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
439 mathmath: TangentBundle.tangentMap_tangentBundle_pure, Pretrivialization.symm_proj_apply, ContMDiffRiemannianMetric.contMDiff, FiberPrebundle.mem_pretrivializationAt_source, mdifferentiableAt_proj, mdifferentiableAt_sub_section, Trivialization.pullback_symm_apply_proj, MDifferentiableOn.finsum_section_of_locallyFinite, contMDiffOn_zeroSection, FiberBundle.trivialization_mem_atlas', MDifferentiableAt.clm_bundle_apply, mdifferentiableWithinAt_totalSpace, contMDiffWithinAt_vectorSpace_iff_contDiffWithinAt, ContMDiffOn.mlieBracketWithin_vectorField, ContMDiffWithinAt.mlieBracketWithin_vectorField, mdifferentiableAt_iff_localFrame_coeff, Trivialization.contMDiffOn_section_baseSet_iff, Trivialization.coordChangeL_apply, MDifferentiableOn.smul_const_section, FiberBundle.mem_baseSet_trivializationAt', tangentMap_chart, ContMDiffWithinAt.sub_section, Pretrivialization.IsLinear.linear, Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm, contMDiff_addInvariantVectorField, Pullback.continuous_lift, mdifferentiableOn_symm_coordChangeL, mdifferentiableWithinAt_smul_const_section, mdifferentiableAt_add_section, ContMDiffAt.finsum_section_of_locallyFinite, contMDiffWithinAt_zeroSection, TotalSpace.toProd_apply, contMDiff_mulInvariantVectorField, Trivialization.contMDiffOn, FiberPrebundle.continuous_trivChange, MDifferentiableWithinAt.finsum_section_of_locallyFinite, VectorBundleCore.mem_localTriv_target, IsLocalFrameOn.contMDiffAt_of_coeff, FiberBundle.continuousAt_totalSpace, VectorBundleCore.continuous_proj, FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas, Trivialization.symm_proj_apply, mdifferentiableWithinAt_section, Pretrivialization.continuousLinearMap_apply, Trivialization.Prod.continuous_inv_fun, contMDiffOn_baseSet_iff_localFrame_coeff, ContMDiffWithinAt.coordChange, FiberPrebundle.continuous_proj, FiberBundle.continuousWithinAt_totalSpace, Trivialization.localFrame_coeff_eq_coeff, ContMDiffWithinAt.add_section, ContMDiffAt.add_section, FiberBundleCore.mk_mem_localTrivAt_source, ContMDiff.coordChange, MemTrivializationAtlas.out, FiberBundleCore.mem_localTrivAt_source, Trivialization.prod_symm_apply_proj, ContinuousWithinAt.clm_apply_of_inCoordinates, MDifferentiable.clm_bundle_apply₂, VectorPrebundle.mk_contMDiffCoordChange, Trivialization.symm_apply, tangentBundle_model_space_coe_chartAt, contMDiffAt_zeroSection, contMDiffAt_mulInvariantVectorField, MDifferentiable.clm_bundle_apply, Trivial.fiberBundle_trivializationAtlas', Trivialization.prod_symm_apply, VectorBundleCore.localTriv_symm_fst, MDifferentiableWithinAt.clm_bundle_apply₂, mdifferentiableAt_addInvariantVectorField, ContMDiff.sum_section_of_locallyFinite, MDifferentiableAt.mpullback_vectorField, VectorBundleCore.localTrivAt_apply, ContMDiffWithinAt.sum_section, TangentBundle.trivializationAt_fst, MDifferentiableWithinAt.coordChange, Trivialization.mk_symm, ContinuousRiemannianMetric.continuous, FiberBundle.pullback_trivializationAtlas', MDifferentiableAt.sum_section, Trivialization.mdifferentiableAt_section_iff, Trivialization.continuousLinearEquivAt_apply', TotalSpace.toProd_symm_apply_proj, Trivial.isInducing_toProd, Trivial.trivialization_target, mdifferentiableAt_totalSpace, mdifferentiable_mulInvariantVectorField, FiberBundle.chartedSpace_chartAt, Trivialization.symm_apply_apply, ContMDiffOn.finsum_section_of_locallyFinite, Trivialization.IsLinear.linear, MDifferentiableOn.sum_section_of_locallyFinite, Trivialization.comp_continuousLinearEquivAt_eq_coord_change, ContinuousAt.clm_apply_of_inCoordinates, ContMDiffWithinAt.neg_section, memTrivializationAtlas_iff, Trivialization.prod_baseSet, ContMDiffWithinAt.mpullbackWithin_vectorField_inter, mdifferentiableAt_hom_bundle, IsLocalFrameOn.mdifferentiableAt_of_coeff_aux, Trivialization.prod_symm_apply_snd, FiberBundle.Prod.isInducing_diag, FiberBundle.isOpenMap_proj, MDifferentiable.coordChange, FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp, Trivialization.continuousLinearEquivAt_prod, Trivialization.prod_apply', TangentBundle.trivializationAt_apply, mdifferentiableWithinAt_neg_section, Trivialization.prod_source, FiberBundle.totalSpaceMk_isClosedEmbedding, TotalSpace.isManifold, ContMDiffAt.const_smul_section, contMDiff_vectorSpace_iff_contDiff, ContMDiffOn.mpullbackWithin_vectorField_inter, ContMDiffWithinAt.change_section_trivialization, FiberPrebundle.totalSpaceMk_isInducing, ContMDiffAt.clm_apply_of_inCoordinates, Trivialization.mdifferentiableWithinAt_totalSpace_iff, Continuous.clm_bundle_apply, MDifferentiable.mpullback_vectorField, mdifferentiable_smul_const_section, VectorBundleCore.isOpenMap_proj, Trivialization.mdifferentiableOn_section_baseSet_iff, UniqueMDiffWithinAt.bundle_preimage', FiberBundle.exists_contMDiffOn_extend, MDifferentiableAt.finsum_section_of_locallyFinite, IsContMDiffRiemannianBundle.exists_contMDiff, ContMDiffAt.clm_bundle_apply, mdifferentiableAt_neg_section, Trivialization.contMDiffOn_localFrame_baseSet, FiberBundle.continuous_totalSpaceMk, VectorBundleCore.baseSet_at, Trivialization.apply_symm_apply_eq_coordChangeL, MDifferentiableAt.clm_bundle_apply₂, ContMDiffWithinAt.mpullback_vectorField_of_mem_nhdsWithin_of_eq, FiberBundle.writtenInExtChartAt_trivializationAt, Trivialization.coe_linearMapAt_of_mem, IsLocalFrameOn.mdifferentiableOn_of_coeff, ContMDiffWithinAt.mpullback_vectorField_preimage, pullbackTopology_def, ContinuousOn.clm_bundle_apply₂, Pretrivialization.apply_mk_symm, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter_of_eq, MDifferentiableAt.smul_section, Trivialization.prod_target, ContMDiffWithinAt.mpullback_vectorField_preimage_of_eq, Pretrivialization.coe_linearMapAt_of_mem, FiberBundle.mdifferentiableAt_extend, contMDiffOn_baseSet_localFrame_coeff, Trivialization.continuousOn_symm, FiberBundleCore.localTrivAt_apply, Trivialization.toPretrivialization.isLinear, Pretrivialization.continuousOn_continuousLinearMapCoordChange, FiberBundle.isQuotientMap_proj, Trivial.homeomorphProd_symm_apply_snd, IsContinuousRiemannianBundle.exists_continuous, VectorBundle.continuousLinearEquivAt_apply, VectorBundleCore.mem_source_at, trivializationAt_model_space_apply, Trivialization.coe_coordChangeL, MDifferentiableWithinAt.clm_apply_of_inCoordinates, contMDiffOn_continuousLinearMapCoordChange, VectorPrebundle.pretrivialization_mem_atlas, mdifferentiableWithinAt_proj, IsLocalFrameOn.contMDiffOn, VectorBundle.continuousOn_coordChange', MDifferentiableOn.clm_bundle_apply, Trivialization.pullback_baseSet, FiberBundle.chartedSpace'_chartAt, Trivialization.mdifferentiableOn_section_iff, Trivial.trivialization_apply, FiberBundleCore.mem_localTrivAt_target, ContMDiffWithinAt.finsum_section_of_locallyFinite, contMDiffWithinAt_totalSpace, TangentBundle.trivializationAt_source, Trivialization.contMDiffWithinAt_iff, mdifferentiableWithinAt_sub_section, mdifferentiableOn_proj, Pretrivialization.continuousLinearMap_symm_apply, FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter, mdifferentiableWithinAt_add_section, ContMDiffWithinAt.mpullbackWithin_vectorField', Trivialization.pullback_source, tangentBundle_model_space_coe_chartAt_symm, tangentBundleModelSpaceHomeomorph_coe, MDifferentiable.sum_section, Pretrivialization.coe_coe_fst, ContMDiffWithinAt.clm_bundle_apply₂, contMDiffOn_symm_coordChangeL, IsLocalFrameOn.contMDiffAt, contMDiffAt_hom_bundle, MDifferentiableAt.coordChange, MDifferentiable.sum_section_of_locallyFinite, mdifferentiableOn_baseSet_localFrame_coeff, FiberBundle.extChartAt, FiberBundle.trivializationAt_proj_fst, MDifferentiableWithinAt.sum_section_of_locallyFinite, Trivialization.zeroSection, ContMDiffAt.mlieBracket_vectorField, ContMDiffOn.add_section, Trivialization.linearEquivAt_apply, ContMDiff.finsum_section_of_locallyFinite, Pullback.continuous_proj, contMDiffWithinAt_proj, FiberBundle.totalSpaceMk_isEmbedding, Pretrivialization.symm_coe_proj, VectorBundleCore.localTriv_apply, contMDiffAt_totalSpace, FiberBundle.prod_trivializationAtlas', MDifferentiableAt.clm_apply_of_inCoordinates, ContMDiffWithinAt.clm_bundle_apply, ContMDiff.clm_bundle_apply, UniqueMDiffOn.tangentBundle_proj_preimage, tangentBundle_model_space_chartAt, Trivialization.linear, ContMDiff.clm_bundle_apply₂, Trivialization.isLocalFrameOn_localFrame_baseSet, Pullback.continuous_totalSpaceMk, Trivialization.symm_coe_proj, Trivialization.baseSet_continuousLinearMap, MDifferentiableOn.sum_section, VectorPrebundle.exists_coordChange, FiberBundle.contMDiffAt_extend', ContMDiffFiberwiseLinear.hasGroupoid, contMDiffOn_proj, contMDiff_proj, mdifferentiableOn_continuousLinearMapCoordChange, Trivialization.continuousLinearEquivAt_apply, FiberPrebundle.continuous_totalSpaceMk, MDifferentiableAt.sum_section_of_locallyFinite, Trivialization.mk_mem_target, MDifferentiableOn.mpullback_vectorField_preimage, Trivialization.linearMapAt_apply, VectorPrebundle.totalSpaceMk_isInducing, Trivialization.apply_eq_prod_continuousLinearEquivAt, ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem_of_eq, Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, FiberPrebundle.isOpen_source, VectorPrebundle.mem_trivialization_at_source, FiberBundle.isCoveringMap, tangentMap_chart_symm, UniqueMDiffWithinAt.bundle_preimage, contMDiffAt_vectorSpace_iff_contDiffAt, ContMDiff.neg_section, mdifferentiableWithinAt_zeroSection, TotalSpace.range_mk, mdifferentiable_addInvariantVectorField, FiberBundle.map_proj_nhds, VectorPrebundle.mem_base_pretrivializationAt, VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm, Trivialization.contMDiff_iff, ContMDiffWithinAt.sum_section_of_locallyFinite, Trivialization.continuousLinearMap_apply, VectorBundleCore.localTrivAt_apply_mk, hom_trivializationAt_source, TotalSpace.mk_injective, ContinuousAt.clm_bundle_apply₂, TotalSpace.toProd_symm_apply_snd, mdifferentiableOn_coordChangeL, FiberBundle.mem_trivializationAt_proj_source, VectorPrebundle.mk_coordChange, inducing_pullbackTotalSpaceEmbedding, contMDiff_zeroSection, ContDiff.mlieBracket_vectorField, FiberBundle.extChartAt_target, contMDiffAt_localFrame_of_mem, ContinuousWithinAt.clm_bundle_apply₂, Trivialization.coordChangeL_symm_apply, ContMDiffVectorBundle.contMDiffOn_coordChangeL, mdifferentiableAt_section, Trivialization.contMDiffAt_section_iff, mdifferentiable_proj, ContMDiffWithinAt.clm_apply_of_inCoordinates, Trivial.homeomorphProd_symm_apply_proj, FiberBundle.writtenInExtChartAt_trivializationAt_symm, ContMDiffAt.sub_section, tangentBundleModelSpaceHomeomorph_coe_symm, tangentBundleCore_localTriv_baseSet, MDifferentiableWithinAt.mpullback_vectorField_preimage, MDifferentiableWithinAt.clm_bundle_apply, ContMDiffSection.contMDiff, FiberBundle.trivialization_mem_atlas, Trivialization.coe_coe_fst, UniqueMDiffOn.bundle_preimage, Trivialization.contMDiffAt_iff, ContMDiffOn.const_smul_section, contMDiffAt_section, Trivialization.contMDiffWithinAt_section, Trivialization.Prod.continuous_to_fun, Trivialization.pullback_symm_apply_snd, MDifferentiableOn.coordChange, Trivialization.continuousOn, MDifferentiableOn.mpullbackWithin_vectorField_inter, ContMDiffAt.clm_bundle_apply₂, ContMDiffWithinAt.mpullbackWithin_vectorField_inter_of_eq, FiberBundle.totalSpaceMk_isInducing', mDifferentiableOn_sub_section, FiberBundle.exists_trivialization_Icc_subset, TangentBundle.trivializationAt_baseSet, Trivialization.contMDiffOn_symm, mdifferentiableAt_mulInvariantVectorField, contMDiffAt_iff_localFrame_coeff, FiberBundle.continuous_proj, ContMDiffOn.sum_section_of_locallyFinite, VectorPrebundle.continuousOn_coordChange, ContMDiffCovariantDerivativeOn.contMDiff, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter, ContMDiffOn.contMDiffOn_tangentMapWithin, FiberBundle.surjective_proj, mdifferentiable_sub_section, ContMDiffWithinAt.const_smul_section, mdifferentiable_neg_section, ContMDiffAt.sum_section_of_locallyFinite, Pretrivialization.linearEquivAt_apply, MDifferentiable.finsum_section_of_locallyFinite, VectorBundleCore.mem_localTriv_source, ContMDiffOn.mpullback_vectorField_preimage, IsLocalFrameOn.contMDiffOn_of_coeff, Continuous.clm_bundle_apply₂, continuousAt_hom_bundle, VectorPrebundle.contMDiffOn_contMDiffCoordChange, Trivialization.mk_coordChangeL, FiberBundle.exists_mdifferentiableOn_extend, ContMDiffOn.coordChange, Trivialization.coordChangeL_apply', ContMDiff.sub_section, MDifferentiableWithinAt.mpullback_vectorField_preimage_of_eq, continuousWithinAt_hom_bundle, ContMDiffWithinAt.mpullbackWithin_vectorField, contMDiffAt_proj, contMDiffAt_addInvariantVectorField, Pretrivialization.symm_apply, MDifferentiableAt.smul_const_section, ContMDiffAt.mpullback_vectorField_preimage, ContMDiffOn.smul_section_of_tsupport, ContMDiffWithinAt.smul_section, mdifferentiable_add_section, Pretrivialization.mk_mem_target, hom_chart, MDifferentiableOn.smul_section, VectorPrebundle.totalSpaceMk_preimage_source, hom_trivializationAt_apply, contMDiffWithinAt_section, Pretrivialization.coe_mem_source, VectorPrebundle.IsContMDiff.exists_contMDiffCoordChange, mdifferentiableOn_zeroSection, Pretrivialization.symm_apply_apply_mk, Trivialization.symm_apply_apply_mk, ContMDiffSection.mdifferentiable', FiberBundleCore.mem_localTrivAt_baseSet, contMDiffOn_coordChangeL, ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq, hom_trivializationAt_baseSet, ContMDiffOn.sub_section, ContinuousAt.clm_bundle_apply, ContMDiffSection.mdifferentiableAt, FiberBundle.totalSpaceMk_isInducing, VectorBundleCore.mem_localTrivAt_baseSet, MDifferentiableOn.smul_section_of_tsupport, VectorPrebundle.coordChange_apply, FiberBundle.mem_baseSet_trivializationAt, ContMDiff.add_section, ContMDiffAt.coordChange, ContMDiffOn.sum_section, Trivialization.mdifferentiableAt_snd_comp_iff₂, Trivial.trivialization_baseSet, ContMDiffOn.neg_section, ContMDiffSection.mdifferentiable, Trivialization.contMDiffOn_iff, Trivial.trivialization_symm_apply_proj, mdifferentiable_smul_section, mdifferentiable_zeroSection, Trivialization.mdifferentiableWithinAt_section_iff, ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem, VectorField.eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm, IsLocalFrameOn.mdifferentiableAt_of_coeff, Trivialization.contMDiffWithinAt_snd_comp_iff₂, ContMDiffAt.sum_section, ContMDiff.const_smul_section, contMDiffOn_vectorSpace_iff_contDiffOn, Trivialization.coe_linearMapAt, ContMDiffOn.clm_bundle_apply, Trivialization.contMDiffOn_section_iff, mdifferentiableOn_add_section, FiberBundle.chartedSpace_chartAt_symm_fst, ContMDiffSection.contMDiff_toFun, ContMDiffAt.smul_section, VectorPrebundle.continuous_totalSpaceMk, FiberPrebundle.totalSpaceMk_preimage_source, Pretrivialization.coe_linearMapAt, Trivialization.prod_apply, Trivialization.mdifferentiableWithinAt_snd_comp_iff₂, FiberPrebundle.mem_base_pretrivializationAt, ContMDiffAt.neg_section, Pretrivialization.mk_symm, TotalSpace.exists, Trivial.homeomorphProd_apply, mdifferentiableWithinAt_hom_bundle, IsLocalFrameOn.contMDiffAt_of_coeff_aux, MDifferentiableWithinAt.change_section_trivialization, ContinuousOn.clm_bundle_apply, Trivialization.contMDiffOn_symm_trans, Trivialization.pullback_apply, hom_trivializationAt_target, Trivialization.coe_coordChangeL', Trivial.trivialization_source, mdifferentiableAt_zeroSection, Trivialization.apply_mk_symm, ContMDiff.sum_section, mdifferentiableOn_neg_section, Trivialization.pullback_target, FiberBundleCore.localTrivAt_apply_mk, Pretrivialization.continuousLinearMapCoordChange_apply, Trivialization.mdifferentiableAt_totalSpace_iff, VectorPrebundle.contMDiffCoordChange_apply, continuousOn_coordChange, Pretrivialization.linearMapAt_apply, ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq', contMDiffWithinAt_hom_bundle, Pretrivialization.linear, Trivial.trivialization_symm_apply_snd, Trivialization.coe_mem_source, MDifferentiableOn.clm_bundle_apply₂, ContMDiffOn.clm_bundle_apply₂, ContinuousWithinAt.clm_bundle_apply, FiberBundleCore.continuous_totalSpaceMk, ContMDiff.mpullback_vectorField, Trivialization.Bundle.Trivialization.mdifferentiable, MDifferentiableWithinAt.sum_section, ContMDiff.smul_section, TangentBundle.trivializationAt_target, ContMDiffWithinAt.mpullback_vectorField_of_mem_nhdsWithin, MDifferentiableWithinAt.smul_section, ContMDiffOn.smul_section, FiberBundleCore.localTrivAt_snd, contMDiffOn_iff_localFrame_coeff, FiberPrebundle.continuousOn_of_comp_right, 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
167 mathmath: Bundle.ContMDiffRiemannianMetric.contMDiff, mdifferentiableAt_sub_section, MDifferentiableOn.finsum_section_of_locallyFinite, Bundle.Pullback.lift_mk, MDifferentiableAt.clm_bundle_apply, contMDiffWithinAt_vectorSpace_iff_contDiffWithinAt, ContMDiffOn.mlieBracketWithin_vectorField, ContMDiffWithinAt.mlieBracketWithin_vectorField, mdifferentiableAt_iff_localFrame_coeff, Bundle.Trivialization.contMDiffOn_section_baseSet_iff, MDifferentiableOn.smul_const_section, ContMDiffWithinAt.sub_section, mdifferentiableWithinAt_smul_const_section, mdifferentiableAt_add_section, ContMDiffAt.finsum_section_of_locallyFinite, MDifferentiableWithinAt.finsum_section_of_locallyFinite, IsLocalFrameOn.contMDiffAt_of_coeff, mdifferentiableWithinAt_section, mk_inj, contMDiffOn_baseSet_iff_localFrame_coeff, ContMDiffWithinAt.add_section, ContMDiffAt.add_section, MDifferentiable.clm_bundle_apply₂, MDifferentiable.clm_bundle_apply, MDifferentiableWithinAt.clm_bundle_apply₂, ContMDiff.sum_section_of_locallyFinite, MDifferentiableAt.mpullback_vectorField, ContMDiffWithinAt.sum_section, Bundle.ContinuousRiemannianMetric.continuous, MDifferentiableAt.sum_section, mk_cast, Bundle.Trivialization.mdifferentiableAt_section_iff, ContMDiffOn.finsum_section_of_locallyFinite, MDifferentiableOn.sum_section_of_locallyFinite, ContMDiffWithinAt.neg_section, ContMDiffWithinAt.mpullbackWithin_vectorField_inter, IsLocalFrameOn.mdifferentiableAt_of_coeff_aux, mdifferentiableWithinAt_neg_section, ContMDiffAt.const_smul_section, contMDiff_vectorSpace_iff_contDiff, ContMDiffOn.mpullbackWithin_vectorField_inter, Continuous.clm_bundle_apply, MDifferentiable.mpullback_vectorField, mdifferentiable_smul_const_section, Bundle.Trivialization.mdifferentiableOn_section_baseSet_iff, FiberBundle.exists_contMDiffOn_extend, MDifferentiableAt.finsum_section_of_locallyFinite, IsContMDiffRiemannianBundle.exists_contMDiff, ContMDiffAt.clm_bundle_apply, mdifferentiableAt_neg_section, Bundle.Trivialization.contMDiffOn_localFrame_baseSet, MDifferentiableAt.clm_bundle_apply₂, ContMDiffWithinAt.mpullback_vectorField_of_mem_nhdsWithin_of_eq, IsLocalFrameOn.mdifferentiableOn_of_coeff, ContMDiffWithinAt.mpullback_vectorField_preimage, ContinuousOn.clm_bundle_apply₂, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter_of_eq, MDifferentiableAt.smul_section, ContMDiffWithinAt.mpullback_vectorField_preimage_of_eq, FiberBundle.mdifferentiableAt_extend, Bundle.Trivialization.continuousOn_symm, IsContinuousRiemannianBundle.exists_continuous, IsLocalFrameOn.contMDiffOn, MDifferentiableOn.clm_bundle_apply, Bundle.Trivialization.mdifferentiableOn_section_iff, ContMDiffWithinAt.finsum_section_of_locallyFinite, mdifferentiableWithinAt_sub_section, mdifferentiableWithinAt_add_section, ContMDiffWithinAt.mpullbackWithin_vectorField', MDifferentiable.sum_section, ContMDiffWithinAt.clm_bundle_apply₂, IsLocalFrameOn.contMDiffAt, MDifferentiable.sum_section_of_locallyFinite, MDifferentiableWithinAt.sum_section_of_locallyFinite, ContMDiffAt.mlieBracket_vectorField, ContMDiffOn.add_section, ContMDiff.finsum_section_of_locallyFinite, ContMDiffWithinAt.clm_bundle_apply, ContMDiff.clm_bundle_apply, ContMDiff.clm_bundle_apply₂, MDifferentiableOn.sum_section, FiberBundle.contMDiffAt_extend', MDifferentiableAt.sum_section_of_locallyFinite, MDifferentiableOn.mpullback_vectorField_preimage, ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem_of_eq, contMDiffAt_vectorSpace_iff_contDiffAt, ContMDiff.neg_section, VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm, ContMDiffWithinAt.sum_section_of_locallyFinite, ContinuousAt.clm_bundle_apply₂, ContDiff.mlieBracket_vectorField, contMDiffAt_localFrame_of_mem, ContinuousWithinAt.clm_bundle_apply₂, mdifferentiableAt_section, Bundle.Trivialization.contMDiffAt_section_iff, ContMDiffAt.sub_section, MDifferentiableWithinAt.mpullback_vectorField_preimage, MDifferentiableWithinAt.clm_bundle_apply, ContMDiffSection.contMDiff, ContMDiffOn.const_smul_section, Bundle.contMDiffAt_section, Bundle.Trivialization.contMDiffWithinAt_section, MDifferentiableOn.mpullbackWithin_vectorField_inter, ContMDiffAt.clm_bundle_apply₂, ContMDiffWithinAt.mpullbackWithin_vectorField_inter_of_eq, mDifferentiableOn_sub_section, contMDiffAt_iff_localFrame_coeff, ContMDiffOn.sum_section_of_locallyFinite, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter, mdifferentiable_sub_section, ContMDiffWithinAt.const_smul_section, mdifferentiable_neg_section, ContMDiffAt.sum_section_of_locallyFinite, MDifferentiable.finsum_section_of_locallyFinite, ContMDiffOn.mpullback_vectorField_preimage, IsLocalFrameOn.contMDiffOn_of_coeff, Continuous.clm_bundle_apply₂, FiberBundle.exists_mdifferentiableOn_extend, ContMDiff.sub_section, MDifferentiableWithinAt.mpullback_vectorField_preimage_of_eq, ContMDiffWithinAt.mpullbackWithin_vectorField, MDifferentiableAt.smul_const_section, ContMDiffAt.mpullback_vectorField_preimage, ContMDiffOn.smul_section_of_tsupport, ContMDiffWithinAt.smul_section, mdifferentiable_add_section, MDifferentiableOn.smul_section, Bundle.contMDiffWithinAt_section, ContMDiffSection.mdifferentiable', ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq, ContMDiffOn.sub_section, ContinuousAt.clm_bundle_apply, ContMDiffSection.mdifferentiableAt, MDifferentiableOn.smul_section_of_tsupport, ContMDiff.add_section, ContMDiffOn.sum_section, ContMDiffOn.neg_section, ContMDiffSection.mdifferentiable, mdifferentiable_smul_section, Bundle.Trivialization.mdifferentiableWithinAt_section_iff, ContMDiffWithinAt.mpullbackWithin_vectorField_of_mem, VectorField.eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm, IsLocalFrameOn.mdifferentiableAt_of_coeff, ContMDiffAt.sum_section, ContMDiff.const_smul_section, contMDiffOn_vectorSpace_iff_contDiffOn, ContMDiffOn.clm_bundle_apply, Bundle.Trivialization.contMDiffOn_section_iff, mdifferentiableOn_add_section, ContMDiffSection.contMDiff_toFun, ContMDiffAt.smul_section, ContMDiffAt.neg_section, IsLocalFrameOn.contMDiffAt_of_coeff_aux, ContinuousOn.clm_bundle_apply, ContMDiff.sum_section, mdifferentiableOn_neg_section, ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq', MDifferentiableOn.clm_bundle_apply₂, ContMDiffOn.clm_bundle_apply₂, ContinuousWithinAt.clm_bundle_apply, ContMDiff.mpullback_vectorField, MDifferentiableWithinAt.sum_section, ContMDiff.smul_section, ContMDiffWithinAt.mpullback_vectorField_of_mem_nhdsWithin, MDifferentiableWithinAt.smul_section, ContMDiffOn.smul_section, contMDiffOn_iff_localFrame_coeff
proj 📖CompOp
267 mathmath: TangentBundle.tangentMap_tangentBundle_pure, Bundle.Pretrivialization.symm_proj_apply, FiberPrebundle.mem_pretrivializationAt_source, Bundle.mdifferentiableAt_proj, Bundle.Trivialization.pullback_symm_apply_proj, FiberBundle.trivialization_mem_atlas', mdifferentiableWithinAt_totalSpace, tangentMapWithin_prodSnd, Bundle.Trivialization.contMDiffOn_section_baseSet_iff, Bundle.Trivialization.coordChangeL_apply, FiberBundle.mem_baseSet_trivializationAt', tangentMap_chart, Bundle.Pretrivialization.IsLinear.linear, Bundle.Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm, mdifferentiableOn_symm_coordChangeL, TangentBundle.coe_chartAt_fst, toProd_apply, Bundle.Trivialization.contMDiffOn, FiberPrebundle.continuous_trivChange, VectorBundleCore.mem_localTriv_target, Bundle.zeroSection_proj, FiberBundle.continuousAt_totalSpace, FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas, Bundle.Trivialization.symm_proj_apply, mdifferentiableWithinAt_section, Bundle.Pretrivialization.continuousLinearMap_apply, Bundle.Trivialization.Prod.continuous_inv_fun, contMDiffOn_baseSet_iff_localFrame_coeff, ContMDiffWithinAt.coordChange, FiberPrebundle.continuous_proj, FiberBundle.continuousWithinAt_totalSpace, Bundle.Trivialization.localFrame_coeff_eq_coeff, FiberBundleCore.mk_mem_localTrivAt_source, ContMDiff.coordChange, Bundle.zeroSection_snd, MemTrivializationAtlas.out, FiberBundleCore.mem_localTrivAt_source, Bundle.Trivialization.prod_symm_apply_proj, VectorPrebundle.mk_contMDiffCoordChange, Bundle.Trivialization.symm_apply, equivTangentBundleProd_apply, Bundle.Trivial.fiberBundle_trivializationAtlas', Bundle.Trivialization.prod_symm_apply, VectorBundleCore.localTriv_symm_fst, VectorBundleCore.localTrivAt_apply, TangentBundle.trivializationAt_fst, MDifferentiableWithinAt.coordChange, Bundle.Trivialization.mk_symm, FiberBundle.pullback_trivializationAtlas', Bundle.Trivialization.mdifferentiableAt_section_iff, Bundle.Trivialization.continuousLinearEquivAt_apply', toProd_symm_apply_proj, Bundle.Trivial.trivialization_target, mdifferentiableAt_totalSpace, FiberBundle.chartedSpace_chartAt, Bundle.Trivialization.symm_apply_apply, Bundle.Trivialization.IsLinear.linear, Bundle.Trivialization.comp_continuousLinearEquivAt_eq_coord_change, memTrivializationAtlas_iff, Bundle.Trivialization.prod_baseSet, mdifferentiableAt_hom_bundle, FiberBundleCore.mem_localTrivAsPartialEquiv_source, Bundle.Trivialization.prod_symm_apply_snd, eta, FiberBundle.Prod.isInducing_diag, FiberBundle.isOpenMap_proj, MDifferentiable.coordChange, Bundle.Trivialization.continuousLinearEquivAt_prod, Bundle.Trivialization.prod_apply', TangentBundle.trivializationAt_apply, Bundle.Trivialization.prod_source, ContMDiffWithinAt.change_section_trivialization, FiberPrebundle.totalSpaceMk_isInducing, Bundle.Trivialization.mdifferentiableWithinAt_totalSpace_iff, Bundle.Trivialization.mdifferentiableOn_section_baseSet_iff, UniqueMDiffWithinAt.bundle_preimage', tangentMapWithin_proj, tangentMap_prodSnd, Bundle.Trivialization.contMDiffOn_localFrame_baseSet, VectorBundleCore.baseSet_at, Bundle.Trivialization.apply_symm_apply_eq_coordChangeL, FiberBundle.writtenInExtChartAt_trivializationAt, Bundle.Trivialization.coe_linearMapAt_of_mem, pullbackTopology_def, Bundle.Pretrivialization.apply_mk_symm, ContMDiffOn.continuousOn_tangentMapWithin, FiberBundleCore.localTriv_apply, Bundle.Trivialization.prod_target, Bundle.Pretrivialization.coe_linearMapAt_of_mem, contMDiffOn_baseSet_localFrame_coeff, Bundle.Trivialization.continuousOn_symm, FiberBundleCore.mem_localTriv_source, FiberBundleCore.localTrivAt_apply, Bundle.Trivialization.toPretrivialization.isLinear, Bundle.Pretrivialization.continuousOn_continuousLinearMapCoordChange, FiberBundle.isQuotientMap_proj, VectorBundle.continuousLinearEquivAt_apply, FiberBundleCore.localTrivAsPartialEquiv_apply, VectorBundleCore.mem_source_at, trivializationAt_model_space_apply, Bundle.Trivialization.coe_coordChangeL, contMDiffOn_continuousLinearMapCoordChange, VectorPrebundle.pretrivialization_mem_atlas, Bundle.mdifferentiableWithinAt_proj, VectorBundle.continuousOn_coordChange', Bundle.Trivialization.pullback_baseSet, FiberBundle.chartedSpace'_chartAt, Bundle.Trivialization.mdifferentiableOn_section_iff, Bundle.Trivial.trivialization_apply, tangentMapWithin_prodFst, FiberBundleCore.mem_localTrivAt_target, Bundle.contMDiffWithinAt_totalSpace, TangentBundle.trivializationAt_source, Bundle.Trivialization.contMDiffWithinAt_iff, Bundle.mdifferentiableOn_proj, Bundle.Pretrivialization.continuousLinearMap_symm_apply, FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter, Bundle.Trivialization.pullback_source, Bundle.Pretrivialization.coe_coe_fst, TangentBundle.chartAt, contMDiffOn_symm_coordChangeL, contMDiffAt_hom_bundle, MDifferentiableAt.coordChange, mdifferentiableOn_baseSet_localFrame_coeff, FiberBundle.extChartAt, FiberBundle.trivializationAt_proj_fst, Bundle.Trivialization.zeroSection, Bundle.Trivialization.linearEquivAt_apply, Pullback.continuous_proj, Bundle.contMDiffWithinAt_proj, Bundle.Pretrivialization.symm_coe_proj, VectorBundleCore.localTriv_apply, Bundle.contMDiffAt_totalSpace, FiberBundle.prod_trivializationAtlas', UniqueMDiffOn.tangentBundle_proj_preimage, TangentBundle.mem_chart_target_iff, Bundle.Trivialization.linear, Bundle.Trivialization.isLocalFrameOn_localFrame_baseSet, Bundle.Trivialization.symm_coe_proj, Bundle.Trivialization.baseSet_continuousLinearMap, VectorPrebundle.exists_coordChange, Bundle.contMDiffOn_proj, Bundle.contMDiff_proj, mdifferentiableOn_continuousLinearMapCoordChange, Bundle.Trivialization.continuousLinearEquivAt_apply, Bundle.Trivialization.mk_mem_target, Bundle.Trivialization.linearMapAt_apply, VectorPrebundle.totalSpaceMk_isInducing, Bundle.Trivialization.apply_eq_prod_continuousLinearEquivAt, Bundle.Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, FiberPrebundle.isOpen_source, VectorPrebundle.mem_trivialization_at_source, FiberBundle.isCoveringMap, tangentMap_chart_symm, UniqueMDiffWithinAt.bundle_preimage, range_mk, FiberBundle.map_proj_nhds, VectorPrebundle.mem_base_pretrivializationAt, Bundle.Trivialization.contMDiff_iff, Bundle.Trivialization.continuousLinearMap_apply, VectorBundleCore.localTrivAt_apply_mk, hom_trivializationAt_source, mdifferentiableOn_coordChangeL, FiberBundle.mem_trivializationAt_proj_source, VectorPrebundle.mk_coordChange, FiberBundle.extChartAt_target, Bundle.Trivialization.coordChangeL_symm_apply, ContMDiffVectorBundle.contMDiffOn_coordChangeL, mdifferentiableAt_section, Bundle.Trivialization.contMDiffAt_section_iff, Bundle.mdifferentiable_proj, Bundle.Trivial.homeomorphProd_symm_apply_proj, FiberBundle.writtenInExtChartAt_trivializationAt_symm, tangentBundleCore_localTriv_baseSet, FiberBundle.trivialization_mem_atlas, Bundle.Trivialization.coe_coe_fst, UniqueMDiffOn.bundle_preimage, Bundle.Trivialization.contMDiffAt_iff, Bundle.contMDiffAt_section, Bundle.Trivialization.contMDiffWithinAt_section, Bundle.Trivialization.Prod.continuous_to_fun, tangentMap_proj, Bundle.Trivialization.pullback_symm_apply_snd, MDifferentiableOn.coordChange, Bundle.Trivialization.continuousOn, FiberBundle.exists_trivialization_Icc_subset, TangentBundle.trivializationAt_baseSet, Bundle.Trivialization.contMDiffOn_symm, FiberBundle.continuous_proj, VectorPrebundle.continuousOn_coordChange, tangentMap_prodFst, ContMDiffOn.contMDiffOn_tangentMapWithin, FiberBundle.surjective_proj, Bundle.Pretrivialization.linearEquivAt_apply, VectorBundleCore.mem_localTriv_source, continuousAt_hom_bundle, VectorPrebundle.contMDiffOn_contMDiffCoordChange, Bundle.Trivialization.mk_coordChangeL, ContMDiffOn.coordChange, Bundle.Trivialization.coordChangeL_apply', continuousWithinAt_hom_bundle, Bundle.contMDiffAt_proj, Bundle.Pretrivialization.symm_apply, Bundle.Pretrivialization.mk_mem_target, hom_chart, VectorPrebundle.totalSpaceMk_preimage_source, TangentBundle.coe_chartAt_symm_fst, hom_trivializationAt_apply, Bundle.contMDiffWithinAt_section, Bundle.Pretrivialization.coe_mem_source, VectorPrebundle.IsContMDiff.exists_contMDiffCoordChange, Bundle.Pretrivialization.symm_apply_apply_mk, Bundle.Trivialization.symm_apply_apply_mk, FiberBundleCore.mem_localTrivAt_baseSet, contMDiffOn_coordChangeL, hom_trivializationAt_baseSet, ext_iff, VectorBundleCore.mem_localTrivAt_baseSet, VectorPrebundle.coordChange_apply, FiberBundle.mem_baseSet_trivializationAt, ContMDiffAt.coordChange, Bundle.Trivialization.mdifferentiableAt_snd_comp_iff₂, Bundle.Trivial.trivialization_baseSet, equivTangentBundleProd_symm_apply_proj, Bundle.Trivialization.contMDiffOn_iff, Bundle.Trivial.trivialization_symm_apply_proj, Bundle.Trivialization.mdifferentiableWithinAt_section_iff, TangentBundle.chartAt_toPartialEquiv, Bundle.Trivialization.contMDiffWithinAt_snd_comp_iff₂, tangentMap_prod_right, Bundle.Trivialization.coe_linearMapAt, Bundle.Trivialization.contMDiffOn_section_iff, tangentMap_prod_left, FiberBundle.chartedSpace_chartAt_symm_fst, FiberPrebundle.totalSpaceMk_preimage_source, Bundle.Pretrivialization.coe_linearMapAt, Bundle.Trivialization.prod_apply, Bundle.Trivialization.mdifferentiableWithinAt_snd_comp_iff₂, FiberPrebundle.mem_base_pretrivializationAt, Bundle.Pretrivialization.mk_symm, Bundle.Trivial.homeomorphProd_apply, mdifferentiableWithinAt_hom_bundle, MDifferentiableWithinAt.change_section_trivialization, Bundle.Trivialization.contMDiffOn_symm_trans, Bundle.Trivialization.pullback_apply, hom_trivializationAt_target, Bundle.Trivialization.coe_coordChangeL', Bundle.Trivial.trivialization_source, Bundle.Trivialization.apply_mk_symm, Bundle.Trivialization.pullback_target, FiberBundleCore.localTrivAt_apply_mk, Bundle.Pretrivialization.continuousLinearMapCoordChange_apply, Bundle.Trivialization.mdifferentiableAt_totalSpace_iff, VectorPrebundle.contMDiffCoordChange_apply, continuousOn_coordChange, Bundle.Pretrivialization.linearMapAt_apply, contMDiffWithinAt_hom_bundle, Bundle.Pretrivialization.linear, Bundle.Trivial.trivialization_symm_apply_snd, Bundle.Trivialization.coe_mem_source, Bundle.Trivialization.Bundle.Trivialization.mdifferentiable, TangentBundle.mem_chart_source_iff, Bundle.Pullback.lift_proj, TangentBundle.trivializationAt_target, FiberBundleCore.localTrivAt_snd, FiberPrebundle.continuousOn_of_comp_right, FiberPrebundle.pretrivialization_mem_atlas
snd 📖CompOp
48 mathmath: TangentBundle.tangentMap_tangentBundle_pure, Bundle.Pretrivialization.symm_proj_apply, tangentMapWithin_prodSnd, tangentMap_snd, toProd_apply, Bundle.Trivialization.symm_proj_apply, Bundle.Pretrivialization.continuousLinearMap_apply, Bundle.zeroSection_snd, Bundle.Trivialization.symm_apply, equivTangentBundleProd_apply, VectorBundleCore.localTrivAt_apply, Bundle.Trivialization.continuousLinearEquivAt_apply', tangentMapWithin_snd, mdifferentiableAt_hom_bundle, Bundle.Trivialization.prod_symm_apply_snd, 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, Bundle.Trivial.trivialization_apply, tangentMapWithin_prodFst, equivTangentBundleProd_symm_apply_snd, contMDiffAt_hom_bundle, VectorBundleCore.localTriv_apply, Bundle.Trivialization.continuousLinearMap_apply, toProd_symm_apply_snd, Bundle.Trivialization.pullback_symm_apply_snd, tangentMap_prodFst, continuousAt_hom_bundle, continuousWithinAt_hom_bundle, Bundle.Pretrivialization.symm_apply, hom_chart, hom_trivializationAt_apply, ext_iff, Bundle.Pullback.lift_snd, 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 📖mathematicalBundle.TotalSpace
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

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