Documentation Verification Report

Bundle

📁 Source: Mathlib/Data/Bundle.lean

Statistics

MetricCount
DefinitionsTotalSpace, mk', proj, snd, toProd, trivialSnd, instCoeTCTotalSpace, instInhabitedTotalSpaceOfDefault, pullbackTotalSpaceEmbedding, termπ__, «term_*ᵖ_», «term_×ᵇ_»
12
Theoremslift_mk, lift_proj, lift_snd, eta, exists, ext, ext_iff, mk_cast, mk_inj, mk_injective, range_mk, toProd_apply, toProd_symm_apply_proj, toProd_symm_apply_snd, instNonemptyPullback
15
Total27

Bundle

Definitions

NameCategoryTheorems
TotalSpace 📖CompData
218 mathmath: TangentBundle.tangentMap_tangentBundle_pure, ContMDiffRiemannianMetric.contMDiff, FiberPrebundle.mem_pretrivializationAt_source, mdifferentiableAt_proj, Trivialization.pullback_source, contMDiffOn_zeroSection, FiberBundle.trivialization_mem_atlas', mdifferentiableWithinAt_totalSpace, contMDiffWithinAt_vectorSpace_iff_contDiffWithinAt, FiberBundle.mem_baseSet_trivializationAt', tangentMap_chart, contMDiff_addInvariantVectorField, Pullback.continuous_lift, Trivialization.mdifferentiableOn_section_baseSet_iff, mdifferentiableOn_symm_coordChangeL, Trivialization.prod_apply, contMDiffWithinAt_zeroSection, TotalSpace.toProd_apply, contMDiff_mulInvariantVectorField, VectorBundleCore.mem_localTriv_target, IsLocalFrameOn.contMDiffAt_of_coeff, FiberBundle.continuousAt_totalSpace, VectorBundleCore.continuous_proj, Trivialization.pullback_target, mdifferentiableWithinAt_section, Trivialization.contMDiffOn_symm_trans, FiberPrebundle.continuous_proj, FiberBundle.continuousWithinAt_totalSpace, FiberBundleCore.mk_mem_localTrivAt_source, MemTrivializationAtlas.out, FiberBundleCore.mem_localTrivAt_source, Trivialization.Prod.continuous_inv_fun, Trivialization.continuousOn, tangentBundle_model_space_coe_chartAt, contMDiffAt_zeroSection, contMDiffAt_mulInvariantVectorField, Pretrivialization.coe_linearMapAt, Trivial.fiberBundle_trivializationAtlas', VectorBundleCore.localTriv_symm_fst, mdifferentiableAt_addInvariantVectorField, VectorBundleCore.localTrivAt_apply, TangentBundle.trivializationAt_fst, ContinuousRiemannianMetric.continuous, FiberBundle.pullback_trivializationAtlas', TotalSpace.toProd_symm_apply_proj, Trivial.isInducing_toProd, Trivial.trivialization_target, mdifferentiableAt_totalSpace, mdifferentiable_mulInvariantVectorField, FiberBundle.chartedSpace_chartAt, memTrivializationAtlas_iff, mdifferentiableAt_hom_bundle, IsLocalFrameOn.mdifferentiableAt_of_coeff_aux, FiberBundle.Prod.isInducing_diag, FiberBundle.isOpenMap_proj, TangentBundle.trivializationAt_apply, FiberBundle.totalSpaceMk_isClosedEmbedding, TotalSpace.isManifold, contMDiff_vectorSpace_iff_contDiff, FiberPrebundle.totalSpaceMk_isInducing, Trivialization.prod_symm_apply, VectorBundleCore.isOpenMap_proj, UniqueMDiffWithinAt.bundle_preimage', IsContMDiffRiemannianBundle.exists_contMDiff, FiberBundle.continuous_totalSpaceMk, VectorBundleCore.baseSet_at, Trivialization.toPretrivialization.isLinear, IsLocalFrameOn.mdifferentiableOn_of_coeff, pullbackTopology_def, Trivialization.contMDiffOn_localFrame_baseSet, FiberBundleCore.localTrivAt_apply, FiberBundle.isQuotientMap_proj, Trivial.homeomorphProd_symm_apply_snd, IsContinuousRiemannianBundle.exists_continuous, VectorBundleCore.mem_source_at, trivializationAt_model_space_apply, contMDiffOn_continuousLinearMapCoordChange, VectorPrebundle.pretrivialization_mem_atlas, mdifferentiableWithinAt_proj, IsLocalFrameOn.contMDiffOn, VectorBundle.continuousOn_coordChange', Pretrivialization.linearMapAt_apply, FiberBundle.chartedSpace'_chartAt, Trivial.trivialization_apply, FiberBundleCore.mem_localTrivAt_target, contMDiffWithinAt_totalSpace, TangentBundle.trivializationAt_source, Trivialization.coe_mem_source, mdifferentiableOn_proj, Trivialization.coe_linearMapAt, tangentBundle_model_space_coe_chartAt_symm, tangentBundleModelSpaceHomeomorph_coe, Trivialization.baseSet_continuousLinearMap, contMDiffOn_symm_coordChangeL, IsLocalFrameOn.contMDiffAt, contMDiffAt_hom_bundle, FiberBundle.extChartAt, FiberBundle.trivializationAt_proj_fst, Pullback.continuous_proj, Trivialization.continuousLinearMap_apply, contMDiffWithinAt_proj, FiberBundle.totalSpaceMk_isEmbedding, VectorBundleCore.localTriv_apply, contMDiffAt_totalSpace, FiberBundle.prod_trivializationAtlas', Pretrivialization.coe_mem_source, Trivialization.contMDiffOn_section_baseSet_iff, UniqueMDiffOn.tangentBundle_proj_preimage, tangentBundle_model_space_chartAt, Trivialization.pullback_apply, Pullback.continuous_totalSpaceMk, ContMDiffFiberwiseLinear.hasGroupoid, contMDiffOn_proj, contMDiff_proj, mdifferentiableOn_continuousLinearMapCoordChange, FiberPrebundle.continuous_totalSpaceMk, Trivialization.pullback_symm_apply_snd, VectorPrebundle.totalSpaceMk_isInducing, Trivialization.prod_symm_apply_proj, Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, FiberPrebundle.isOpen_source, Trivialization.prod_symm_apply_snd, VectorPrebundle.mem_trivialization_at_source, Trivialization.pullback_baseSet, FiberBundle.isCoveringMap, tangentMap_chart_symm, UniqueMDiffWithinAt.bundle_preimage, contMDiffAt_vectorSpace_iff_contDiffAt, mdifferentiableWithinAt_zeroSection, TotalSpace.range_mk, mdifferentiable_addInvariantVectorField, Trivialization.Prod.continuous_to_fun, FiberBundle.map_proj_nhds, VectorPrebundle.mem_base_pretrivializationAt, VectorBundleCore.localTrivAt_apply_mk, hom_trivializationAt_source, TotalSpace.mk_injective, TotalSpace.toProd_symm_apply_snd, Trivialization.prod_target, mdifferentiableOn_coordChangeL, FiberBundle.mem_trivializationAt_proj_source, inducing_pullbackTotalSpaceEmbedding, contMDiff_zeroSection, FiberBundle.extChartAt_target, ContMDiffVectorBundle.contMDiffOn_coordChangeL, mdifferentiableAt_section, mdifferentiable_proj, Trivial.homeomorphProd_symm_apply_proj, tangentBundleModelSpaceHomeomorph_coe_symm, Pretrivialization.continuousLinearMap_apply, tangentBundleCore_localTriv_baseSet, ContMDiffSection.contMDiff, FiberBundle.trivialization_mem_atlas, Trivialization.prod_baseSet, UniqueMDiffOn.bundle_preimage, contMDiffAt_section, FiberBundle.totalSpaceMk_isInducing', FiberBundle.exists_trivialization_Icc_subset, Pretrivialization.continuousOn_continuousLinearMapCoordChange, TangentBundle.trivializationAt_baseSet, mdifferentiableAt_mulInvariantVectorField, Trivialization.pullback_symm_apply_proj, FiberBundle.continuous_proj, FiberBundle.surjective_proj, VectorBundleCore.mem_localTriv_source, Pretrivialization.continuousLinearMap_symm_apply, IsLocalFrameOn.contMDiffOn_of_coeff, continuousAt_hom_bundle, Trivialization.mk_mem_target, Trivialization.contMDiffOn_symm, continuousWithinAt_hom_bundle, contMDiffAt_proj, contMDiffAt_addInvariantVectorField, hom_chart, VectorPrebundle.totalSpaceMk_preimage_source, hom_trivializationAt_apply, contMDiffWithinAt_section, mdifferentiableOn_zeroSection, ContMDiffSection.mdifferentiable', FiberBundleCore.mem_localTrivAt_baseSet, contMDiffOn_coordChangeL, hom_trivializationAt_baseSet, ContMDiffSection.mdifferentiableAt, FiberBundle.totalSpaceMk_isInducing, VectorBundleCore.mem_localTrivAt_baseSet, FiberBundle.mem_baseSet_trivializationAt, Trivial.trivialization_baseSet, ContMDiffSection.mdifferentiable, Trivial.trivialization_symm_apply_proj, mdifferentiable_zeroSection, IsLocalFrameOn.mdifferentiableAt_of_coeff, Trivialization.linearMapAt_apply, contMDiffOn_vectorSpace_iff_contDiffOn, Trivialization.isLocalFrameOn_localFrame_baseSet, ContMDiffSection.contMDiff_toFun, VectorPrebundle.continuous_totalSpaceMk, Trivialization.continuousOn_symm, FiberPrebundle.totalSpaceMk_preimage_source, Pretrivialization.mk_mem_target, FiberPrebundle.mem_base_pretrivializationAt, Trivial.homeomorphProd_apply, mdifferentiableWithinAt_hom_bundle, IsLocalFrameOn.contMDiffAt_of_coeff_aux, Trivialization.contMDiffOn, contMDiffOn_section_of_mem_baseSet₀, hom_trivializationAt_target, Trivial.trivialization_source, mdifferentiableAt_zeroSection, Trivialization.mdifferentiable, FiberBundleCore.localTrivAt_apply_mk, continuousOn_coordChange, contMDiffWithinAt_hom_bundle, Trivialization.prod_source, Trivial.trivialization_symm_apply_snd, FiberBundleCore.continuous_totalSpaceMk, TangentBundle.trivializationAt_target, FiberBundleCore.localTrivAt_snd, FiberPrebundle.pretrivialization_mem_atlas
instCoeTCTotalSpace 📖CompOp
instInhabitedTotalSpaceOfDefault 📖CompOp
pullbackTotalSpaceEmbedding 📖CompOp
1 mathmath: inducing_pullbackTotalSpaceEmbedding
termπ__ 📖CompOp
«term_*ᵖ_» 📖CompOp
«term_×ᵇ_» 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
instNonemptyPullback 📖mathematicalPullback

Bundle.Pullback

Theorems

NameKindAssumesProvesValidatesDepends On
lift_mk 📖mathematicallift
Bundle.TotalSpace.mk'
Bundle.Pullback
lift_proj 📖mathematicalBundle.TotalSpace.proj
lift
Bundle.Pullback
lift_snd 📖mathematicalBundle.TotalSpace.snd
lift
Bundle.Pullback

Bundle.TotalSpace

Definitions

NameCategoryTheorems
mk' 📖CompOp
38 mathmath: Bundle.ContMDiffRiemannianMetric.contMDiff, Trivialization.mdifferentiableAt_section_iff, Bundle.Pullback.lift_mk, Trivialization.mdifferentiableOn_section_baseSet_iff, IsLocalFrameOn.contMDiffAt_of_coeff, mdifferentiableWithinAt_section, mk_inj, Trivialization.contMDiffOn_section_iff, Bundle.ContinuousRiemannianMetric.continuous, mk_cast, IsLocalFrameOn.mdifferentiableAt_of_coeff_aux, IsContMDiffRiemannianBundle.exists_contMDiff, IsLocalFrameOn.mdifferentiableOn_of_coeff, Trivialization.contMDiffOn_localFrame_baseSet, IsContinuousRiemannianBundle.exists_continuous, IsLocalFrameOn.contMDiffOn, IsLocalFrameOn.contMDiffAt, Trivialization.contMDiffOn_section_baseSet_iff, contMDiffAt_section_of_mem_baseSet, Trivialization.mdifferentiableWithinAt_section_iff, contMDiffAt_localFrame_of_mem, mdifferentiableAt_section, ContMDiffSection.contMDiff, Bundle.contMDiffAt_section, Trivialization.mdifferentiableOn_section_iff, Trivialization.contMDiffAt_section_iff, IsLocalFrameOn.contMDiffOn_of_coeff, Bundle.contMDiffWithinAt_section, ContMDiffSection.mdifferentiable', ContMDiffSection.mdifferentiableAt, ContMDiffSection.mdifferentiable, IsLocalFrameOn.mdifferentiableAt_of_coeff, ContMDiffSection.contMDiff_toFun, Trivialization.continuousOn_symm, IsLocalFrameOn.contMDiffAt_of_coeff_aux, contMDiffOn_section_of_mem_baseSet₀, Trivialization.contMDiffWithinAt_section, contMDiffOn_section_of_mem_baseSet
proj 📖CompOp
182 mathmath: TangentBundle.tangentMap_tangentBundle_pure, FiberPrebundle.mem_pretrivializationAt_source, Bundle.mdifferentiableAt_proj, Trivialization.pullback_source, FiberBundle.trivialization_mem_atlas', mdifferentiableWithinAt_totalSpace, FiberBundle.mem_baseSet_trivializationAt', Trivialization.mdifferentiableOn_section_baseSet_iff, mdifferentiableOn_symm_coordChangeL, TangentBundle.coe_chartAt_fst, Trivialization.prod_apply, toProd_apply, VectorBundleCore.mem_localTriv_target, Bundle.zeroSection_proj, FiberBundle.continuousAt_totalSpace, Trivialization.pullback_target, mdifferentiableWithinAt_section, Trivialization.contMDiffOn_symm_trans, FiberPrebundle.continuous_proj, FiberBundle.continuousWithinAt_totalSpace, FiberBundleCore.mk_mem_localTrivAt_source, Bundle.zeroSection_snd, MemTrivializationAtlas.out, FiberBundleCore.mem_localTrivAt_source, Trivialization.Prod.continuous_inv_fun, Trivialization.continuousOn, Pretrivialization.coe_linearMapAt, equivTangentBundleProd_apply, Bundle.Trivial.fiberBundle_trivializationAtlas', VectorBundleCore.localTriv_symm_fst, VectorBundleCore.localTrivAt_apply, TangentBundle.trivializationAt_fst, FiberBundle.pullback_trivializationAtlas', toProd_symm_apply_proj, Bundle.Trivial.trivialization_target, mdifferentiableAt_totalSpace, FiberBundle.chartedSpace_chartAt, memTrivializationAtlas_iff, mdifferentiableAt_hom_bundle, FiberBundleCore.mem_localTrivAsPartialEquiv_source, eta, FiberBundle.Prod.isInducing_diag, FiberBundle.isOpenMap_proj, TangentBundle.trivializationAt_apply, FiberPrebundle.totalSpaceMk_isInducing, Trivialization.prod_symm_apply, UniqueMDiffWithinAt.bundle_preimage', tangentMapWithin_proj, tangentMap_prodSnd, VectorBundleCore.baseSet_at, Trivialization.toPretrivialization.isLinear, FiberBundle.writtenInExtChartAt_trivializationAt, pullbackTopology_def, ContMDiffOn.continuousOn_tangentMapWithin, FiberBundleCore.localTriv_apply, Trivialization.contMDiffOn_localFrame_baseSet, FiberBundleCore.mem_localTriv_source, FiberBundleCore.localTrivAt_apply, FiberBundle.isQuotientMap_proj, FiberBundleCore.localTrivAsPartialEquiv_apply, VectorBundleCore.mem_source_at, trivializationAt_model_space_apply, contMDiffOn_continuousLinearMapCoordChange, VectorPrebundle.pretrivialization_mem_atlas, Bundle.mdifferentiableWithinAt_proj, VectorBundle.continuousOn_coordChange', Pretrivialization.linearMapAt_apply, FiberBundle.chartedSpace'_chartAt, Bundle.Trivial.trivialization_apply, FiberBundleCore.mem_localTrivAt_target, Bundle.contMDiffWithinAt_totalSpace, TangentBundle.trivializationAt_source, Trivialization.coe_mem_source, Bundle.mdifferentiableOn_proj, Trivialization.coe_linearMapAt, TangentBundle.chartAt, Trivialization.baseSet_continuousLinearMap, contMDiffOn_symm_coordChangeL, contMDiffAt_hom_bundle, FiberBundle.extChartAt, FiberBundle.trivializationAt_proj_fst, Pullback.continuous_proj, Trivialization.continuousLinearMap_apply, Bundle.contMDiffWithinAt_proj, VectorBundleCore.localTriv_apply, Bundle.contMDiffAt_totalSpace, FiberBundle.prod_trivializationAtlas', Pretrivialization.coe_mem_source, Trivialization.contMDiffOn_section_baseSet_iff, UniqueMDiffOn.tangentBundle_proj_preimage, TangentBundle.mem_chart_target_iff, Trivialization.pullback_apply, Bundle.contMDiffOn_proj, Bundle.contMDiff_proj, mdifferentiableOn_continuousLinearMapCoordChange, Trivialization.pullback_symm_apply_snd, VectorPrebundle.totalSpaceMk_isInducing, Trivialization.prod_symm_apply_proj, Bundle.Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, FiberPrebundle.isOpen_source, Trivialization.prod_symm_apply_snd, VectorPrebundle.mem_trivialization_at_source, Trivialization.pullback_baseSet, FiberBundle.isCoveringMap, range_mk, Trivialization.Prod.continuous_to_fun, FiberBundle.map_proj_nhds, VectorPrebundle.mem_base_pretrivializationAt, VectorBundleCore.localTrivAt_apply_mk, hom_trivializationAt_source, Trivialization.prod_target, mdifferentiableOn_coordChangeL, FiberBundle.mem_trivializationAt_proj_source, FiberBundle.extChartAt_target, ContMDiffVectorBundle.contMDiffOn_coordChangeL, mdifferentiableAt_section, Bundle.mdifferentiable_proj, Bundle.Trivial.homeomorphProd_symm_apply_proj, FiberBundle.writtenInExtChartAt_trivializationAt_symm, Pretrivialization.continuousLinearMap_apply, tangentBundleCore_localTriv_baseSet, FiberBundle.trivialization_mem_atlas, Trivialization.prod_baseSet, UniqueMDiffOn.bundle_preimage, Bundle.contMDiffAt_section, tangentMap_proj, FiberBundle.exists_trivialization_Icc_subset, Pretrivialization.continuousOn_continuousLinearMapCoordChange, TangentBundle.trivializationAt_baseSet, Trivialization.pullback_symm_apply_proj, FiberBundle.continuous_proj, tangentMap_prodFst, ContMDiffOn.contMDiffOn_tangentMapWithin, FiberBundle.surjective_proj, VectorBundleCore.mem_localTriv_source, Pretrivialization.continuousLinearMap_symm_apply, continuousAt_hom_bundle, Trivialization.mk_mem_target, Trivialization.contMDiffOn_symm, continuousWithinAt_hom_bundle, Bundle.contMDiffAt_proj, hom_chart, VectorPrebundle.totalSpaceMk_preimage_source, TangentBundle.coe_chartAt_symm_fst, hom_trivializationAt_apply, Bundle.contMDiffWithinAt_section, FiberBundleCore.mem_localTrivAt_baseSet, contMDiffOn_coordChangeL, hom_trivializationAt_baseSet, ext_iff, VectorBundleCore.mem_localTrivAt_baseSet, FiberBundle.mem_baseSet_trivializationAt, Bundle.Trivial.trivialization_baseSet, equivTangentBundleProd_symm_apply_proj, Bundle.Trivial.trivialization_symm_apply_proj, TangentBundle.chartAt_toPartialEquiv, Trivialization.linearMapAt_apply, tangentMap_prod_right, tangentMap_prod_left, Trivialization.isLocalFrameOn_localFrame_baseSet, FiberBundle.chartedSpace_chartAt_symm_fst, Trivialization.continuousOn_symm, FiberPrebundle.totalSpaceMk_preimage_source, Pretrivialization.mk_mem_target, FiberPrebundle.mem_base_pretrivializationAt, Bundle.Trivial.homeomorphProd_apply, mdifferentiableWithinAt_hom_bundle, Trivialization.contMDiffOn, contMDiffOn_section_of_mem_baseSet₀, hom_trivializationAt_target, Bundle.Trivial.trivialization_source, Trivialization.mdifferentiable, FiberBundleCore.localTrivAt_apply_mk, continuousOn_coordChange, contMDiffWithinAt_hom_bundle, Trivialization.prod_source, Bundle.Trivial.trivialization_symm_apply_snd, TangentBundle.mem_chart_source_iff, Bundle.Pullback.lift_proj, TangentBundle.trivializationAt_target, FiberBundleCore.localTrivAt_snd, FiberPrebundle.pretrivialization_mem_atlas
snd 📖CompOp
48 mathmath: TangentBundle.tangentMap_tangentBundle_pure, tangentMapWithin_prodSnd, tangentMap_snd, toProd_apply, Bundle.zeroSection_snd, equivTangentBundleProd_apply, VectorBundleCore.localTrivAt_apply, tangentMapWithin_snd, mdifferentiableAt_hom_bundle, eta, FiberBundle.Prod.isInducing_diag, TangentBundle.trivializationAt_apply, tangentMap_prodSnd, FiberBundleCore.localTriv_apply, FiberBundleCore.localTrivAt_apply, Bundle.Trivial.homeomorphProd_symm_apply_snd, FiberBundleCore.localTrivAsPartialEquiv_apply, trivializationAt_model_space_apply, Trivialization.continuousLinearEquivAt_apply', Pretrivialization.symm_apply, Bundle.Trivial.trivialization_apply, tangentMapWithin_prodFst, equivTangentBundleProd_symm_apply_snd, contMDiffAt_hom_bundle, Trivialization.symm_apply, Trivialization.continuousLinearMap_apply, VectorBundleCore.localTriv_apply, Pretrivialization.symm_proj_apply, Trivialization.pullback_symm_apply_snd, Trivialization.prod_symm_apply_snd, toProd_symm_apply_snd, Pretrivialization.continuousLinearMap_apply, tangentMap_prodFst, continuousAt_hom_bundle, continuousWithinAt_hom_bundle, hom_chart, hom_trivializationAt_apply, ext_iff, Bundle.Pullback.lift_snd, Trivialization.symm_proj_apply, tangentMap_prod_right, tangentMap_prod_left, Bundle.Trivial.homeomorphProd_apply, mdifferentiableWithinAt_hom_bundle, contMDiff_snd_tangentBundle_modelSpace, contMDiffWithinAt_hom_bundle, Bundle.Trivial.trivialization_symm_apply_snd, FiberBundleCore.localTrivAt_snd
toProd 📖CompOp
11 mathmath: tangentMap_chart, toProd_apply, tangentBundle_model_space_coe_chartAt, toProd_symm_apply_proj, Bundle.Trivial.isInducing_toProd, tangentBundle_model_space_coe_chartAt_symm, tangentBundleModelSpaceHomeomorph_coe, tangentBundle_model_space_chartAt, tangentMap_chart_symm, toProd_symm_apply_snd, tangentBundleModelSpaceHomeomorph_coe_symm
trivialSnd 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
eta 📖mathematicalproj
snd
exists 📖
ext 📖proj
snd
ext_iff 📖mathematicalproj
snd
ext
mk_cast 📖mathematicalmk'
mk_inj 📖mathematicalmk'
mk_injective 📖mathematicalBundle.TotalSpace
range_mk 📖mathematicalSet.range
Bundle.TotalSpace
Set.preimage
proj
Set
Set.instSingletonSet
Set.Subset.antisymm
toProd_apply 📖mathematicalDFunLike.coe
Equiv
Bundle.TotalSpace
EquivLike.toFunLike
Equiv.instEquivLike
toProd
proj
snd
toProd_symm_apply_proj 📖mathematicalproj
DFunLike.coe
Equiv
Bundle.TotalSpace
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
toProd
toProd_symm_apply_snd 📖mathematicalsnd
DFunLike.coe
Equiv
Bundle.TotalSpace
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
toProd

---

← Back to Index