Documentation Verification Report

PartitionOfUnity

πŸ“ Source: Mathlib/Geometry/Manifold/PartitionOfUnity.lean

Statistics

MetricCount
DefinitionstoSmoothPartitionOfUnity, SmoothBumpCovering, IsSubordinate, c, fintype, ind, instCoeFunForallSmoothBumpFunctionC, toBumpCovering, toFun, toSmoothPartitionOfUnity, SmoothPartitionOfUnity, IsSubordinate, finsupport, fintsupport, instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop, instInhabited, single, toFun, toPartitionOfUnity
19
TheoremstoSmoothPartitionOfUnity, coe_toSmoothPartitionOfUnity, contMDiff_toPartitionOfUnity, toSmoothPartitionOfUnity_toPartitionOfUnity, exists_contMDiffMap_forall_closedBall_subset, exists_smooth_forall_closedBall_subset, exists_contMDiff_support_eq, exists_contMDiff_support_eq_aux, exists_msmooth_support_eq, exists_msmooth_support_eq_aux, exists_contMDiffMap_forall_closedBall_subset, exists_contMDiffMap_forall_closedEBall_subset, exists_smooth_forall_closedBall_subset, support_subset, toBumpCovering, toSmoothPartitionOfUnity, apply_ind, c_mem', eventuallyEq_one, eventuallyEq_one', exists_finset_toSmoothPartitionOfUnity_eventuallyEq, exists_isSubordinate, isSubordinate_toBumpCovering, locallyFinite, locallyFinite', mem_chartAt_ind_source, mem_chartAt_source_of_eq_one, mem_extChartAt_ind_source, mem_extChartAt_source_of_eq_one, mem_support_ind, point_finite, sum_toSmoothPartitionOfUnity_eq, support_toSmoothPartitionOfUnity_subset, toSmoothPartitionOfUnity_apply, toSmoothPartitionOfUnity_eq_mul_prod, toSmoothPartitionOfUnity_zero_of_zero, contMDiff_finsum_smul, toPartitionOfUnity, coe_finsupport, contDiffAt_finsum, contMDiffAt_finsum, contMDiff_finsum_smul, contMDiff_smul, contMDiff_sum, eventually_finsupport_subset, eventually_fintsupport_subset, exists_isSubordinate, exists_isSubordinate_chartAt_source, exists_isSubordinate_chartAt_source_of_isClosed, exists_pos_of_mem, finite_tsupport, finsum_smul_mem_convex, finsupport_subset_fintsupport, isSubordinate_toPartitionOfUnity, le_one, locallyFinite, locallyFinite', mem_finsupport, mem_fintsupport_iff, nonneg, nonneg', sum_eq_one, sum_eq_one', sum_finsupport, sum_finsupport', sum_finsupport_smul_eq_finsum, sum_le_one, sum_le_one', sum_nonneg, toPartitionOfUnity_toFun, exists_contMDiffMap_forall_mem_convex_of_local, exists_contMDiffMap_forall_mem_convex_of_local_const, exists_contMDiffMap_one_nhds_of_subset_interior, exists_contMDiffMap_zero_one_nhds_of_isClosed, exists_contMDiffMap_zero_one_of_isClosed, exists_contMDiffOn_forall_mem_convex_of_local, exists_contMDiffOn_section_forall_mem_convex_of_local, exists_contMDiffSection_forall_mem_convex_of_local, exists_contMDiff_support_eq_eq_one_iff, exists_contMDiff_zero_iff_one_iff_of_isClosed, exists_msmooth_support_eq_eq_one_iff, exists_msmooth_zero_iff_one_iff_of_isClosed, exists_smooth_forall_mem_convex_of_local, exists_smooth_forall_mem_convex_of_local_const, exists_smooth_one_nhds_of_subset_interior, exists_smooth_section_forall_mem_convex_of_local, exists_smooth_zero_one_nhds_of_isClosed, exists_smooth_zero_one_of_isClosed
88
Total107

BumpCovering

Definitions

NameCategoryTheorems
toSmoothPartitionOfUnity πŸ“–CompOp
3 mathmath: IsSubordinate.toSmoothPartitionOfUnity, toSmoothPartitionOfUnity_toPartitionOfUnity, coe_toSmoothPartitionOfUnity

Theorems

NameKindAssumesProvesValidatesDepends On
coe_toSmoothPartitionOfUnity πŸ“–mathematicalContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
DFunLike.coe
ContinuousMap
ContinuousMap.instFunLike
BumpCovering
instFunLikeContinuousMapReal
ContMDiffMap
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
PartitionOfUnity
PartitionOfUnity.instFunLikeContinuousMapReal
toPartitionOfUnity
β€”β€”
contMDiff_toPartitionOfUnity πŸ“–mathematicalContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
DFunLike.coe
ContinuousMap
ContinuousMap.instFunLike
BumpCovering
instFunLikeContinuousMapReal
PartitionOfUnity
PartitionOfUnity.instFunLikeContinuousMapReal
toPartitionOfUnity
β€”ContMDiff.mul
instContMDiffMulOfTopWithTopENat
ContMDiffRing.toContMDiffMul
instFieldContMDiffRing
contMDiff_finprod_cond
ContMDiff.sub
instNormedSpaceLieAddGroup
contMDiff_const
Function.mulSupport_one_sub
locallyFinite
toSmoothPartitionOfUnity_toPartitionOfUnity πŸ“–mathematicalContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
DFunLike.coe
ContinuousMap
ContinuousMap.instFunLike
BumpCovering
instFunLikeContinuousMapReal
SmoothPartitionOfUnity.toPartitionOfUnity
toSmoothPartitionOfUnity
toPartitionOfUnity
β€”β€”

BumpCovering.IsSubordinate

Theorems

NameKindAssumesProvesValidatesDepends On
toSmoothPartitionOfUnity πŸ“–mathematicalBumpCovering.IsSubordinate
ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
DFunLike.coe
ContinuousMap
ContinuousMap.instFunLike
BumpCovering
BumpCovering.instFunLikeContinuousMapReal
SmoothPartitionOfUnity.IsSubordinate
BumpCovering.toSmoothPartitionOfUnity
β€”toPartitionOfUnity

Emetric

Theorems

NameKindAssumesProvesValidatesDepends On
exists_contMDiffMap_forall_closedBall_subset πŸ“–mathematicalIsClosed
UniformSpace.toTopologicalSpace
PseudoEMetricSpace.toUniformSpace
EMetricSpace.toPseudoEMetricSpace
IsOpen
Set
Set.instHasSubset
LocallyFinite
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Metric.closedEBall
ENNReal.ofReal
β€”Metric.exists_contMDiffMap_forall_closedEBall_subset
exists_smooth_forall_closedBall_subset πŸ“–mathematicalIsClosed
UniformSpace.toTopologicalSpace
PseudoEMetricSpace.toUniformSpace
EMetricSpace.toPseudoEMetricSpace
IsOpen
Set
Set.instHasSubset
LocallyFinite
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Metric.closedEBall
ENNReal.ofReal
β€”Metric.exists_contMDiffMap_forall_closedEBall_subset

IsOpen

Theorems

NameKindAssumesProvesValidatesDepends On
exists_contMDiff_support_eq πŸ“–mathematicalIsOpenFunction.support
Real
Real.instZero
ContMDiff
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Real.instLE
β€”SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source
exists_contMDiff_support_eq_aux
OpenPartialHomeomorph.isOpen_inter_preimage_symm
Set.mem_range_self
mul_nonneg
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
SmoothPartitionOfUnity.nonneg
Function.support_eq_iff
ne_of_gt
SmoothPartitionOfUnity.exists_pos_of_mem
Set.mem_univ
mul_pos
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
lt_of_le_of_ne
Function.mem_support
Set.mem_preimage
Set.preimage_inter
subset_tsupport
OpenPartialHomeomorph.map_source
OpenPartialHomeomorph.left_inv
finsum_pos
Real.instIsOrderedCancelAddMonoid
Set.Finite.subset
LocallyFinite.point_finite
SmoothPartitionOfUnity.locallyFinite
Set.compl_subset_compl
NormMulClass.toNoZeroDivisors
NormedDivisionRing.toNormMulClass
finsum_eq_zero_of_forall_eq_zero
Function.notMem_support
Mathlib.Tactic.Contrapose.contraposeβ‚„
MulZeroClass.mul_zero
MulZeroClass.zero_mul
SmoothPartitionOfUnity.contMDiff_finsum_smul
ContMDiffAt.comp
contMDiffAt_of_mem_maximalAtlas
IsManifold.instOfSomeENatTopOfLEInfty
IsManifold.instLEInftySomeENat
IsManifold.chart_mem_maximalAtlas
finsum_nonneg
Real.instIsOrderedAddMonoid
exists_contMDiff_support_eq_aux πŸ“–mathematicalIsOpenFunction.support
Real
Real.instZero
ContMDiff
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
chartedSpaceSelf
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
WithTop.some
ENat
Set
Set.instHasSubset
Set.range
Set.Icc
Real.instPreorder
Real.instOne
β€”Continuous.isOpen_preimage
ModelWithCorners.continuous_symm
exists_contDiff_support_eq
Function.support_comp_eq_preimage
Set.preimage_comp
ModelWithCorners.symm_comp_self
ContDiff.comp_contMDiff
contMDiff_model
Set.Subset.trans
Set.range_comp_subset_range
exists_msmooth_support_eq πŸ“–mathematicalIsOpenFunction.support
Real
Real.instZero
ContMDiff
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Real.instLE
β€”exists_contMDiff_support_eq
exists_msmooth_support_eq_aux πŸ“–mathematicalIsOpenFunction.support
Real
Real.instZero
ContMDiff
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
chartedSpaceSelf
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
WithTop.some
ENat
Set
Set.instHasSubset
Set.range
Set.Icc
Real.instPreorder
Real.instOne
β€”exists_contMDiff_support_eq_aux

Metric

Theorems

NameKindAssumesProvesValidatesDepends On
exists_contMDiffMap_forall_closedBall_subset πŸ“–mathematicalIsClosed
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
MetricSpace.toPseudoMetricSpace
IsOpen
Set
Set.instHasSubset
LocallyFinite
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
closedBall
β€”exists_contMDiffMap_forall_closedEBall_subset
closedEBall_ofReal
LT.lt.le
exists_contMDiffMap_forall_closedEBall_subset πŸ“–mathematicalIsClosed
UniformSpace.toTopologicalSpace
PseudoEMetricSpace.toUniformSpace
EMetricSpace.toPseudoEMetricSpace
IsOpen
Set
Set.instHasSubset
LocallyFinite
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
closedEBall
ENNReal.ofReal
β€”forall_swap
exists_contMDiffMap_forall_mem_convex_of_local_const
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
exists_forall_closedEBall_subset_auxβ‚‚
exists_forall_closedEBall_subset_aux₁
exists_smooth_forall_closedBall_subset πŸ“–mathematicalIsClosed
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
MetricSpace.toPseudoMetricSpace
IsOpen
Set
Set.instHasSubset
LocallyFinite
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
closedBall
β€”exists_contMDiffMap_forall_closedBall_subset

SmoothBumpCovering

Definitions

NameCategoryTheorems
IsSubordinate πŸ“–MathDef
2 mathmath: exists_isSubordinate, isSubordinate_toBumpCovering
c πŸ“–CompOp
21 mathmath: toSmoothPartitionOfUnity_eq_mul_prod, mem_support_ind, IsSubordinate.support_subset, toSmoothPartitionOfUnity_apply, locallyFinite', IsSubordinate.toBumpCovering, sum_toSmoothPartitionOfUnity_eq, point_finite, mem_chartAt_ind_source, eventuallyEq_one, exists_finset_toSmoothPartitionOfUnity_eventuallyEq, c_mem', mem_extChartAt_ind_source, comp_embeddingPiTangent_mfderiv, locallyFinite, eventuallyEq_one', apply_ind, support_toSmoothPartitionOfUnity_subset, IsSubordinate.toSmoothPartitionOfUnity, isSubordinate_toBumpCovering, embeddingPiTangent_coe
fintype πŸ“–CompOpβ€”
ind πŸ“–CompOp
6 mathmath: mem_support_ind, mem_chartAt_ind_source, eventuallyEq_one, mem_extChartAt_ind_source, comp_embeddingPiTangent_mfderiv, apply_ind
instCoeFunForallSmoothBumpFunctionC πŸ“–CompOpβ€”
toBumpCovering πŸ“–CompOp
2 mathmath: IsSubordinate.toBumpCovering, isSubordinate_toBumpCovering
toFun πŸ“–CompOp
14 mathmath: toSmoothPartitionOfUnity_eq_mul_prod, mem_support_ind, IsSubordinate.support_subset, toSmoothPartitionOfUnity_apply, locallyFinite', sum_toSmoothPartitionOfUnity_eq, point_finite, eventuallyEq_one, exists_finset_toSmoothPartitionOfUnity_eventuallyEq, locallyFinite, eventuallyEq_one', apply_ind, support_toSmoothPartitionOfUnity_subset, embeddingPiTangent_coe
toSmoothPartitionOfUnity πŸ“–CompOp
7 mathmath: toSmoothPartitionOfUnity_eq_mul_prod, toSmoothPartitionOfUnity_apply, sum_toSmoothPartitionOfUnity_eq, exists_finset_toSmoothPartitionOfUnity_eventuallyEq, support_toSmoothPartitionOfUnity_subset, IsSubordinate.toSmoothPartitionOfUnity, toSmoothPartitionOfUnity_zero_of_zero

Theorems

NameKindAssumesProvesValidatesDepends On
apply_ind πŸ“–mathematicalSet
Set.instMembership
SmoothBumpFunction.toFun
c
ind
toFun
Real
Real.instOne
β€”Filter.EventuallyEq.eq_of_nhds
eventuallyEq_one
c_mem' πŸ“–mathematicalβ€”Set
Set.univ
Set.instMembership
c
β€”β€”
eventuallyEq_one πŸ“–mathematicalSet
Set.instMembership
Filter.EventuallyEq
Real
nhds
SmoothBumpFunction.toFun
c
ind
toFun
Pi.instOne
Real.instOne
β€”eventuallyEq_one'
eventuallyEq_one' πŸ“–mathematicalSet
Set.univ
Set.instMembership
Filter.EventuallyEq
Real
nhds
SmoothBumpFunction.toFun
c
toFun
Pi.instOne
Real.instOne
β€”β€”
exists_finset_toSmoothPartitionOfUnity_eventuallyEq πŸ“–mathematicalβ€”Filter.EventuallyEq
Real
nhds
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
Pi.instMul
Real.instMul
SmoothBumpFunction.toFun
c
toFun
Finset.prod
Pi.commMonoid
Real.instCommMonoid
Finset.filter
WellOrderingRel
Pi.instSub
Real.instSub
Pi.instOne
Real.instOne
β€”IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
IsTopologicalAddGroup.to_continuousSub
instIsTopologicalAddGroupReal
ContinuousMap.coe_prod
Finset.prod_congr
BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
exists_isSubordinate πŸ“–mathematicalSet
Filter
Filter.instMembership
nhds
IsSubordinateβ€”SmoothBumpFunction.nhds_basis_support
refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
instWeaklyLocallyCompactSpaceOfLocallyCompactSpace
ChartedSpace.locallyCompactSpace
ModelWithCorners.locallyCompactSpace
locallyCompact_of_proper
FiniteDimensional.proper_real
exists_subset_iUnion_closed_subset
T4Space.toNormalSpace
T4Space.of_paracompactSpace_t2Space
paracompact_of_locallyCompact_sigmaCompact
SmoothBumpFunction.isOpen_support
LocallyFinite.point_finite
SmoothBumpFunction.support_updateRIn
SmoothBumpFunction.eventuallyEq_one_of_dist_lt
SmoothBumpFunction.support_subset_source
Set.mem_iUnion
SmoothBumpFunction.exists_r_pos_lt_subset_ball
isSubordinate_toBumpCovering πŸ“–mathematicalβ€”BumpCovering.IsSubordinate
toBumpCovering
c
IsSubordinate
β€”β€”
locallyFinite πŸ“–mathematicalβ€”LocallyFinite
Function.support
Real
Real.instZero
SmoothBumpFunction.toFun
c
toFun
β€”locallyFinite'
locallyFinite' πŸ“–mathematicalβ€”LocallyFinite
Function.support
Real
Real.instZero
SmoothBumpFunction.toFun
c
toFun
β€”β€”
mem_chartAt_ind_source πŸ“–mathematicalSet
Set.instMembership
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
chartAt
c
ind
β€”mem_chartAt_source_of_eq_one
apply_ind
mem_chartAt_source_of_eq_one πŸ“–mathematicalSmoothBumpFunction.toFun
c
toFun
Real
Real.instOne
Set
Set.instMembership
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
chartAt
β€”SmoothBumpFunction.support_subset_source
NeZero.charZero_one
RCLike.charZero_rclike
mem_extChartAt_ind_source πŸ“–mathematicalSet
Set.instMembership
PartialEquiv.source
extChartAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
c
ind
β€”mem_extChartAt_source_of_eq_one
apply_ind
mem_extChartAt_source_of_eq_one πŸ“–mathematicalSmoothBumpFunction.toFun
c
toFun
Real
Real.instOne
Set
Set.instMembership
PartialEquiv.source
extChartAt
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
β€”extChartAt_source
mem_chartAt_source_of_eq_one
mem_support_ind πŸ“–mathematicalSet
Set.instMembership
Function.support
Real
Real.instZero
SmoothBumpFunction.toFun
c
ind
toFun
β€”apply_ind
NeZero.charZero_one
RCLike.charZero_rclike
point_finite πŸ“–mathematicalβ€”Set.Finite
setOf
Real
SmoothBumpFunction.toFun
c
toFun
Real.instZero
β€”LocallyFinite.point_finite
locallyFinite
sum_toSmoothPartitionOfUnity_eq πŸ“–mathematicalβ€”finsum
Real
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
Real.instSub
Real.instOne
finprod
Real.instCommMonoid
SmoothBumpFunction.toFun
c
toFun
β€”BumpCovering.sum_toPartitionOfUnity_eq
support_toSmoothPartitionOfUnity_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Function.support
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
SmoothBumpFunction.toFun
c
toFun
β€”BumpCovering.support_toPartitionOfUnity_subset
toSmoothPartitionOfUnity_apply πŸ“–mathematicalβ€”DFunLike.coe
ContMDiffMap
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
Real.instMul
SmoothBumpFunction.toFun
c
toFun
finprod
Real.instCommMonoid
WellOrderingRel
Real.instSub
Real.instOne
β€”β€”
toSmoothPartitionOfUnity_eq_mul_prod πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
ContMDiffMap
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
Real.instMul
SmoothBumpFunction.toFun
c
toFun
Finset.prod
Real.instCommMonoid
Finset.filter
WellOrderingRel
Real.instSub
Real.instOne
β€”BumpCovering.toPartitionOfUnity_eq_mul_prod
toSmoothPartitionOfUnity_zero_of_zero πŸ“–mathematicalSmoothBumpFunction.toFun
c
toFun
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
toSmoothPartitionOfUnity
β€”BumpCovering.toPartitionOfUnity_zero_of_zero

SmoothBumpCovering.IsSubordinate

Theorems

NameKindAssumesProvesValidatesDepends On
support_subset πŸ“–mathematicalSmoothBumpCovering.IsSubordinateSet
Set.instHasSubset
Function.support
Real
Real.instZero
SmoothBumpFunction.toFun
SmoothBumpCovering.c
SmoothBumpCovering.toFun
β€”Set.Subset.trans
subset_closure
toBumpCovering πŸ“–mathematicalSmoothBumpCovering.IsSubordinateBumpCovering.IsSubordinate
SmoothBumpCovering.toBumpCovering
SmoothBumpCovering.c
β€”SmoothBumpCovering.isSubordinate_toBumpCovering
toSmoothPartitionOfUnity πŸ“–mathematicalSmoothBumpCovering.IsSubordinateSmoothPartitionOfUnity.IsSubordinate
SmoothBumpCovering.toSmoothPartitionOfUnity
SmoothBumpCovering.c
β€”BumpCovering.IsSubordinate.toPartitionOfUnity
toBumpCovering

SmoothPartitionOfUnity

Definitions

NameCategoryTheorems
IsSubordinate πŸ“–MathDef
6 mathmath: exists_isSubordinate, exists_isSubordinate_chartAt_source, BumpCovering.IsSubordinate.toSmoothPartitionOfUnity, exists_isSubordinate_chartAt_source_of_isClosed, SmoothBumpCovering.IsSubordinate.toSmoothPartitionOfUnity, isSubordinate_toPartitionOfUnity
finsupport πŸ“–CompOp
6 mathmath: mem_finsupport, coe_finsupport, finsupport_subset_fintsupport, sum_finsupport_smul_eq_finsum, sum_finsupport, eventually_finsupport_subset
fintsupport πŸ“–CompOp
4 mathmath: eventually_fintsupport_subset, finsupport_subset_fintsupport, mem_fintsupport_iff, eventually_finsupport_subset
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop πŸ“–CompOp
29 mathmath: SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod, contMDiffAt_finsum, locallyFinite, nonneg, sum_finsupport', SmoothBumpCovering.toSmoothPartitionOfUnity_apply, contDiffAt_finsum, le_one, SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq, mem_finsupport, coe_finsupport, finsum_smul_mem_convex, sum_nonneg, SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq, sum_eq_one, exists_pos_of_mem, sum_finsupport_smul_eq_finsum, contMDiff_finsum_smul, sum_finsupport, SmoothBumpCovering.support_toSmoothPartitionOfUnity_subset, finite_tsupport, BumpCovering.coe_toSmoothPartitionOfUnity, mem_fintsupport_iff, toPartitionOfUnity_toFun, contMDiff_smul, IsSubordinate.contMDiff_finsum_smul, sum_le_one, SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero, contMDiff_sum
instInhabited πŸ“–CompOpβ€”
single πŸ“–CompOpβ€”
toFun πŸ“–CompOp
4 mathmath: locallyFinite', sum_eq_one', sum_le_one', nonneg'
toPartitionOfUnity πŸ“–CompOp
4 mathmath: IsSubordinate.toPartitionOfUnity, BumpCovering.toSmoothPartitionOfUnity_toPartitionOfUnity, isSubordinate_toPartitionOfUnity, toPartitionOfUnity_toFun

Theorems

NameKindAssumesProvesValidatesDepends On
coe_finsupport πŸ“–mathematicalβ€”SetLike.coe
Finset
Finset.instSetLike
finsupport
Function.support
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”PartitionOfUnity.coe_finsupport
contDiffAt_finsum πŸ“–mathematicalContDiffAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
WithTop.some
ENat
finsum
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
NormedSpace.toModule
Real.normedField
DFunLike.coe
ContMDiffMap
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
modelWithCornersSelf
chartedSpaceSelf
Real.pseudoMetricSpace
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”contMDiffAt_finsum
contMDiffAt_finsum πŸ“–mathematicalContMDiffAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
finsum
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
NormedSpace.toModule
Real.normedField
DFunLike.coe
ContMDiffMap
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
Real.pseudoMetricSpace
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”contMDiffAt_finsum
instContMDiffAddSelf
LocallyFinite.smul_left
locallyFinite
ContMDiffAt.smul
ContMDiff.contMDiffAt
ContMDiff.of_le
ContMDiffMap.contMDiff
le_top
contMDiffAt_of_notMem
Set.compl_subset_compl
tsupport_smul_subset_left
contMDiff_finsum_smul πŸ“–mathematicalContMDiffAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
ContMDiff
finsum
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
NormedSpace.toModule
Real.normedField
DFunLike.coe
ContMDiffMap
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
Real.pseudoMetricSpace
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”contMDiff_finsum
instContMDiffAddSelf
contMDiff_smul
LocallyFinite.subset
locallyFinite
Function.support_smul_subset_left
contMDiff_smul πŸ“–mathematicalContMDiffAt
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
ContMDiff
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
ESeminormedAddCommMonoid.toAddCommMonoid
NormedSpace.toModule
Real.normedField
DFunLike.coe
ContMDiffMap
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
Real.pseudoMetricSpace
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”contMDiff_of_tsupport
ContMDiffAt.smul
ContMDiffAt.of_le
ContMDiff.contMDiffAt
ContMDiffMap.contMDiff
le_top
tsupport_smul_subset_left
contMDiff_sum πŸ“–mathematicalβ€”ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
finsum
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”contMDiff_finsum
instContMDiffAddSelf
ContMDiffMap.contMDiff
locallyFinite
eventually_finsupport_subset πŸ“–mathematicalβ€”Filter.Eventually
Finset
Finset.instHasSubset
finsupport
fintsupport
nhds
β€”PartitionOfUnity.eventually_finsupport_subset
eventually_fintsupport_subset πŸ“–mathematicalβ€”Filter.Eventually
Finset
Finset.instHasSubset
fintsupport
nhds
β€”PartitionOfUnity.eventually_fintsupport_subset
exists_isSubordinate πŸ“–mathematicalIsOpen
Set
Set.instHasSubset
Set.iUnion
IsSubordinateβ€”BumpCovering.exists_isSubordinate_of_prop
T4Space.toNormalSpace
T4Space.of_paracompactSpace_t2Space
paracompact_of_locallyCompact_sigmaCompact
instWeaklyLocallyCompactSpaceOfLocallyCompactSpace
ChartedSpace.locallyCompactSpace
ModelWithCorners.locallyCompactSpace
locallyCompact_of_proper
FiniteDimensional.proper_real
exists_contMDiffMap_zero_one_of_isClosed
ContMDiffMap.instContinuousMapClass
ContMDiffMap.contMDiff
BumpCovering.IsSubordinate.toSmoothPartitionOfUnity
exists_isSubordinate_chartAt_source πŸ“–mathematicalβ€”IsSubordinate
Set.univ
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
chartAt
β€”exists_isSubordinate
isClosed_univ
OpenPartialHomeomorph.open_source
Set.mem_iUnion_of_mem
mem_chart_source
exists_isSubordinate_chartAt_source_of_isClosed πŸ“–mathematicalβ€”IsSubordinate
Set.Elem
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
chartAt
Set
Set.instMembership
β€”exists_isSubordinate
OpenPartialHomeomorph.open_source
Set.mem_iUnion_of_mem
mem_chart_source
exists_pos_of_mem πŸ“–mathematicalSet
Set.instMembership
Real
Real.instLT
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”le_antisymm
nonneg
sum_eq_one
finsum_zero
NeZero.charZero_one
RCLike.charZero_rclike
finite_tsupport πŸ“–mathematicalβ€”Set.Finite
setOf
Set
Set.instMembership
tsupport
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”PartitionOfUnity.finite_tsupport
finsum_smul_mem_convex πŸ“–mathematicalSet
Set.instMembership
Convex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
finsum
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”Convex.finsum_mem
Real.instIsStrictOrderedRing
nonneg
sum_eq_one
finsupport_subset_fintsupport πŸ“–mathematicalβ€”Finset
Finset.instHasSubset
finsupport
fintsupport
β€”PartitionOfUnity.finsupport_subset_fintsupport
isSubordinate_toPartitionOfUnity πŸ“–mathematicalβ€”PartitionOfUnity.IsSubordinate
toPartitionOfUnity
IsSubordinate
β€”β€”
le_one πŸ“–mathematicalβ€”Real
Real.instLE
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
Real.instOne
β€”PartitionOfUnity.le_one
locallyFinite πŸ“–mathematicalβ€”LocallyFinite
Function.support
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”locallyFinite'
locallyFinite' πŸ“–mathematicalβ€”LocallyFinite
Function.support
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
toFun
β€”β€”
mem_finsupport πŸ“–mathematicalβ€”Finset
Finset.instMembership
finsupport
Set
Set.instMembership
Function.support
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”PartitionOfUnity.mem_finsupport
mem_fintsupport_iff πŸ“–mathematicalβ€”Finset
Finset.instMembership
fintsupport
Set
Set.instMembership
tsupport
Real
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”Set.Finite.mem_toFinset
finite_tsupport
nonneg πŸ“–mathematicalβ€”Real
Real.instLE
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”nonneg'
nonneg' πŸ“–mathematicalβ€”Real
Real.instLE
Real.instZero
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
toFun
β€”β€”
sum_eq_one πŸ“–mathematicalSet
Set.instMembership
finsum
Real
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
Real.instOne
β€”sum_eq_one'
sum_eq_one' πŸ“–mathematicalSet
Set.univ
Set.instMembership
finsum
Real
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
toFun
Real.instOne
β€”β€”
sum_finsupport πŸ“–mathematicalSet
Set.instMembership
Finset.sum
Real
Real.instAddCommMonoid
finsupport
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
Real.instOne
β€”PartitionOfUnity.sum_finsupport
sum_finsupport' πŸ“–mathematicalSet
Set.instMembership
Finset
Finset.instHasSubset
finsupport
Finset.sum
Real
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
Real.instOne
β€”PartitionOfUnity.sum_finsupport'
sum_finsupport_smul_eq_finsum πŸ“–mathematicalβ€”Finset.sum
AddCommGroup.toAddCommMonoid
finsupport
Real
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
finsum
β€”PartitionOfUnity.sum_finsupport_smul_eq_finsum
sum_le_one πŸ“–mathematicalβ€”Real
Real.instLE
finsum
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
Real.instOne
β€”sum_le_one'
sum_le_one' πŸ“–mathematicalβ€”Real
Real.instLE
finsum
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
toFun
Real.instOne
β€”β€”
sum_nonneg πŸ“–mathematicalβ€”Real
Real.instLE
Real.instZero
finsum
Real.instAddCommMonoid
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”PartitionOfUnity.sum_nonneg
toPartitionOfUnity_toFun πŸ“–mathematicalβ€”DFunLike.coe
PartitionOfUnity
ContinuousMap
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
PartitionOfUnity.instFunLikeContinuousMapReal
toPartitionOfUnity
toContinuousMap
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”β€”

SmoothPartitionOfUnity.IsSubordinate

Theorems

NameKindAssumesProvesValidatesDepends On
contMDiff_finsum_smul πŸ“–mathematicalSmoothPartitionOfUnity.IsSubordinate
IsOpen
ContMDiffOn
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
ContMDiff
finsum
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Real.semiring
NormedSpace.toModule
Real.normedField
DFunLike.coe
ContMDiffMap
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
RCLike.toInnerProductSpaceReal
Real.pseudoMetricSpace
Top.top
instTopENat
ContMDiffMap.instFunLike
SmoothPartitionOfUnity
SmoothPartitionOfUnity.instFunLikeContMDiffMapRealModelWithCornersSelfSomeENatTop
β€”SmoothPartitionOfUnity.contMDiff_finsum_smul
ContMDiffOn.contMDiffAt
IsOpen.mem_nhds
toPartitionOfUnity πŸ“–mathematicalSmoothPartitionOfUnity.IsSubordinatePartitionOfUnity.IsSubordinate
SmoothPartitionOfUnity.toPartitionOfUnity
β€”SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity

(root)

Definitions

NameCategoryTheorems
SmoothBumpCovering πŸ“–CompDataβ€”
SmoothPartitionOfUnity πŸ“–CompData
29 mathmath: SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod, SmoothPartitionOfUnity.contMDiffAt_finsum, SmoothPartitionOfUnity.locallyFinite, SmoothPartitionOfUnity.nonneg, SmoothPartitionOfUnity.sum_finsupport', SmoothBumpCovering.toSmoothPartitionOfUnity_apply, SmoothPartitionOfUnity.contDiffAt_finsum, SmoothPartitionOfUnity.le_one, SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq, SmoothPartitionOfUnity.mem_finsupport, SmoothPartitionOfUnity.coe_finsupport, SmoothPartitionOfUnity.finsum_smul_mem_convex, SmoothPartitionOfUnity.sum_nonneg, SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq, SmoothPartitionOfUnity.sum_eq_one, SmoothPartitionOfUnity.exists_pos_of_mem, SmoothPartitionOfUnity.sum_finsupport_smul_eq_finsum, SmoothPartitionOfUnity.contMDiff_finsum_smul, SmoothPartitionOfUnity.sum_finsupport, SmoothBumpCovering.support_toSmoothPartitionOfUnity_subset, SmoothPartitionOfUnity.finite_tsupport, BumpCovering.coe_toSmoothPartitionOfUnity, SmoothPartitionOfUnity.mem_fintsupport_iff, SmoothPartitionOfUnity.toPartitionOfUnity_toFun, SmoothPartitionOfUnity.contMDiff_smul, SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul, SmoothPartitionOfUnity.sum_le_one, SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero, SmoothPartitionOfUnity.contMDiff_sum

Theorems

NameKindAssumesProvesValidatesDepends On
exists_contMDiffMap_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
Set.instMembership
DFunLike.coe
ContMDiffMap
ContMDiffMap.instFunLike
β€”exists_contMDiffSection_forall_mem_convex_of_local
Bundle.Trivial.vectorBundle
Bundle.contMDiffWithinAt_section
Bundle.contMDiffAt_section
ContMDiffSection.contMDiff
exists_contMDiffMap_forall_mem_convex_of_local_const πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
Filter.Eventually
Set
Set.instMembership
nhds
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
β€”exists_contMDiffMap_forall_mem_convex_of_local
contMDiffOn_const
exists_contMDiffMap_one_nhds_of_subset_interior πŸ“–mathematicalSet
Set.instHasSubset
interior
Filter.Eventually
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Real.instOne
nhdsSet
Real.instZero
Set.instMembership
Set.Icc
Real.instPreorder
β€”exists_contMDiffMap_zero_one_nhds_of_isClosed
IsOpen.isClosed_compl
isOpen_interior
Set.subset_compl_iff_disjoint_left
compl_compl
Filter.Eventually.self_of_nhdsSet
interior_subset
exists_contMDiffMap_zero_one_nhds_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Filter.Eventually
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Real.instZero
nhdsSet
Real.instOne
Set.instMembership
Set.Icc
Real.instPreorder
β€”normal_exists_closure_subset
IsClosed.isOpen_compl
Set.subset_compl_iff_disjoint_left
Disjoint.symm
isClosed_closure
Set.subset_compl_comm
exists_contMDiffMap_zero_one_of_isClosed
Filter.eventually_of_mem
Filter.mem_of_superset
IsOpen.mem_nhdsSet
subset_closure
exists_contMDiffMap_zero_one_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.EqOn
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Pi.instZero
Real.instZero
Pi.instOne
Real.instOne
Set.instMembership
Set.Icc
Real.instPreorder
β€”IsOpen.mem_nhds
IsClosed.isOpen_compl
Set.disjoint_right
SmoothBumpCovering.exists_isSubordinate
ContMDiff.of_le
SmoothPartitionOfUnity.contMDiff_sum
SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero
Function.notMem_support
Set.subset_compl_comm
SmoothBumpCovering.IsSubordinate.support_subset
finsum_zero
SmoothPartitionOfUnity.sum_eq_one
SmoothPartitionOfUnity.sum_nonneg
SmoothPartitionOfUnity.sum_le_one
exists_contMDiffOn_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
Set.instMembership
DFunLike.coe
ContMDiffMap
ContMDiffMap.instFunLike
β€”exists_contMDiffMap_forall_mem_convex_of_local
exists_contMDiffOn_section_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Prod.normedAddCommGroup
Prod.normedSpace
NontriviallyNormedField.toNormedField
NormedAddCommGroup.toSeminormedAddCommGroup
ModelProd
instTopologicalSpaceModelProd
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
ModelWithCorners.prod
modelWithCornersSelf
Bundle.TotalSpace
FiberBundle.chartedSpace
WithTop.some
ENat
Bundle.TotalSpace.mk'
Set.instMembership
DFunLike.coe
ContMDiffSection
ContMDiffSection.instDFunLike
β€”exists_contMDiffSection_forall_mem_convex_of_local
exists_contMDiffSection_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Prod.normedAddCommGroup
Prod.normedSpace
NontriviallyNormedField.toNormedField
NormedAddCommGroup.toSeminormedAddCommGroup
ModelProd
instTopologicalSpaceModelProd
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
ModelWithCorners.prod
modelWithCornersSelf
Bundle.TotalSpace
FiberBundle.chartedSpace
WithTop.some
ENat
Bundle.TotalSpace.mk'
Set.instMembership
DFunLike.coe
ContMDiffSection
ContMDiffSection.instDFunLike
β€”mem_interior_iff_mem_nhds
SmoothPartitionOfUnity.exists_isSubordinate
isClosed_univ
isOpen_interior
ContMDiffOn.smul_section_of_tsupport
ContMDiff.contMDiffOn
ContMDiff.of_le
ContMDiffMap.contMDiff
sup_eq_left
ContMDiffOn.mono
interior_subset
ContMDiff.finsum_section_of_locallyFinite
LocallyFinite.subset
SmoothPartitionOfUnity.locallyFinite
Function.support.eq_1
Set.mem_setOf_eq
left_ne_zero_of_smul
Convex.finsum_mem
Real.instIsStrictOrderedRing
SmoothPartitionOfUnity.nonneg
SmoothPartitionOfUnity.sum_eq_one
Set.mem_univ
subset_closure
Function.mem_support
exists_contMDiff_support_eq_eq_one_iff πŸ“–mathematicalIsOpen
Set
Set.instHasSubset
ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Set.range
Set.Icc
Real.instPreorder
Real.instZero
Real.instOne
Function.support
Set.instMembership
β€”IsOpen.exists_contMDiff_support_eq
IsClosed.isOpen_compl
lt_of_le_of_ne
lt_of_not_ge
Mathlib.Tactic.Linarith.lt_irrefl
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.sub_congr
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Ring.atom_pf
Mathlib.Tactic.Ring.sub_pf
Mathlib.Tactic.Ring.neg_add
Mathlib.Tactic.Ring.neg_mul
Mathlib.Tactic.Ring.neg_one_mul
Mathlib.Meta.NormNum.IsInt.to_raw_eq
Mathlib.Meta.NormNum.isInt_mul
Mathlib.Meta.NormNum.IsInt.of_raw
Mathlib.Meta.NormNum.IsNat.to_isInt
Mathlib.Meta.NormNum.IsNat.of_raw
Mathlib.Tactic.Ring.neg_zero
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_add_zero
Mathlib.Tactic.Ring.add_pf_add_overlap_zero
Mathlib.Tactic.Ring.add_overlap_pf_zero
Mathlib.Meta.NormNum.IsInt.to_isNat
Mathlib.Meta.NormNum.isInt_add
Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
Real.instIsStrictOrderedRing
Mathlib.Tactic.Linarith.sub_neg_of_lt
Real.instIsOrderedRing
Mathlib.Tactic.Linarith.sub_nonpos_of_le
Function.mem_support
Mathlib.Tactic.Contrapose.contraposeβ‚‚
HasSubset.Subset.trans
Set.instIsTransSubset
Eq.subset
Set.instReflSubset
Mathlib.Tactic.Ring.add_pf_add_gt
ContMDiff.divβ‚€
instContMDiffInvβ‚€ModelWithCornersSelf
instContMDiffMulOfTopWithTopENat
ContMDiffRing.toContMDiffMul
instFieldContMDiffRing
ContMDiff.add
instContMDiffAddSelf
ne_of_gt
Set.range_subset_iff
div_nonneg
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
LT.lt.le
div_le_one_of_leβ‚€
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instZeroLEOneClass
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
Set.eq_univ_of_forall
LT.lt.ne'
Function.support_div
Set.inter_univ
div_eq_one_iff_eq
exists_contMDiff_zero_iff_one_iff_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Set.instHasSubset
Set.range
Set.Icc
Real.instPreorder
Real.instZero
Real.instOne
Set.instMembership
β€”exists_contMDiff_support_eq_eq_one_iff
IsClosed.isOpen_compl
Disjoint.subset_compl_left
exists_msmooth_support_eq_eq_one_iff πŸ“–mathematicalIsOpen
Set
Set.instHasSubset
ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Set.range
Set.Icc
Real.instPreorder
Real.instZero
Real.instOne
Function.support
Set.instMembership
β€”exists_contMDiff_support_eq_eq_one_iff
exists_msmooth_zero_iff_one_iff_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
ContMDiff
Real
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
Set.instHasSubset
Set.range
Set.Icc
Real.instPreorder
Real.instZero
Real.instOne
Set.instMembership
β€”exists_contMDiff_zero_iff_one_iff_of_isClosed
exists_smooth_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
Set.instMembership
DFunLike.coe
ContMDiffMap
ContMDiffMap.instFunLike
β€”exists_contMDiffMap_forall_mem_convex_of_local
exists_smooth_forall_mem_convex_of_local_const πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
ESeminormedAddCommMonoid.toAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
ESeminormedAddMonoid.toAddMonoid
ESeminormedAddCommMonoid.toESeminormedAddMonoid
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
NormedSpace.toModule
Real.normedField
Filter.Eventually
Set
Set.instMembership
nhds
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
modelWithCornersSelf
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
β€”exists_contMDiffMap_forall_mem_convex_of_local_const
exists_smooth_one_nhds_of_subset_interior πŸ“–mathematicalSet
Set.instHasSubset
interior
Filter.Eventually
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Real.instOne
nhdsSet
Real.instZero
Set.instMembership
Set.Icc
Real.instPreorder
β€”exists_contMDiffMap_one_nhds_of_subset_interior
exists_smooth_section_forall_mem_convex_of_local πŸ“–mathematicalConvex
Real
Real.semiring
Real.partialOrder
AddCommGroup.toAddCommMonoid
SMulZeroClass.toSMul
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
DistribSMul.toSMulZeroClass
DistribMulAction.toDistribSMul
Real.instMonoid
Module.toDistribMulAction
Set
Filter
Filter.instMembership
nhds
ContMDiffOn
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Prod.normedAddCommGroup
Prod.normedSpace
NontriviallyNormedField.toNormedField
NormedAddCommGroup.toSeminormedAddCommGroup
ModelProd
instTopologicalSpaceModelProd
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
ModelWithCorners.prod
modelWithCornersSelf
Bundle.TotalSpace
FiberBundle.chartedSpace
WithTop.some
ENat
Bundle.TotalSpace.mk'
Set.instMembership
DFunLike.coe
ContMDiffSection
ContMDiffSection.instDFunLike
β€”exists_contMDiffSection_forall_mem_convex_of_local
exists_smooth_zero_one_nhds_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Filter.Eventually
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Real.instZero
nhdsSet
Real.instOne
Set.instMembership
Set.Icc
Real.instPreorder
β€”exists_contMDiffMap_zero_one_nhds_of_isClosed
exists_smooth_zero_one_of_isClosed πŸ“–mathematicalDisjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.EqOn
Real
DFunLike.coe
ContMDiffMap
DenselyNormedField.toNontriviallyNormedField
Real.denselyNormedField
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
modelWithCornersSelf
Real.pseudoMetricSpace
chartedSpaceSelf
WithTop.some
ENat
ContMDiffMap.instFunLike
Pi.instZero
Real.instZero
Pi.instOne
Real.instOne
Set.instMembership
Set.Icc
Real.instPreorder
β€”exists_contMDiffMap_zero_one_of_isClosed

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