Documentation Verification Report

Basic

πŸ“ Source: Mathlib/Geometry/Euclidean/Sphere/Basic.lean

Statistics

MetricCount
DefinitionsConcyclic, Cospherical, IsDiameter, center, ofDiameter, radius, instCoeSphereSet, instMembershipSphere
8
TheoremsCoplanar, Cospherical, subset, affineIndependent, affineIndependent_of_mem_of_ne, affineIndependent_of_ne, subset, dist_left_right, dist_left_right_div_two, left_eq_of_isDiameter, left_eq_right_iff, left_mem, left_ne_right_iff_radius_ne_zero, left_ne_right_iff_radius_pos, midpoint_eq_center, ofDiameter_eq, pointReflection_center_left, pointReflection_center_right, right_eq_of_isDiameter, right_mem, sbtw, wbtw, center_eq_iff_eq_of_mem, center_mem_iff, center_ne_iff_ne_of_mem, coe_mk, cospherical, ext, ext_iff, isDiameter_comm, isDiameter_iff_left_mem_and_midpoint_eq_center, isDiameter_iff_left_mem_and_pointReflection_center_left, isDiameter_iff_mem_and_mem_and_dist, isDiameter_iff_mem_and_mem_and_wbtw, isDiameter_iff_ofDiameter_eq, isDiameter_iff_right_mem_and_midpoint_eq_center, isDiameter_iff_right_mem_and_pointReflection_center_right, isDiameter_ofDiameter, mem_coe, mem_coe', mk_center, mk_center_radius, mk_radius, ne_iff, nonempty_iff, radius_nonneg_of_mem, concyclic_empty, concyclic_pair, concyclic_singleton, cospherical_def, cospherical_empty, cospherical_iff_exists_sphere, cospherical_pair, cospherical_singleton, dist_center_eq_dist_center_of_mem_sphere, dist_center_eq_dist_center_of_mem_sphere', dist_of_mem_subset_mk_sphere, dist_of_mem_subset_sphere, eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two, eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two, inner_nonneg_of_dist_le_radius, inner_pos_of_dist_lt_radius, inner_pos_or_eq_of_dist_le_radius, inner_vsub_vsub_of_mem_sphere_of_mem_sphere, instNonemptySphere, mem_sphere, mem_sphere', sbtw_of_collinear_of_dist_center_lt_radius, subset_sphere, wbtw_of_collinear_of_dist_center_le_radius
70
Total78

EuclideanGeometry

Definitions

NameCategoryTheorems
Concyclic πŸ“–CompData
5 mathmath: concyclic_empty, concyclic_of_two_zsmul_oangle_eq_of_not_collinear, concyclic_pair, concyclic_or_collinear_of_two_zsmul_oangle_eq, concyclic_singleton
Cospherical πŸ“–MathDef
11 mathmath: cospherical_pair, cospherical_def, cospherical_singleton, Sphere.cospherical, cospherical_or_collinear_of_two_zsmul_oangle_eq, cospherical_iff_exists_mem_of_complete, cospherical_of_two_zsmul_oangle_eq_of_not_collinear, Concyclic.Cospherical, cospherical_iff_exists_mem_of_finiteDimensional, cospherical_iff_exists_sphere, cospherical_empty
instCoeSphereSet πŸ“–CompOpβ€”
instMembershipSphere πŸ“–CompOp
39 mathmath: subset_sphere, Sphere.thales_theorem, Sphere.isDiameter_iff_right_mem_and_midpoint_eq_center, Sphere.secondInter_mem, Sphere.IsExtTangentAt.mem_left, Affine.Simplex.mem_circumsphere, Sphere.isTangent_orthRadius_iff_mem, Sphere.isDiameter_iff_right_mem_and_pointReflection_center_right, Sphere.isTangentAt_orthRadius_iff_mem, Sphere.IsDiameter.right_mem, Sphere.IsDiameter.left_mem, Affine.Triangle.altitudeFoot_mem_ninePointCircle, Affine.Simplex.eulerPoint_mem_ninePointCircle, Sphere.isExtTangentAt_center_iff, Affine.Simplex.ExcenterExists.touchpoint_mem_exsphere, Affine.Triangle.mem_circumsphere_of_two_zsmul_oangle_eq, Sphere.power_eq_zero_iff_mem_sphere, Sphere.IsTangentAt.mem_and_mem_iff_eq, Sphere.isIntTangentAt_self_iff_mem, mem_sphere, Sphere.IsIntTangentAt.mem_left, mem_sphere', Sphere.mem_coe, Sphere.isDiameter_iff_mem_and_mem_and_dist, Sphere.IsTangentAt.mem_sphere, Sphere.isDiameter_iff_mem_and_mem_and_wbtw, Sphere.IsExtTangentAt.mem_right, Affine.Simplex.faceOppositeCentroid_mem_ninePointCircle, Sphere.isIntTangentAt_center_iff, Sphere.mem_tangentSet_iff, Sphere.mem_coe', Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle, Sphere.IsIntTangentAt.mem_right, Sphere.center_mem_iff, Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter, Sphere.isDiameter_iff_left_mem_and_midpoint_eq_center, Sphere.angle_eq_pi_div_two_iff_mem_sphere_ofDiameter, Sphere.isDiameter_iff_left_mem_and_pointReflection_center_left, Affine.Simplex.touchpoint_mem_insphere

Theorems

NameKindAssumesProvesValidatesDepends On
concyclic_empty πŸ“–mathematicalβ€”Concyclic
Set
Set.instEmptyCollection
β€”cospherical_empty
AddTorsor.nonempty
coplanar_empty
concyclic_pair πŸ“–mathematicalβ€”Concyclic
Set
Set.instInsert
Set.instSingletonSet
β€”cospherical_pair
coplanar_pair
concyclic_singleton πŸ“–mathematicalβ€”Concyclic
Set
Set.instSingletonSet
β€”cospherical_singleton
coplanar_singleton
cospherical_def πŸ“–mathematicalβ€”Cospherical
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
β€”β€”
cospherical_empty πŸ“–mathematicalβ€”Cospherical
Set
Set.instEmptyCollection
β€”β€”
cospherical_iff_exists_sphere πŸ“–mathematicalβ€”Cospherical
Set
Set.instHasSubset
Metric.sphere
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
β€”β€”
cospherical_pair πŸ“–mathematicalβ€”Cospherical
Set
Set.instInsert
Set.instSingletonSet
β€”RCLike.charZero_rclike
Nat.instAtLeastTwoHAddOfNat
dist_comm
dist_midpoint_left
dist_midpoint_right
cospherical_singleton πŸ“–mathematicalβ€”Cospherical
Set
Set.instSingletonSet
β€”dist_self
dist_center_eq_dist_center_of_mem_sphere πŸ“–mathematicalSphere
instMembershipSphere
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
β€”mem_sphere
dist_center_eq_dist_center_of_mem_sphere' πŸ“–mathematicalSphere
instMembershipSphere
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
β€”mem_sphere'
dist_of_mem_subset_mk_sphere πŸ“–mathematicalSet
Set.instMembership
Set.instHasSubset
Metric.sphere
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Dist.dist
PseudoMetricSpace.toDist
β€”dist_of_mem_subset_sphere
dist_of_mem_subset_sphere πŸ“–mathematicalSet
Set.instMembership
Set.instHasSubset
Metric.sphere
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Dist.dist
PseudoMetricSpace.toDist
β€”mem_sphere
Sphere.mem_coe
Set.mem_of_mem_of_subset
eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two πŸ“–β€”Module.finrank
Real
Real.semiring
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
NormedSpace.toModule
Real.normedField
InnerProductSpace.toNormedSpace
Real.instRCLike
Sphere
instMembershipSphere
β€”β€”eq_of_dist_eq_of_dist_eq_of_finrank_eq_two
Sphere.center_ne_iff_ne_of_mem
eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two πŸ“–β€”Module.finrank
Real
Submodule
Ring.toSemiring
Real.instRing
AddCommGroup.toAddCommMonoid
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
SetLike.instMembership
Submodule.setLike
AffineSubspace.direction
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Real.semiring
Submodule.addCommMonoid
Submodule.module
AffineSubspace
AffineSubspace.instSetLike
Sphere.center
Sphere
instMembershipSphere
β€”β€”eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two
Sphere.center_ne_iff_ne_of_mem
inner_nonneg_of_dist_le_radius πŸ“–mathematicalSphere
instMembershipSphere
Real
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Real.instZero
Inner.inner
InnerProductSpace.toInner
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
VSub.vsub
AddTorsor.toVSub
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddTorsor.toAddTorsor
β€”inner_pos_or_eq_of_dist_le_radius
LT.lt.le
vsub_self
inner_zero_left
inner_pos_of_dist_lt_radius πŸ“–mathematicalSphere
instMembershipSphere
Real
Real.instLT
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Real.instZero
Inner.inner
InnerProductSpace.toInner
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
VSub.vsub
AddTorsor.toVSub
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddTorsor.toAddTorsor
β€”LT.lt.ne
mem_sphere
inner_pos_or_eq_of_dist_le_radius
LT.lt.le
inner_pos_or_eq_of_dist_le_radius πŸ“–mathematicalSphere
instMembershipSphere
Real
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Real.instLT
Real.instZero
Inner.inner
InnerProductSpace.toInner
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
VSub.vsub
AddTorsor.toVSub
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddTorsor.toAddTorsor
β€”vsub_sub_vsub_cancel_right
inner_sub_left
real_inner_self_eq_norm_mul_norm
sub_pos
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
lt_of_le_of_ne
LE.le.trans
real_inner_le_norm
mul_le_mul_of_nonneg_right
IsOrderedRing.toMulPosMono
Real.instIsOrderedRing
dist_eq_norm_vsub
mem_sphere
norm_nonneg
LE.le.lt_or_eq
LT.lt.ne
LE.le.trans_lt
mul_lt_mul_of_pos_right
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
norm_pos_iff
vsub_ne_zero
LE.le.not_gt
dist_nonneg
dist_self
inner_eq_norm_mul_iff_real
sub_eq_zero
smul_sub
smul_ne_zero_iff
instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors
Real.instIsDomain
GroupWithZero.toNoZeroSMulDivisors
IsDomain.toIsCancelMulZero
norm_ne_zero_iff
vsub_self
norm_zero
inner_vsub_vsub_of_mem_sphere_of_mem_sphere πŸ“–mathematicalSphere
instMembershipSphere
Inner.inner
Real
InnerProductSpace.toInner
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
VSub.vsub
AddTorsor.toVSub
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Sphere.center
Real.instZero
β€”inner_vsub_vsub_of_dist_eq_of_dist_eq
dist_center_eq_dist_center_of_mem_sphere
instNonemptySphere πŸ“–mathematicalβ€”Sphereβ€”β€”
mem_sphere πŸ“–mathematicalβ€”Sphere
instMembershipSphere
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
β€”β€”
mem_sphere' πŸ“–mathematicalβ€”Sphere
instMembershipSphere
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
β€”Metric.mem_sphere'
sbtw_of_collinear_of_dist_center_lt_radius πŸ“–mathematicalCollinear
Real
Real.instDivisionRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Set
Set.instInsert
Set.instSingletonSet
Sphere
instMembershipSphere
Real.instLT
Dist.dist
PseudoMetricSpace.toDist
Sphere.center
Sphere.radius
Sbtw
Real.instRing
Real.partialOrder
β€”Collinear.sbtw_of_dist_eq_of_dist_lt
UniformConvexSpace.toStrictConvexSpace
InnerProductSpace.toUniformConvexSpace
subset_sphere πŸ“–mathematicalβ€”Set
Set.instHasSubset
Metric.sphere
MetricSpace.toPseudoMetricSpace
Sphere.center
Sphere.radius
Sphere
instMembershipSphere
β€”β€”
wbtw_of_collinear_of_dist_center_le_radius πŸ“–mathematicalCollinear
Real
Real.instDivisionRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Set
Set.instInsert
Set.instSingletonSet
Sphere
instMembershipSphere
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
Sphere.center
Sphere.radius
Wbtw
Real.instRing
Real.partialOrder
β€”Collinear.wbtw_of_dist_eq_of_dist_le
UniformConvexSpace.toStrictConvexSpace
InnerProductSpace.toUniformConvexSpace

EuclideanGeometry.Concyclic

Theorems

NameKindAssumesProvesValidatesDepends On
Coplanar πŸ“–mathematicalEuclideanGeometry.ConcyclicCoplanar
Real
Real.instDivisionRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
β€”β€”
Cospherical πŸ“–mathematicalEuclideanGeometry.ConcyclicEuclideanGeometry.Cosphericalβ€”β€”
subset πŸ“–β€”Set
Set.instHasSubset
EuclideanGeometry.Concyclic
β€”β€”EuclideanGeometry.Cospherical.subset
Cospherical
Coplanar.subset
Coplanar

EuclideanGeometry.Cospherical

Theorems

NameKindAssumesProvesValidatesDepends On
affineIndependent πŸ“–mathematicalEuclideanGeometry.Cospherical
Set
Set.instHasSubset
Set.range
AffineIndependent
Real
Real.instRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
β€”affineIndependent_iff_not_collinear
collinear_iff_of_mem
Set.mem_range_self
smul_zero
zero_vadd
Set.forall_mem_range
Set.mem_of_mem_of_subset
smul_eq_zero
instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors
Real.instIsDomain
GroupWithZero.toNoZeroSMulDivisors
IsDomain.toIsCancelMulZero
vadd_vsub
vsub_eq_zero_iff_eq
Nat.instAtLeastTwoHAddOfNat
EuclideanGeometry.dist_smul_vadd_eq_dist
zero_smul
neg_mul
inner_self_eq_norm_sq_to_K
affineIndependent_of_mem_of_ne πŸ“–mathematicalEuclideanGeometry.Cospherical
Set
Set.instMembership
AffineIndependent
Real
Real.instRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Matrix.vecCons
Matrix.vecEmpty
β€”affineIndependent
Matrix.range_cons
Matrix.range_empty
Set.union_empty
Set.union_singleton
Set.union_insert
Fin.cons_injective_iff
Matrix.cons_val_fin_one
affineIndependent_of_ne πŸ“–mathematicalEuclideanGeometry.Cospherical
Set
Set.instInsert
Set.instSingletonSet
AffineIndependent
Real
Real.instRing
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
Matrix.vecCons
Matrix.vecEmpty
β€”affineIndependent_of_mem_of_ne
Set.mem_insert
Set.mem_insert_of_mem
Set.mem_singleton
subset πŸ“–β€”Set
Set.instHasSubset
EuclideanGeometry.Cospherical
β€”β€”β€”

EuclideanGeometry.Sphere

Definitions

NameCategoryTheorems
IsDiameter πŸ“–CompData
11 mathmath: isDiameter_iff_right_mem_and_midpoint_eq_center, isDiameter_iff_right_mem_and_pointReflection_center_right, isDiameter_ofDiameter, isDiameter_comm, isDiameter_of_angle_eq_pi_div_two, isDiameter_iff_mem_and_mem_and_dist, isDiameter_iff_ofDiameter_eq, isDiameter_iff_mem_and_mem_and_wbtw, Affine.Simplex.isDiameter_ninePointCircle, isDiameter_iff_left_mem_and_midpoint_eq_center, isDiameter_iff_left_mem_and_pointReflection_center_left
center πŸ“–CompOp
90 mathmath: EuclideanGeometry.subset_sphere, isExtTangent_iff_dist_center, isDiameter_iff_right_mem_and_midpoint_eq_center, secondInter_dist, IsExtTangent.dist_center, mem_orthRadius_iff_inner_right, mem_commonExtTangents_iff, isDiameter_iff_right_mem_and_pointReflection_center_right, Affine.Simplex.circumsphere_center, nonempty_iff, IsTangentAt.angle_eq_pi_div_two, IsDiameter.wbtw, oangle_eq_pi_sub_two_zsmul_oangle_center_left, IsExtTangentAt.wbtw, Affine.Simplex.ninePointCircle_restrict, orthRadius_center, coe_secondInter, power_pos_iff_radius_lt_dist_center, isTangent_iff_isTangentAt_orthogonalProjection, mk_center_radius, tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center, IsTangentAt.inner_right_eq_zero_of_mem, IsDiameter.pointReflection_center_left, IsTangentAt.radius_lt_dist_center, IsTangentAt.inner_left_eq_zero_of_mem, abs_oangle_center_left_toReal_lt_pi_div_two, Affine.Simplex.circumsphere_unique_dist_eq, EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere', dist_div_cos_oangle_center_div_two_eq_radius, power_neg_iff_dist_center_lt_radius, isExtTangentAt_center_iff, inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center, abs_oangle_center_right_toReal_lt_pi_div_two, cospherical, direction_orthRadius_le_iff, IsDiameter.sbtw, IsTangent.infDist_eq_radius, IsIntTangentAt.wbtw, dist_div_cos_oangle_center_eq_two_mul_radius, dist_orthogonalProjection_eq_radius_iff_isTangent, EuclideanGeometry.mem_sphere, IsTangent.eq_orthRadius_or_eq_orthRadius_pointReflection_of_parallel_orthRadius, IsDiameter.pointReflection_center_right, power_nonneg_iff_radius_le_dist_center, Affine.Simplex.ninePointCircle_center_mem_affineSpan, oangle_eq_pi_sub_two_zsmul_oangle_center_right, power_nonpos_iff_dist_center_le_radius, EuclideanGeometry.mem_sphere', mem_coe, center_mem_orthRadius_iff, Affine.Simplex.midpoint_faceOppositeCentroid_eulerPoint, IsTangentAt.eq_orthogonalProjection, IsDiameter.midpoint_eq_center, isDiameter_iff_mem_and_mem_and_wbtw, coe_mk, IsTangent.isTangentAt, two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi, orthRadius_le_orthRadius_iff, IsTangentAt.dist_sq_eq_of_mem, isIntTangentAt_center_iff, oangle_center_eq_two_zsmul_oangle, isTangentAt_center_iff, isIntTangent_iff_dist_center, Affine.Simplex.exsphere_center, IsTangentAt_iff_angle_eq_pi_div_two, mem_coe', mem_commonIntTangents_iff, Affine.Simplex.ninePointCircle_map, center_mem_iff, mk_center, direction_orthRadius, AffineIndependent.existsUnique_dist_eq, IsIntTangent.dist_center, orthRadius_parallel_orthRadius_iff, EuclideanGeometry.inner_vsub_vsub_of_mem_sphere_of_mem_sphere, Affine.Simplex.insphere_center, IsTangent.radius_le_dist_center, secondInter_eq_lineMap, secondInter_map, isDiameter_iff_left_mem_and_midpoint_eq_center, EuclideanGeometry.cospherical_iff_exists_sphere, isDiameter_iff_left_mem_and_pointReflection_center_left, Affine.Simplex.ninePointCircle_center, dist_orthogonalProjection_eq_radius_iff_isTangentAt, mem_orthRadius_iff_inner_left, secondInter_eq_self_iff, center_eq_iff_eq_of_mem, ext_iff, infDist_eq_radius_iff_isTangent, EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere
ofDiameter πŸ“–CompOp
4 mathmath: isDiameter_ofDiameter, isDiameter_iff_ofDiameter_eq, IsDiameter.ofDiameter_eq, angle_eq_pi_div_two_iff_mem_sphere_ofDiameter
radius πŸ“–CompOp
48 mathmath: EuclideanGeometry.subset_sphere, isExtTangent_iff_dist_center, Affine.Simplex.insphere_radius, IsExtTangent.dist_center, IsDiameter.dist_left_right, Affine.Simplex.circumsphere_radius, dist_div_sin_oangle_div_two_eq_radius, Affine.Simplex.ninePointCircle_radius, nonempty_iff, Affine.Simplex.ninePointCircle_restrict, coe_secondInter, mk_center_radius, dist_div_sin_oangle_eq_two_mul_radius, IsDiameter.left_ne_right_iff_radius_pos, IsTangentAt.radius_lt_dist_center, Affine.Simplex.circumsphere_unique_dist_eq, dist_div_cos_oangle_center_div_two_eq_radius, isExtTangentAt_center_iff, cospherical, IsTangent.infDist_eq_radius, dist_div_cos_oangle_center_eq_two_mul_radius, IsDiameter.left_eq_right_iff, mk_radius, dist_orthogonalProjection_eq_radius_iff_isTangent, EuclideanGeometry.mem_sphere, EuclideanGeometry.mem_sphere', radius_nonneg_of_mem, mem_coe, isDiameter_iff_mem_and_mem_and_dist, coe_mk, IsTangentAt.dist_sq_eq_of_mem, isIntTangentAt_center_iff, isTangentAt_center_iff, isIntTangent_iff_dist_center, mem_coe', Affine.Simplex.ninePointCircle_map, center_mem_iff, AffineIndependent.existsUnique_dist_eq, IsIntTangent.dist_center, Affine.Simplex.exsphere_radius, IsTangent.radius_le_dist_center, isIntTangent_self_iff, secondInter_map, EuclideanGeometry.cospherical_iff_exists_sphere, dist_orthogonalProjection_eq_radius_iff_isTangentAt, IsDiameter.dist_left_right_div_two, ext_iff, infDist_eq_radius_iff_isTangent

Theorems

NameKindAssumesProvesValidatesDepends On
center_eq_iff_eq_of_mem πŸ“–mathematicalEuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
centerβ€”ext
EuclideanGeometry.mem_sphere
center_mem_iff πŸ“–mathematicalβ€”EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
center
radius
Real
Real.instZero
β€”dist_self
center_ne_iff_ne_of_mem πŸ“–β€”EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
β€”β€”Iff.not
center_eq_iff_eq_of_mem
coe_mk πŸ“–mathematicalβ€”Metric.sphere
MetricSpace.toPseudoMetricSpace
center
radius
β€”β€”
cospherical πŸ“–mathematicalβ€”EuclideanGeometry.Cospherical
Metric.sphere
MetricSpace.toPseudoMetricSpace
center
radius
β€”EuclideanGeometry.cospherical_iff_exists_sphere
Set.Subset.rfl
ext πŸ“–β€”center
radius
β€”β€”β€”
ext_iff πŸ“–mathematicalβ€”center
radius
β€”ext
isDiameter_comm πŸ“–mathematicalβ€”IsDiameterβ€”IsDiameter.symm
isDiameter_iff_left_mem_and_midpoint_eq_center πŸ“–mathematicalβ€”IsDiameter
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
midpoint
Real
Real.instRing
invertibleTwo
Semifield.toDivisionSemiring
Field.toSemifield
Real.instField
RCLike.charZero_rclike
Real.instRCLike
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
center
β€”RCLike.charZero_rclike
IsDiameter.left_mem
IsDiameter.midpoint_eq_center
isDiameter_iff_left_mem_and_pointReflection_center_left πŸ“–mathematicalβ€”IsDiameter
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.pointReflection
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
center
β€”IsDiameter.left_mem
IsDiameter.pointReflection_center_left
RCLike.charZero_rclike
midpoint_pointReflection_right
isDiameter_iff_mem_and_mem_and_dist πŸ“–mathematicalβ€”IsDiameter
InnerProductSpace.toNormedSpace
Real
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Real.instMul
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
radius
β€”Nat.instAtLeastTwoHAddOfNat
IsDiameter.left_mem
IsDiameter.right_mem
IsDiameter.dist_left_right
RCLike.charZero_rclike
midpoint_eq_iff
AffineEquiv.pointReflection_apply
eq_vadd_iff_vsub_eq
eq_of_norm_eq_of_norm_add_eq
UniformConvexSpace.toStrictConvexSpace
InnerProductSpace.toUniformConvexSpace
EuclideanGeometry.mem_sphere'
EuclideanGeometry.mem_sphere
vsub_add_vsub_cancel
dist_comm
two_mul
isDiameter_iff_mem_and_mem_and_wbtw πŸ“–mathematicalβ€”IsDiameter
InnerProductSpace.toNormedSpace
Real
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
Wbtw
Real.instRing
Real.partialOrder
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
center
β€”IsDiameter.left_mem
IsDiameter.right_mem
IsDiameter.wbtw
Wbtw.dist_add_dist
Nat.instAtLeastTwoHAddOfNat
isDiameter_iff_mem_and_mem_and_dist
two_mul
EuclideanGeometry.mem_sphere'
EuclideanGeometry.mem_sphere
isDiameter_iff_ofDiameter_eq πŸ“–mathematicalβ€”IsDiameter
ofDiameter
β€”IsDiameter.ofDiameter_eq
isDiameter_ofDiameter
isDiameter_iff_right_mem_and_midpoint_eq_center πŸ“–mathematicalβ€”IsDiameter
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
midpoint
Real
Real.instRing
invertibleTwo
Semifield.toDivisionSemiring
Field.toSemifield
Real.instField
RCLike.charZero_rclike
Real.instRCLike
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
center
β€”RCLike.charZero_rclike
IsDiameter.right_mem
IsDiameter.midpoint_eq_center
IsDiameter.symm
midpoint_comm
isDiameter_iff_right_mem_and_pointReflection_center_right πŸ“–mathematicalβ€”IsDiameter
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.pointReflection
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
center
β€”isDiameter_comm
isDiameter_iff_left_mem_and_pointReflection_center_left
isDiameter_ofDiameter πŸ“–mathematicalβ€”IsDiameter
ofDiameter
β€”RCLike.charZero_rclike
Nat.instAtLeastTwoHAddOfNat
dist_left_midpoint
Real.norm_ofNat
inv_mul_eq_div
mem_coe πŸ“–mathematicalβ€”Set
Set.instMembership
Metric.sphere
MetricSpace.toPseudoMetricSpace
center
radius
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
β€”β€”
mem_coe' πŸ“–mathematicalβ€”Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
center
radius
EuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
β€”β€”
mk_center πŸ“–mathematicalβ€”centerβ€”β€”
mk_center_radius πŸ“–mathematicalβ€”center
radius
β€”ext
mk_radius πŸ“–mathematicalβ€”radiusβ€”β€”
ne_iff πŸ“–β€”β€”β€”β€”not_and_or
ext_iff
nonempty_iff πŸ“–mathematicalβ€”Set.Nonempty
Metric.sphere
MetricSpace.toPseudoMetricSpace
center
radius
Real
Real.instLE
Real.instZero
β€”radius_nonneg_of_mem
NormedSpace.sphere_nonempty
EMetric.instNontrivialTopologyOfNontrivial
dist_vadd_left
sub_zero
radius_nonneg_of_mem πŸ“–mathematicalEuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
Real
Real.instLE
Real.instZero
radius
β€”Metric.nonneg_of_mem_sphere

EuclideanGeometry.Sphere.IsDiameter

Theorems

NameKindAssumesProvesValidatesDepends On
dist_left_right πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterDist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Real
Real.instMul
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
EuclideanGeometry.Sphere.radius
β€”Nat.instAtLeastTwoHAddOfNat
EuclideanGeometry.mem_sphere
left_mem
RCLike.charZero_rclike
midpoint_eq_center
dist_left_midpoint
Real.norm_ofNat
mul_inv_cancel_leftβ‚€
dist_left_right_div_two πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterReal
DivInvMonoid.toDiv
Real.instDivInvMonoid
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
EuclideanGeometry.Sphere.radius
β€”Nat.instAtLeastTwoHAddOfNat
dist_left_right
mul_div_cancel_leftβ‚€
EuclideanDomain.toMulDivCancelClass
RCLike.charZero_rclike
left_eq_of_isDiameter πŸ“–β€”EuclideanGeometry.Sphere.IsDiameterβ€”β€”pointReflection_center_right
left_eq_right_iff πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterEuclideanGeometry.Sphere.radius
Real
Real.instZero
β€”dist_eq_zero
Nat.instAtLeastTwoHAddOfNat
dist_left_right
NormMulClass.toNoZeroDivisors
NormedDivisionRing.toNormMulClass
RCLike.charZero_rclike
left_mem πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterEuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
β€”β€”
left_ne_right_iff_radius_ne_zero πŸ“–β€”EuclideanGeometry.Sphere.IsDiameterβ€”β€”Iff.not
left_eq_right_iff
left_ne_right_iff_radius_pos πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterReal
Real.instLT
Real.instZero
EuclideanGeometry.Sphere.radius
β€”left_ne_right_iff_radius_ne_zero
lt_iff_le_and_ne
EuclideanGeometry.Sphere.radius_nonneg_of_mem
left_mem
midpoint_eq_center πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiametermidpoint
Real
Real.instRing
invertibleTwo
Semifield.toDivisionSemiring
Field.toSemifield
Real.instField
RCLike.charZero_rclike
Real.instRCLike
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
EuclideanGeometry.Sphere.center
β€”β€”
ofDiameter_eq πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterEuclideanGeometry.Sphere.ofDiameterβ€”EuclideanGeometry.Sphere.ext
RCLike.charZero_rclike
midpoint_eq_center
Nat.instAtLeastTwoHAddOfNat
dist_left_right_div_two
pointReflection_center_left πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.pointReflection
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
EuclideanGeometry.Sphere.center
β€”RCLike.charZero_rclike
midpoint_eq_center
Equiv.pointReflection_midpoint_left
pointReflection_center_right πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterDFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.pointReflection
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
EuclideanGeometry.Sphere.center
β€”RCLike.charZero_rclike
midpoint_eq_center
Equiv.pointReflection_midpoint_right
right_eq_of_isDiameter πŸ“–β€”EuclideanGeometry.Sphere.IsDiameterβ€”β€”pointReflection_center_left
right_mem πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterEuclideanGeometry.Sphere
EuclideanGeometry.instMembershipSphere
β€”EuclideanGeometry.mem_sphere
left_mem
RCLike.charZero_rclike
midpoint_eq_center
dist_left_midpoint_eq_dist_right_midpoint
sbtw πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterSbtw
Real
Real.instRing
Real.partialOrder
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
EuclideanGeometry.Sphere.center
β€”RCLike.charZero_rclike
midpoint_eq_center
sbtw_midpoint_of_ne
Real.instIsStrictOrderedRing
left_ne_right_iff_radius_ne_zero
wbtw πŸ“–mathematicalEuclideanGeometry.Sphere.IsDiameterWbtw
Real
Real.instRing
Real.partialOrder
NormedAddCommGroup.toAddCommGroup
NormedSpace.toModule
Real.normedField
NormedAddCommGroup.toSeminormedAddCommGroup
NormedAddTorsor.toAddTorsor
MetricSpace.toPseudoMetricSpace
EuclideanGeometry.Sphere.center
β€”RCLike.charZero_rclike
midpoint_eq_center
wbtw_midpoint
Real.instIsStrictOrderedRing

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