Documentation Verification Report

AffineSubspace

📁 Source: Mathlib/Topology/Algebra/AffineSubspace.lean

Statistics

Metric	Count
Definitions `AffineSubspace`, `subtypeA`	2
Theorems `coe_subtypeA`, `instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection`, `isClosed_direction_iff`, `subtypeA_toAffineMap`	4
Total	6

AffineSubspace

Definitions

Name	Category	Theorems
`subtypeA` 📖	CompOp	3 math math: `coe_subtypeA`, `toContinuousAffineMap_subtypeₐᵢ`, `subtypeA_toAffineMap`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`coe_subtypeA` 📖	mathematical	—	`DFunLike.coe` `ContinuousAffineMap` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `Submodule.setLike` `direction` `AffineSubspace` `instSetLike` `Submodule.addCommGroup` `Submodule.module` `instTopologicalSpaceSubtype` `toAddTorsor` `ContinuousAffineMap.instFunLike` `subtypeA`	—	—
`instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection` 📖	mathematical	—	`IsTopologicalAddTorsor` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `Submodule.setLike` `direction` `AffineSubspace` `instSetLike` `AddCommGroup.toAddGroup` `Submodule.addCommGroup` `instTopologicalSpaceSubtype` `toAddTorsor`	—	`Topology.IsEmbedding.continuous_iff` `Topology.IsEmbedding.subtypeVal` `Continuous.vadd` `IsTopologicalAddTorsor.toContinuousVAdd` `Continuous.comp'` `continuous_subtype_val` `Continuous.fst` `continuous_id'` `Continuous.snd` `Continuous.vsub`
`isClosed_direction_iff` 📖	mathematical	—	`IsClosed` `SetLike.coe` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `Submodule.setLike` `direction` `AffineSubspace` `instSetLike`	—	`eq_bot_or_nonempty` `direction_bot` `Homeomorph.isClosed_image` `coe_direction_eq_vsub_set_right` `Set.image_congr`
`subtypeA_toAffineMap` 📖	mathematical	—	`ContinuousAffineMap.toAffineMap` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `Submodule.setLike` `direction` `AffineSubspace` `instSetLike` `Submodule.addCommGroup` `Submodule.module` `instTopologicalSpaceSubtype` `toAddTorsor` `subtypeA` `subtype`	—	—

(root)

Definitions

Name	Category	Theorems
`AffineSubspace` 📖	CompData	375 math math: `AffineSubspace.direction_inf`, `Affine.Simplex.faceOppositeCentroid_mem_affineSpan_face`, `AffineSubspace.coe_vsub`, `Affine.Simplex.orthogonalProjectionSpan_map`, `EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal`, `AffineMap.restrict.bijective`, `Isometry.preimage_perpBisector`, `Submodule.mem_toAffineSubspace`, `Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_face`, `AffineSubspace.self_mem_mk'`, `affineSpan_le_toAffineSubspace_span`, `AffineSubspace.mem_perpBisector_iff_inner_eq_zero'`, `AffineSubspace.map_pointwise_vadd`, `Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_faceOpposite`, `AffineBasis.tot`, `AffineSubspace.smul_mem_smul_iff₀`, `Affine.Simplex.circumcenter_mem_affineSpan`, `Affine.Simplex.excenter_singleton_mem_affineSpan_range`, `IsOpen.exists_subset_affineIndependent_span_eq_top`, `EuclideanGeometry.orthogonalProjection_congr`, `AffineSubspace.mem_sInf_iff`, `AffineMap.eqOn_affineSpan`, `EuclideanGeometry.Sphere.mem_orthRadius_iff_inner_right`, `AffineSubspace.left_mem_perpBisector`, `EuclideanGeometry.preimage_inversion_perpBisector_inversion`, `AffineSubspace.SSameSide.right_notMem`, `AffineSubspace.comap_bot`, `AffineMap.lineMap_rev_mem_affineSpan_pair`, `vadd_right_mem_affineSpan_pair`, `EuclideanGeometry.orthogonalProjection_contLinear`, `EuclideanGeometry.Sphere.mem_commonExtTangents_iff`, `AffineSubspace.WSameSide.nonempty`, `AffineSubspace.coe_subtypeA`, `Affine.Simplex.point_mem_median`, `AffineSubspace.subtype_apply`, `EuclideanGeometry.Sphere.mem_tangentsFrom_iff`, `AffineSubspace.vadd_mem_mk'`, `bot_lt_affineSpan`, `EuclideanGeometry.Sphere.mem_commonTangents_iff`, `AffineEquiv.span_eq_top_iff`, `smul_vsub_rev_vadd_mem_affineSpan_pair`, `affineSpan_singleton_union_vadd_eq_top_of_span_eq_top`, `interior_convexHull_nonempty_iff_affineSpan_eq_top`, `EuclideanGeometry.inter_eq_singleton_orthogonalProjection`, `EuclideanGeometry.inversion_mem_perpBisector_inversion_iff'`, `Sbtw.left_mem_affineSpan`, `AffineSubspace.linear_topEquiv`, `Affine.Simplex.orthogonalProjectionSpan_eq_point`, `AffineSubspace.not_wSameSide_bot`, `Affine.Simplex.affineSpan_face_le`, `EuclideanGeometry.orthogonalProjection_map`, `Collinear.mem_affineSpan_of_mem_of_ne`, `Affine.Simplex.ExcenterExists.touchpoint_notMem_affineSpan_of_ne`, `AffineSubspace.wOppSide_iff_exists_wbtw`, `AffineSubspace.SOppSide.nonempty`, `AffineIndependent.mem_affineSpan_iff`, `AffineSubspace.coe_subtypeₐᵢ`, `EuclideanGeometry.Sphere.IsTangentAt.mem_space`, `AffineSubspace.span_iUnion`, `AffineSubspace.mem_perpBisector_iff_inner_eq_zero`, `AffineBasis.tot'`, `EuclideanGeometry.Sphere.commonIntTangents_union_commonExtTangents`, `AffineSubspace.perpBisector_nonempty`, `Affine.Simplex.altitudeFoot_mem_affineSpan_image_compl`, `AffineSubspace.sSameSide_self_iff`, `Affine.Simplex.mongePoint_mem_mongePlane`, `AffineSubspace.wOppSide_self_iff`, `AffineSubspace.coe_eq_univ_iff`, `affineCombination_mem_affineSpan_pair`, `mem_intrinsicFrontier`, `AffineSubspace.le_def'`, `AffineSubspace.affineSpan_eq_sInf`, `finrank_vectorSpan_insert_le`, `AffineSubspace.coe_smul`, `EuclideanGeometry.Sphere.orthRadius_center`, `EuclideanGeometry.orthogonalProjection_apply'`, `EuclideanGeometry.Sphere.coe_secondInter`, `AffineSubspace.not_sSameSide_bot`, `mem_affineSpan_iff_eq_affineCombination`, `EuclideanGeometry.Sphere.isTangent_iff_isTangentAt_orthogonalProjection`, `AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv`, `affineCombination_mem_affineSpan`, `AffineSubspace.le_def`, `AffineSubspace.mem_iInf_iff`, `EuclideanGeometry.preimage_inversion_perpBisector`, `Affine.Simplex.median_reindex`, `AffineSubspace.WOppSide.nonempty`, `AffineSubspace.bot_coe`, `Convex.interior_nonempty_iff_affineSpan_eq_top`, `Isometry.mapsTo_perpBisector`, `AffineSubspace.parallel_bot_iff_eq_bot`, `AffineIsometryEquiv.coe_ofEq_apply`, `AffineSubspace.inf_coe`, `AffineSubspace.isometryEquivMap.toAffineMap_eq`, `AffineSubspace.lt_def`, `AffineSubspace.map_inf_eq`, `AffineSubspace.instNontrivial`, `AffineSubspace.span_union`, `Affine.Simplex.signedInfDist_apply_self`, `Affine.Simplex.centroid_notMem_affineSpan_of_ne_univ`, `AffineSubspace.coe_inf`, `AffineSubspace.wSameSide_and_wOppSide_iff`, `AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection`, `AffineEquiv.linear_ofEq`, `AffineSubspace.coe_pointwise_vadd`, `Affine.Simplex.incenter_notMem_affineSpan_face`, `Affine.Simplex.orthogonalProjectionSpan_congr`, `AffineSubspace.smul_bot`, `EuclideanGeometry.image_inversion_perpBisector`, `Affine.Simplex.mongePlane_def`, `AffineIsometryEquiv.ofEq_symm`, `AffineSubspace.SOppSide.right_notMem`, `AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot`, `EuclideanGeometry.reflection_eq_self_iff`, `Affine.Simplex.altitudeFoot_mem_affineSpan`, `affineSpan_le`, `smul_vsub_vadd_mem_affineSpan_pair`, `right_mem_affineSpan_pair`, `AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection`, `AffineSubspace.mem_top`, `mem_affineSpan_pair_iff_exists_lineMap_rev_eq`, `EuclideanGeometry.image_inversion_sphere_dist_center`, `EuclideanGeometry.orthogonalProjection_subtype`, `finiteDimensional_direction_affineSpan_insert`, `Affine.Simplex.points_mem_affineSpan_face`, `AffineSubspace.closed_of_finiteDimensional`, `AffineSubspace.toContinuousAffineMap_subtypeₐᵢ`, `Affine.Simplex.points_mem_affineSpan_faceOpposite`, `mem_affineSpan`, `AffineSubspace.smul_top`, `Affine.Simplex.circumsphere_unique_dist_eq`, `AffineSubspace.smul_mem_smul_iff_of_isUnit`, `EuclideanGeometry.preimage_inversion_sphere_dist_center`, `AffineSubspace.convex`, `affineSpan_subset_span`, `AffineIndependent.inf_affineSpan_eq_affineSpan_inter`, `Affine.Simplex.orthogonalProjection_circumcenter`, `Affine.Simplex.affineSpan_faceOpposite_le`, `centroid_mem_affineSpan_of_nonempty`, `AffineSubspace.subtypeA_toAffineMap`, `AffineSubspace.vadd_mem_iff_mem_of_mem_direction`, `AffineSubspace.mem_map`, `AffineSubspace.mk'_top`, `AffineSubspace.mem_map_iff_mem_of_injective`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq`, `Affine.Simplex.touchpoint_mem_affineSpan`, `AffineSubspace.coe_vadd`, `Affine.Simplex.altitudeFoot_mem_altitude`, `coe_affineSpan`, `Affine.Triangle.affineSpan_orthocenter_point_le_altitude`, `EuclideanGeometry.orthogonalProjection_affineSpan_singleton`, `AffineSubspace.subtype_injective`, `AffineSubspace.subtype_linear`, `affineCombination_mem_affineSpan_image`, `AffineSubspace.pointwise_vadd_direction`, `AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty`, `AffineSubspace.topEquiv_symm_apply_coe`, `AffineSubspace.isometryEquivMap.coe_apply`, `Affine.Simplex.mem_affineSpan_image_iff`, `AffineSubspace.mem_comap`, `AffineSubspace.nonempty_sup_left`, `centroid_mem_affineSpan_of_cast_card_ne_zero`, `intrinsicClosure_eq_closure_inter_affineSpan`, `AffineMap.map_top_of_surjective`, `EuclideanGeometry.Sphere.IsTangentAt.mem_and_mem_iff_eq`, `AffineSubspace.wSameSide_self_iff`, `AffineSubspace.mem_perpBisector_iff_inner_eq_inner`, `EuclideanGeometry.Sphere.IsTangent.infDist_eq_radius`, `AffineSubspace.mk'_nonempty`, `mem_affineSpan_pair_iff_exists_lineMap_eq`, `EuclideanGeometry.Sphere.IsTangent.notMem_of_dist_lt`, `Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev`, `EuclideanGeometry.exists_dist_eq_circumradius_of_subset_insert_orthocenter`, `EuclideanGeometry.Sphere.IsTangentAt.le_orthRadius`, `EuclideanGeometry.dist_orthogonalProjection_eq_infNndist`, `AffineSubspace.direction_iInf`, `AffineSubspace.coe_eq_bot_iff`, `AffineSubspace.SOppSide.left_notMem`, `MeasureTheory.Measure.addHaar_affineSubspace`, `Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_pair`, `AffineMap.restrict.surjective`, `AffineSubspace.subtypeₐᵢ_linear`, `Affine.Simplex.altitude_reindex`, `mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd`, `mem_affineSpan_iff_exists`, `subset_affineSpan`, `left_mem_affineSpan_pair`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangent`, `AffineSubspace.direction_top`, `Affine.Simplex.span_eq_top`, `EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem`, `AffineSubspace.isEmpty_bot`, `exists_subset_affineIndependent_affineSpan_eq_top`, `AffineSubspace.mem_perpBisector_iff_dist_eq'`, `Affine.Simplex.ninePointCircle_center_mem_affineSpan`, `AffineSubspace.smul_eq_map`, `AffineSubspace.map_comap_le`, `AffineSubspace.pointwise_vadd_eq_map`, `Affine.Simplex.affineCombination_mem_affineSpan_faceOpposite_iff`, `AffineSubspace.perpBisector_eq_top`, `EuclideanGeometry.dist_orthogonalProjection_eq_infDist`, `EuclideanGeometry.orthogonalProjection_mem`, `AffineSubspace.map_inf_le`, `AffineSubspace.midpoint_mem_perpBisector`, `AffineSubspace.direction_eq_vectorSpan`, `AffineSubspace.gc_map_comap`, `AffineSubspace.pointwise_vadd_bot`, `EuclideanGeometry.Sphere.center_mem_orthRadius_iff`, `AffineSubspace.finiteDimensional_sup`, `Affine.Simplex.ExcenterExists.excenter_mem_affineSpan_range`, `AffineSubspace.ext_iff`, `Affine.Triangle.orthocenter_mem_altitude`, `EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff`, `AffineEquiv.ofEq_symm`, `AffineSubspace.smul_mem_smul_iff`, `AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one`, `AffineSubspace.not_sOppSide_bot`, `AffineSubspace.linear_equivMapOfInjective`, `EuclideanGeometry.Sphere.IsTangentAt.eq_orthogonalProjection`, `IsOpen.exists_between_affineIndependent_span_eq_top`, `AffineSubspace.top_coe`, `Affine.Simplex.points_notMem_affineSpan_faceOpposite`, `EuclideanGeometry.Sphere.secondInter_vsub_mem_affineSpan`, `affineSpan_coe_preimage_eq_top`, `AffineSubspace.pointwise_vadd_top`, `AffineSubspace.nonempty_sup_right`, `AffineSubspace.perpBisector_self`, `AffineSubspace.span_empty`, `EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq`, `AffineSubspace.map_bot`, `affineSpan_eq_top_of_nonempty_interior`, `AffineSubspace.direction_sInf`, `Affine.Simplex.incenter_mem_affineSpan_range`, `EuclideanGeometry.orthogonalProjection_orthogonalProjection`, `vectorSpan_add_self`, `AffineSubspace.instIsSimpleOrderOfSubsingleton`, `AffineSubspace.not_le_iff_exists`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq`, `Affine.Simplex.incenter_notMem_affineSpan_pair`, `AffineSubspace.preimage_coe_affineSpan_singleton`, `AffineSubspace.topEquiv_apply`, `affineSegment_subset_affineSpan`, `AffineSubspace.mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero`, `affineCombination_mem_affineSpan_of_nonempty`, `Affine.Simplex.faceOppositeCentroid_mem_median`, `AffineSubspace.span_univ`, `EuclideanGeometry.inversion_mem_perpBisector_inversion_iff`, `AffineSubspace.map_eq_bot_iff`, `affineSpan_nonempty`, `AffineSubspace.subtypeₐᵢ_linearIsometry`, `Affine.Triangle.orthocenter_mem_affineSpan`, `AffineSubspace.direction_bot`, `Affine.Simplex.mongePoint_mem_affineSpan`, `EuclideanGeometry.Sphere.IsTangent.isTangentAt`, `EuclideanGeometry.Sphere.orthRadius_le_orthRadius_iff`, `AffineSubspace.map_le_iff_le_comap`, `AffineBasis.isUnit_toMatrix_iff`, `AffineSubspace.sup_direction_le`, `AffineSubspace.instNonemptySubtypeMemTop`, `IsClosed.convexHull_subset_affineSpan_isVisible`, `AffineSubspace.mem_perpBisector_iff_dist_eq`, `EuclideanGeometry.reflection_apply'`, `AffineIsometryEquiv.ofEq_rfl`, `mem_fintypeAffineCoords_iff_sum`, `Affine.Simplex.centroid_mem_median`, `affineSpan_eq_bot`, `EuclideanGeometry.Sphere.mem_tangentSet_iff`, `AffineSubspace.comap_supr`, `AffineEquiv.coe_ofEq_apply`, `EuclideanGeometry.Sphere.isTangentAt_center_iff`, `AffineSubspace.le_comap_map`, `AffineSubspace.map_sup`, `AffineSubspace.direction_smul`, `AffineSubspace.comap_top`, `AffineSubspace.instNonemptySubtypeMemMk'`, `AffineSubspace.coe_subtype`, `Affine.Simplex.orthogonalProjectionSpan_reindex`, `Affine.Simplex.touchpoint_empty_notMem_affineSpan_of_ne`, `EuclideanGeometry.orthogonalProjection_linear`, `AffineSubspace.signedInfDist_def`, `AffineSubspace.vadd_mem_pointwise_vadd_iff`, `Affine.Simplex.mem_altitude`, `Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle`, `affineSpan_mono`, `AffineIndependent.injOn_affineCombination_fintypeAffineCoords`, `centroid_mem_affineSpan_of_card_eq_add_one`, `AffineSubspace.mem_mk'`, `AffineSubspace.mem_affineSpan_singleton`, `Affine.Simplex.signedInfDist_affineCombination`, `EuclideanGeometry.Sphere.mem_commonIntTangents_iff`, `AffineSubspace.angle_coe`, `AffineSubspace.mem_inf_iff`, `EuclideanGeometry.orthogonalProjection_eq_self_iff`, `AffineSubspace.mem_perpBisector_iff_inner_eq`, `EuclideanGeometry.reflection_subtype`, `AffineIndependent.injective_affineSpan_image`, `Set.Nonempty.affineSpan`, `AffineSubspace.isometryEquivMap.apply_symm_apply`, `AffineSubspace.SSameSide.left_notMem`, `AffineSubspace.inclusion_rfl`, `AffineSubspace.coe_injective`, `AffineSubspace.SOppSide.exists_sbtw`, `AffineSubspace.coe_iInf`, `AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial`, `EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal`, `Affine.Simplex.setInterior_subset_affineSpan`, `Affine.Simplex.touchpoint_mem_affineSpan_simplex`, `EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace`, `Sbtw.right_mem_affineSpan`, `EuclideanGeometry.orthogonalProjection_apply`, `AffineIndependent.affineSpan_disjoint_of_disjoint`, `AffineIndependent.existsUnique_dist_eq`, `AffineSubspace.pointwise_vadd_span`, `AffineSubspace.signedInfDist_apply_self`, `EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection`, `EuclideanGeometry.orthogonalProjection_mem_orthogonal`, `IsOpen.affineSpan_eq_top`, `AffineSubspace.not_wOppSide_bot`, `Wbtw.mem_affineSpan`, `mem_intrinsicInterior`, `AffineSubspace.nonempty_map`, `centroid_mem_affineSpan_of_card_ne_zero`, `AffineSubspace.equivMapOfInjective_toFun`, `instNonemptySubtypeMemAffineSubspaceAffineSpanOfElem`, `Wbtw.left_mem_affineSpan_of_right_ne`, `EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem`, `AffineSubspace.eq_bot_or_nonempty`, `Affine.Simplex.map_subtype_restrict`, `mem_finsuppAffineCoords_iff_linearCombination`, `Wbtw.right_mem_affineSpan_of_left_ne`, `Affine.Simplex.restrict_map_subtype`, `AffineIndependent.notMem_affineSpan_diff`, `AffineSubspace.subtypeₐᵢ_toAffineMap`, `AffineSubspace.mem_perpBisector_pointReflection_iff_inner_eq_zero`, `AffineSubspace.mem_coe`, `Affine.Simplex.altitudeFoot_mem_affineSpan_faceOpposite`, `AffineMap.lineMap_mem_affineSpan_pair`, `Affine.Simplex.affineSpan_pair_eq_altitude_iff`, `Affine.Simplex.closedInterior_subset_affineSpan`, `AffineSubspace.map_iSup`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq`, `EuclideanGeometry.orthogonalProjection_eq_iff_mem`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq`, `affineSpan_insert_affineSpan`, `AffineSubspace.coe_comap`, `AffineSubspace.bot_parallel_iff_eq_bot`, `AffineSubspace.nonempty_iff_ne_bot`, `AffineSubspace.SSameSide.nonempty`, `affineSpan_insert_zero`, `Affine.Simplex.incenter_notMem_affineSpan_faceOpposite`, `convexHull_subset_affineSpan`, `AffineSubspace.affineSpan_coe`, `AffineSubspace.right_mem_perpBisector`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangentAt`, `EuclideanGeometry.reflection_apply`, `EuclideanGeometry.Sphere.mem_orthRadius_iff_inner_left`, `Affine.Simplex.centroid_mem_affineSpan`, `AffineEquiv.ofEq_rfl`, `AffineSubspace.notMem_bot`, `AffineSubspace.isClosed_direction_iff`, `finiteDimensional_vectorSpan_insert`, `AffineSubspace.coe_affineSpan_singleton`, `EuclideanGeometry.Sphere.infDist_eq_radius_iff_isTangent`, `Affine.Simplex.altitude_def`, `vadd_left_mem_affineSpan_pair`, `intrinsicClosure_subset_affineSpan`, `AffineSubspace.lt_iff_le_and_exists`, `EuclideanGeometry.Sphere.self_mem_orthRadius`, `AffineSubspace.comap_inf`, `AffineSubspace.coe_sInf`, `mem_intrinsicClosure`, `affineSpan_eq_top_iff_nonempty_of_subsingleton`, `EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self`, `AffineSubspace.smul_span`, `AffineSubspace.coe_map`

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