surj with ⟨C, Cpos, hC⟩,
refine is_open_iff.2 (λy yfs, _),
rcases mem_image_iff_bex.1 yfs with ⟨x, xs, fxy⟩,
rcases is_open_iff.1 hs x xs with ⟨ε, εpos, hε⟩,
refine ⟨ε/C, div_pos εpos Cpos, λz hz, _⟩,
rcases hC (z-y) with ⟨w, wim, wnorm⟩,
have : f (x +... | theorem | continuous_linear_map.is_open_map | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"div_pos",
"is_open_map",
"mul_div_cancel'",
"mul_lt_mul_of_pos_left",
"set.mem_image_of_mem"
] | The Banach open mapping theorem: a surjective bounded linear map between Banach spaces is
open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map (surj : surjective f) : quotient_map f | (f.is_open_map surj).to_quotient_map f.continuous surj | theorem | continuous_linear_map.quotient_map | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.affine_map.is_open_map {P Q : Type*}
[metric_space P] [normed_add_torsor E P] [metric_space Q] [normed_add_torsor F Q]
(f : P →ᵃ[𝕜] Q) (hf : continuous f) (surj : surjective f) :
is_open_map f | affine_map.is_open_map_linear_iff.mp $ continuous_linear_map.is_open_map
{ cont := affine_map.continuous_linear_iff.mpr hf, .. f.linear }
(f.linear_surjective_iff.mpr surj) | lemma | affine_map.is_open_map | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"cont",
"continuous",
"continuous_linear_map.is_open_map",
"is_open_map",
"metric_space",
"normed_add_torsor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_preimage (hsurj : surjective f) (s : set F) :
interior (f ⁻¹' s) = f ⁻¹' (interior s) | ((f.is_open_map hsurj).preimage_interior_eq_interior_preimage f.continuous s).symm | lemma | continuous_linear_map.interior_preimage | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_preimage (hsurj : surjective f) (s : set F) :
closure (f ⁻¹' s) = f ⁻¹' (closure s) | ((f.is_open_map hsurj).preimage_closure_eq_closure_preimage f.continuous s).symm | lemma | continuous_linear_map.closure_preimage | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_preimage (hsurj : surjective f) (s : set F) :
frontier (f ⁻¹' s) = f ⁻¹' (frontier s) | ((f.is_open_map hsurj).preimage_frontier_eq_frontier_preimage f.continuous s).symm | lemma | continuous_linear_map.frontier_preimage | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nonlinear_right_inverse_of_surjective (f : E →L[𝕜] F)
(hsurj : linear_map.range f = ⊤) : ∃ (fsymm : nonlinear_right_inverse f), 0 < fsymm.nnnorm | begin
choose C hC fsymm h using exists_preimage_norm_le _ (linear_map.range_eq_top.mp hsurj),
use { to_fun := fsymm,
nnnorm := ⟨C, hC.lt.le⟩,
bound' := λ y, (h y).2,
right_inv' := λ y, (h y).1 },
exact hC
end | lemma | continuous_linear_map.exists_nonlinear_right_inverse_of_surjective | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"bound'",
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonlinear_right_inverse_of_surjective
(f : E →L[𝕜] F) (hsurj : linear_map.range f = ⊤) : nonlinear_right_inverse f | classical.some (exists_nonlinear_right_inverse_of_surjective f hsurj) | def | continuous_linear_map.nonlinear_right_inverse_of_surjective | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | A surjective continuous linear map between Banach spaces admits a (possibly nonlinear)
controlled right inverse. In general, it is not possible to ensure that such a right inverse
is linear (take for instance the map from `E` to `E/F` where `F` is a closed subspace of `E`
without a closed complement. Then it doesn't ha... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonlinear_right_inverse_of_surjective_nnnorm_pos (f : E →L[𝕜] F)
(hsurj : linear_map.range f = ⊤) : 0 < (nonlinear_right_inverse_of_surjective f hsurj).nnnorm | begin
rw nonlinear_right_inverse_of_surjective,
exact classical.some_spec (exists_nonlinear_right_inverse_of_surjective f hsurj)
end | lemma | continuous_linear_map.nonlinear_right_inverse_of_surjective_nnnorm_pos | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_symm (e : E ≃ₗ[𝕜] F) (h : continuous e) :
continuous e.symm | begin
rw continuous_def,
intros s hs,
rw [← e.image_eq_preimage],
rw [← e.coe_coe] at h ⊢,
exact continuous_linear_map.is_open_map ⟨↑e, h⟩ e.surjective s hs
end | theorem | linear_equiv.continuous_symm | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous",
"continuous_def",
"continuous_linear_map.is_open_map"
] | If a bounded linear map is a bijection, then its inverse is also a bounded linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_equiv_of_continuous (e : E ≃ₗ[𝕜] F) (h : continuous e) :
E ≃L[𝕜] F | { continuous_to_fun := h,
continuous_inv_fun := e.continuous_symm h,
..e } | def | linear_equiv.to_continuous_linear_equiv_of_continuous | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous"
] | Associating to a linear equivalence between Banach spaces a continuous linear equivalence when
the direct map is continuous, thanks to the Banach open mapping theorem that ensures that the
inverse map is also continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_to_continuous_linear_equiv_of_continuous (e : E ≃ₗ[𝕜] F) (h : continuous e) :
⇑(e.to_continuous_linear_equiv_of_continuous h) = e | rfl | lemma | linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_to_continuous_linear_equiv_of_continuous_symm (e : E ≃ₗ[𝕜] F)
(h : continuous e) :
⇑(e.to_continuous_linear_equiv_of_continuous h).symm = e.symm | rfl | lemma | linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous_symm | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective (f : E →L[𝕜] F) (hinj : ker f = ⊥)
(hsurj : linear_map.range f = ⊤) : E ≃L[𝕜] F | (linear_equiv.of_bijective ↑f ⟨linear_map.ker_eq_bot.mp hinj, linear_map.range_eq_top.mp hsurj⟩)
.to_continuous_linear_equiv_of_continuous f.continuous | def | continuous_linear_equiv.of_bijective | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_equiv.of_bijective",
"linear_map.range"
] | Convert a bijective continuous linear map `f : E →L[𝕜] F` from a Banach space to a normed space
to a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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