Documentation Verification Report

Defs

📁 Source: Mathlib/Topology/OpenPartialHomeomorph/Defs.lean

Statistics

MetricCount
DefinitionstoOpenPartialHomeomorph, toOpenPartialHomeomorphOfImageEq, toPartialHomeomorph, toPartialHomeomorphOfImageEq, apply, symm_apply, instCoeFunForall, replaceEquiv, toFun', toPartialEquiv
10
TheoremstoOpenPartialHomeomorphOfImageEq_apply, toOpenPartialHomeomorphOfImageEq_source, toOpenPartialHomeomorphOfImageEq_symm_apply, toOpenPartialHomeomorphOfImageEq_target, toOpenPartialHomeomorphOfImageEq_toPartialEquiv, toOpenPartialHomeomorph_apply, toOpenPartialHomeomorph_source, toOpenPartialHomeomorph_symm_apply, toOpenPartialHomeomorph_target, bijOn, coe_coe, coe_coe_symm, continuousOn, continuousOn_invFun, continuousOn_symm, continuousOn_toFun, eq_symm_apply, ext, ext_iff, injOn, invFun_eq_coe, invOn, leftInvOn, left_inv, map_source, map_source'', map_target, mapsTo, mk_coe, mk_coe_symm, open_source, open_target, replaceEquiv_eq_self, rightInvOn, right_inv, surjOn, symm_bijective, symm_mapsTo, symm_source, symm_symm, symm_target, symm_toPartialEquiv, toFun_eq_coe, toPartialEquiv_injective
44
Total54

Homeomorph

Definitions

NameCategoryTheorems
toOpenPartialHomeomorph 📖CompOp
17 mathmath: toOpenPartialHomeomorph_target, Diffeomorph.toPartialHomeomorph_mdifferentiable, Diffeomorph.toOpenPartialHomeomorph_mdifferentiable, toOpenPartialHomeomorph_apply, transPartialHomeomorph_eq_trans, trans_toPartialHomeomorph, refl_toOpenPartialHomeomorph, transOpenPartialHomeomorph_eq_trans, trans_toOpenPartialHomeomorph, symm_toPartialHomeomorph, toOpenPartialHomeomorph_symm_apply, refl_toPartialHomeomorph, Structomorph.mem_groupoid, toOpenPartialHomeomorph_source, symm_toOpenPartialHomeomorph, OpenPartialHomeomorph.transHomeomorph_eq_trans, StructureGroupoid.restriction_mem_maximalAtlas_subtype
toOpenPartialHomeomorphOfImageEq 📖CompOp
5 mathmath: toOpenPartialHomeomorphOfImageEq_target, toOpenPartialHomeomorphOfImageEq_toPartialEquiv, toOpenPartialHomeomorphOfImageEq_source, toOpenPartialHomeomorphOfImageEq_apply, toOpenPartialHomeomorphOfImageEq_symm_apply
toPartialHomeomorph 📖CompOp
toPartialHomeomorphOfImageEq 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
toOpenPartialHomeomorphOfImageEq_apply 📖mathematicalIsOpen
Set.image
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
OpenPartialHomeomorph.toFun'
toOpenPartialHomeomorphOfImageEq
toOpenPartialHomeomorphOfImageEq_source 📖mathematicalIsOpen
Set.image
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
toOpenPartialHomeomorphOfImageEq
toOpenPartialHomeomorphOfImageEq_symm_apply 📖mathematicalIsOpen
Set.image
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
toOpenPartialHomeomorphOfImageEq
symm
toOpenPartialHomeomorphOfImageEq_target 📖mathematicalIsOpen
Set.image
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
toOpenPartialHomeomorphOfImageEq
toOpenPartialHomeomorphOfImageEq_toPartialEquiv 📖mathematicalIsOpen
Set.image
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
OpenPartialHomeomorph.toPartialEquiv
toOpenPartialHomeomorphOfImageEq
Equiv.toPartialEquivOfImageEq
toEquiv
toOpenPartialHomeomorph_apply 📖mathematicalOpenPartialHomeomorph.toFun'
toOpenPartialHomeomorph
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
toOpenPartialHomeomorph_source 📖mathematicalPartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
toOpenPartialHomeomorph
Set.univ
toOpenPartialHomeomorph_symm_apply 📖mathematicalOpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
toOpenPartialHomeomorph
DFunLike.coe
Homeomorph
EquivLike.toFunLike
instEquivLike
symm
toOpenPartialHomeomorph_target 📖mathematicalPartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
toOpenPartialHomeomorph
Set.univ

OpenPartialHomeomorph

Definitions

NameCategoryTheorems
instCoeFunForall 📖CompOp
replaceEquiv 📖CompOp
1 mathmath: replaceEquiv_eq_self
toFun' 📖CompOp
450 mathmath: NumberField.mixedEmbedding.integral_comp_polarSpaceCoord_symm, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply', Trivialization.proj_symm_apply', trans_apply, symm_map_nhds_eq, bijOn, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_fst, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart', NumberField.mixedEmbedding.lintegral_comp_polarCoord_symm, ModularFormClass.differentiableAt_comp_ofComplex, NumberField.mixedEmbedding.measurable_polarCoordReal_symm, map_subtype_source, Trivialization.preimageHomeomorph_symm_apply, UniqueMDiffOn.uniqueMDiffOn_preimage, writtenInExtChartAt_chartAt, IsImage.symm_apply_mem_iff, subtypeRestr_symm_trans_subtypeRestr, Complex.norm_polarCoord_symm, eventually_left_inverse', eventually_ne_nhdsWithin, restr_target, contDiff_unitBallBall_symm, CuspFormClass.zero_at_infty_comp_ofComplex, Complex.integral_comp_pi_polarCoord_symm, symm_image_target_eq_source, subtypeRestr_coe, IsImage.preimage_eq', contMDiffOn_chart, chart_target_mem_nhds, image_eq_target_inter_inv_preimage, image_trans_source, continuousAt, lift_openEmbedding_toFun, tangentMap_chart, TopologicalSpace.Opens.chartAt_subtype_val_symm_eventuallyEq, StructureGroupoid.LocalInvariantProp.liftPropAt_chart_symm, UpperHalfPlane.comp_ofComplex_of_im_pos, UniqueMDiffWithinAt.preimage_PartialHomeomorph, NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isComplex, lift_openEmbedding_trans_apply, continuousOn_univBall_symm, transHomeomorph_symm_apply, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_ne, TangentBundle.coe_chartAt_fst, ext_iff, IsLocalHomeomorphOn.OpenPartialHomeomorph.isLocalHomeomorphOn, EqOnSource.eqOn, ChartedSpace.LiftPropWithinAt.prop, Homeomorph.toOpenPartialHomeomorph_apply, isLittleO_congr, trans'_symm_apply, NumberField.mixedEmbedding.fundamentalCone.compactSet_eq_union_aux₁, NumberField.mixedEmbedding.polarCoordReal_apply, map_nhdsWithin_eq, hasMFDerivAt_extChartAt, trans_source', preimage_interior, coe_coe, coe_ofContinuousOpenRestrict_symm, contMDiffOn_chart_symm, Real.expPartialHomeomorph_apply, Trivialization.contMDiffOn_symm_trans, refl_apply, MDifferentiable.mdifferentiableAt_symm, symm_mapsTo, univUnitBall_symm_apply_zero, HasStrictFDerivAt.eq_implicitFunctionOfComplemented, univUnitBall_apply_zero, UpperHalfPlane.comp_ofComplex, contMDiffOn_isOpenEmbedding_symm, Trivialization.coe_coe, source_inter_preimage_inv_preimage, contDiff_univUnitBall, tangentBundle_model_space_coe_chartAt, StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_groupoid, isLocalStructomorphWithinAt_iff, isLocalStructomorphWithinAt_source_iff, toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply, Complex.integral_comp_polarCoord_symm, HasStrictFDerivAt.localInverse_def, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_apply, prod_symm_apply, NumberField.mixedEmbedding.normAtComplexPlaces_polarSpaceCoord_symm, UpperHalfPlane.ofComplex_apply_of_im_nonpos, VectorBundleCore.localTriv_symm_fst, Unitary.openPartialHomeomorph_symm_apply, NumberField.mixedEmbedding.fundamentalCone.logMap_expMapBasis, toHomeomorphSourceTarget_apply_coe, stereographic_apply_neg, TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq, range_stereographic_symm, eventually_left_inverse, isOpenEmbedding_stereographic_symm, lintegral_comp_pi_polarCoord_symm, NumberField.mixedEmbedding.fundamentalCone.subset_interior_normLeOne, hasMFDerivWithinAt_extChartAt, eventually_nhdsWithin, toFun_eq_coe, trans_source'', Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm, mfderiv_chartAt_eq_tangentCoordChange, mk_coe, Complex.polarCoord_apply, mk_coe_symm, continuous_polarCoord_symm, target_inter_inv_preimage_preimage, Real.coe_tanPartialHomeomorph, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux, Trivialization.domExtend_symm_apply, isOpen_image_iff_of_subset_source, NumberField.mixedEmbedding.fundamentalCone.prod_deriv_expMap_single, UpperHalfPlane.ofComplex_apply_of_im_pos, isOpen_image_source_inter, inv_image_trans_target, leftInvOn, NumberField.mixedEmbedding.fundamentalCone.norm_expMapBasis, FiberwiseLinear.trans_openPartialHomeomorph_apply, isBigO_congr, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply_ker, NumberField.mixedEmbedding.polarCoord_symm_apply, HasStrictFDerivAt.implicitToPartialHomeomorph_self, pi_apply, Trivialization.symm_apply_apply, eventually_right_inverse, integral_comp_polarCoord_symm, TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_coe, isOpen_image_of_subset_source, left_inv, prod_apply, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_aux, symm_image_target_inter_eq, SlashInvariantFormClass.periodic_comp_ofComplex, ChartedSpace.liftPropAt_iff, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_eq, Trivialization.map_target, NumberField.mixedEmbedding.fundamentalCone.prod_expMapBasis_pow, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_closure_subset_compactSet, isOpenEmbedding_restrict, NumberField.mixedEmbedding.fundamentalCone.continuous_expMap, IsImage.iff_preimage_eq', preimage_frontier, coe_ofContinuousOpenRestrict, image_source_eq_target, ImplicitFunctionData.toPartialHomeomorph_coe, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_self, NumberField.mixedEmbedding.fundamentalCone.expMap_smul, continuous_univBall, Topology.IsOpenEmbedding.toPartialHomeomorph_left_inv, continuousWithinAt_iff_continuousWithinAt_comp_right, extend_coe_symm, UpperHalfPlane.ofComplex_apply_eq_of_im_nonpos, image_mem_nhds, restrContDiff_target, UniqueMDiffWithinAt.preimage_openPartialHomeomorph, univBall_apply_zero, continuousAt_iff_continuousAt_comp_right, contDiffOn_univUnitBall_symm, subtypeRestr_symm_apply, restr_apply, map_nhdsWithin_preimage_eq, ChartedSpace.liftProp_iff, image_source_inter_eq, MDifferentiable.comp_symm_deriv, restrContDiff_symm_apply, polarCoord_symm_apply, ImplicitFunctionData.implicitFunction_apply, NumberField.mixedEmbedding.fundamentalCone.hasFDerivAt_expMap, StructureGroupoid.liftPropWithinAt_self_target, IsImage.restr_apply, isLocalStructomorphOn_contDiffGroupoid_iff, tangentBundle_model_space_coe_chartAt_symm, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_coe, contMDiffOn_of_mem_contDiffGroupoid, MDifferentiable.mfderiv_bijective, UpperHalfPlane.ofComplex_apply_eq_ite, toHomeomorphOfSourceEqUnivTargetEqUniv_apply, Homeomorph.unitBall_apply_coe, NumberField.mixedEmbedding.fundamentalCone.continuous_expMapBasis, continuous_iff_continuous_comp_left, map_target, StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_groupoid, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_self, Trivialization.symm_apply, restrContDiff_source, ApproximatesLinearOn.toOpenPartialHomeomorph_coe, Complex.lintegral_comp_polarCoord_symm, stereographic_neg_apply, NumberField.mixedEmbedding.fundamentalCone.expMap_symm_apply, UpperHalfPlane.J_smul, StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_maximalAtlas, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_fst, IsImage.mapsTo, Trivialization.apply_symm_apply, writtenInExtChartAt_chartAt_comp, contDiff_univBall, NumberField.mixedEmbedding.polarSpaceCoord_symm_apply, Complex.measurableEquivRealProd_symm_polarCoord_symm_apply, surjOn, MDifferentiable.range_mfderiv_eq_top, trans_ofSet, contDiffOn_restrContDiff_source, hasFDerivAt_polarCoord_symm, IsImage.restr_symm_apply, StructureGroupoid.LocalInvariantProp.right_invariance', ChartedSpace.isOpen_iff, coe_restrOpen_symm, isOpen_symm_image_iff_of_subset_target, NumberField.mixedEmbedding.fundamentalCone.hasFDerivAt_expMapBasis, NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isReal, contDiff_unitBallBall, UpperHalfPlane.ofComplex_apply, MDifferentiable.mfderiv_surjective, NumberField.mixedEmbedding.fundamentalCone.hasDerivAt_expMap_single, IsImage.symm_mapsTo, StructureGroupoid.LocalInvariantProp.liftPropAt_symm_of_mem_maximalAtlas, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux2, NumberField.mixedEmbedding.polarCoord_symm_eq, image_source_inter_eq', Homeomorph.toOpenPartialHomeomorphOfImageEq_apply, Trivialization.pullback_symm_apply_snd, lift_openEmbedding_symm, Trivialization.prod_symm_apply_proj, Bundle.Trivial.toOpenPartialHomeomorph_trivialization_symm_apply, trans_source, Trivialization.prod_symm_apply_snd, IsImage.leftInvOn_piecewise, UpperHalfPlane.IsZeroAtImInfty.zero_at_infty_comp_ofComplex, trans_target', mdifferentiableOn_atlas, map_source, isOpen_inter_preimage, UpperHalfPlane.eventuallyEq_coe_comp_ofComplex, ofSet_apply, symm_trans_restr, invFun_eq_coe, tangentMap_chart_symm, StructureGroupoid.LocalInvariantProp.liftPropOn_chart, Trivialization.symm_coe_proj, UpperHalfPlane.comp_ofComplex_of_im_le_zero, IsImage.iff_symm_preimage_eq, symm_image_eq_source_inter_preimage, EqOnSource.symm_eqOn_target, NumberField.mixedEmbedding.fundamentalCone.expMap_single_symm_apply, continuousOn, eventually_nhdsWithin', eq_symm_apply, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_pos, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_self, eventually_nhds', SmoothBumpCovering.comp_embeddingPiTangent_mfderiv, continuousAt_iff_continuousAt_comp_left, NumberField.mixedEmbedding.fundamentalCone.expMap_apply, contMDiffAt_symm_of_mem_maximalAtlas, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff, writtenInExtChartAt_sumSwap_eventuallyEq_id, eventually_right_inverse', StructureGroupoid.LocalInvariantProp.right_invariance, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_fst, univUnitBall_apply, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_apply_ker, contDiffOn_restrContDiff_target, extend_coe, extend_symm_continuousWithinAt_comp_right_iff, FiberBundle.writtenInExtChartAt_trivializationAt_symm, pi_symm_apply, Trivialization.apply_symm_apply', StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source, mdifferentiableAt_atlas_symm, restr_symm_apply, chartAt_comp, IsImage.image_eq, Trivialization.mk_coordChange, isLocalStructomorphWithinAt_iff', NumberField.mixedEmbedding.fundamentalCone.injective_expMapBasis, Trivialization.preimageSingletonHomeomorph_symm_apply, preimage_closure, injOn, subtypeRestr_symm_eqOn, NumberField.mixedEmbedding.lintegral_comp_polarCoordReal_symm, Trivialization.mk_symm, NumberField.mixedEmbedding.measurable_polarCoord_symm, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply'', Trivialization.proj_symm_apply, injOn_pi_polarCoord_symm, NumberField.mixedEmbedding.polarCoord_apply, MDifferentiable.symm_comp_deriv, NumberField.mixedEmbedding.hasFDerivAt_polarCoordReal_symm, trans_of_set', restrContDiff_apply, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target, rightInvOn, hasFDerivAt_pi_polarCoord_symm, IsImage.symm_image_eq, coe_trans, Homeomorph.toOpenPartialHomeomorphOfImageEq_symm_apply, Trivialization.pullback_symm_apply_proj, MDifferentiable.mfderiv_injective, lintegral_comp_polarCoord_symm, subtypeRestr_symm_eqOn_of_le, polarCoord_apply, contMDiffOn_of_mem_maximalAtlas, Trivialization.proj_toFun, Trivialization.sourceHomeomorphBaseSetProd_symm_apply, Homeomorph.transOpenPartialHomeomorph_symm_apply, NumberField.mixedEmbedding.polarCoordReal_symm_target_ae_eq_univ, UpperHalfPlane.contMDiffAt_ofComplex, AddCircle.openPartialHomeomorphCoe_symm_apply, ModularFormClass.bounded_at_infty_comp_ofComplex, NumberField.mixedEmbedding.polarSpaceCoord_apply, Homeomorph.toOpenPartialHomeomorph_symm_apply, contDiffOn_univBall_symm, stereographic_apply, writtenInExtChartAt_chartAt_symm_comp, NumberField.mixedEmbedding.fundamentalCone.expMap_add, IsImage.map_nhdsWithin_eq, UpperHalfPlane.contMDiffAt_iff, MDifferentiable.mdifferentiableAt, invOn, IsImage.iff_symm_preimage_eq', NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply, AddCircle.openPartialHomeomorphCoe_apply, writtenInExtChartAt_chartAt_symm, contMDiffAt_of_mem_maximalAtlas, map_source'', Real.expPartialHomeomorph_symm_apply, isBigOWith_congr, Trivialization.contMDiffOn_symm, StructureGroupoid.LocalInvariantProp.liftPropOn_symm_of_mem_maximalAtlas, mapsTo, StructureGroupoid.LocalInvariantProp.left_invariance, EisensteinSeries.eisSummand_extension_differentiableOn, coe_ofContinuousOpen_symm, continuousOn_symm, MDifferentiable.ker_mfderiv_eq_bot, ImplicitFunctionData.toOpenPartialHomeomorph_coe, Real.coe_tanPartialHomeomorph_symm, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux, Complex.lintegral_comp_pi_polarCoord_symm, NumberField.mixedEmbedding.fundamentalCone.setLIntegral_expMapBasis_image, NumberField.mixedEmbedding.fundamentalCone.expMap_single_apply, EisensteinSeries.eisensteinSeries_tendstoLocallyUniformlyOn, Complex.polarCoord_symm_apply, source_preimage_target, stereographic'_symm_apply, hom_chart, HasStrictFDerivAt.toOpenPartialHomeomorph_coe, IsImage.preimage_eq, Trivialization.apply_symm_apply_eq_coordChangeL, TangentBundle.coe_chartAt_symm_fst, NumberField.mixedEmbedding.fundamentalCone.compactSet_ae, NumberField.mixedEmbedding.fundamentalCone.abs_det_fderiv_expMapBasis, ContDiffAt.toOpenPartialHomeomorph_coe, pi_polarCoord_symm_target_ae_eq_univ, preimage_eventuallyEq_target_inter_preimage_inter, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_expMap_symm, ApproximatesLinearOn.toPartialHomeomorph_coe, extChartAt_coe_symm, extChartAt_comp, StructureGroupoid.liftPropWithinAt_self_source, MDifferentiable.range_mfderiv_eq_univ, HasStrictFDerivAt.implicitToPartialHomeomorph_fst, Trivialization.symm_apply_mk_proj, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_right_inv, StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_maximalAtlas, NumberField.mixedEmbedding.fundamentalCone.sum_expMap_symm_apply, to_isOpenEmbedding, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply_ker, tendsto_symm, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_image_preimage_expMapBasis, UpperHalfPlane.mdifferentiableAt_iff, coe_trans_symm, map_nhds_eq, ImplicitFunctionData.toPartialHomeomorph_apply, continuousAt_symm, isOpen_inter_preimage_symm, continuousWithinAt_iff_continuousWithinAt_comp_left, NumberField.mixedEmbedding.fundamentalCone.expMap_sum, Bundle.Trivial.trivialization_symm_apply_proj, nhds_eq_comap_inf_principal, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_normLeOne_eq_image, coe_ofContinuousOpen, ImplicitFunctionData.toOpenPartialHomeomorph_apply, IsImage.symm_preimage_eq', Unitary.openPartialHomeomorph_apply, trans'_apply, StructureGroupoid.LocalInvariantProp.liftPropOn_chart_symm, coe_coe_symm, transHomeomorph_apply, contMDiffOn_symm_of_mem_maximalAtlas, unitBallBall_symm_apply, mem_chart_target, FiberwiseLinear.trans_PartialHomeomorph_apply, UpperHalfPlane.tendsto_comap_im_ofComplex, NumberField.mixedEmbedding.integral_comp_polarCoord_symm, univUnitBall_symm_apply, FiberBundle.chartedSpace_chartAt_symm_fst, NumberField.mixedEmbedding.fundamentalCone.expMap_pos, IsImage.apply_mem_iff, Topology.IsOpenEmbedding.toPartialHomeomorph_right_inv, HasStrictFDerivAt.toPartialHomeomorph_coe, StructureGroupoid.LocalInvariantProp.liftPropAt_chart, IsImage.symm_preimage_eq, coe_restrOpen, continuousOn_iff_continuousOn_comp_right, trans_target, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_left_inv, unitBallBall_apply, NumberField.mixedEmbedding.fundamentalCone.normLeOne_eq_preimage, HasStrictFDerivAt.implicitToPartialHomeomorph_apply_ker, ContDiffAt.toPartialHomeomorph_coe, mdifferentiableOn_atlas_symm, UpperHalfPlane.mdifferentiable_iff, sum_chartAt_inl_apply, NumberField.mixedEmbedding.integral_comp_polarCoordReal_symm, NumberField.mixedEmbedding.fundamentalCone.compactSet_eq_union, extChartAt_coe, mdifferentiableAt_atlas, StructureGroupoid.LocalInvariantProp.left_invariance', integral_comp_pi_polarCoord_symm, NumberField.mixedEmbedding.measurable_polarSpaceCoord_symm, continuousOn_iff_continuousOn_comp_left, isOpen_image_symm_of_subset_target, UpperHalfPlane.mdifferentiableAt_ofComplex, Homeomorph.transOpenPartialHomeomorph_apply, NumberField.mixedEmbedding.fundamentalCone.injective_expMap, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_ne, HasStrictFDerivAt.eq_implicitFunction, univBall_symm_apply_center, Trivialization.coordChangeL_apply', StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart, Bundle.Trivial.trivialization_symm_apply_snd, trans_target'', surjective_stereographic, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_nonneg, source_inter_preimage_target_inter, Topology.IsOpenEmbedding.singletonChartedSpace_chartAt_eq, toHomeomorphSourceTarget_symm_apply_coe, FiberBundleCore.localTriv_symm_apply, Units.chartAt_apply, mem_groupoid_of_pregroupoid, const_apply, IsImage.iff_preimage_eq, lift_openEmbedding_apply, sum_chartAt_inr_apply, right_inv, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply, eventually_nhds, NumberField.mixedEmbedding.lintegral_comp_polarSpaceCoord_symm, ChartedSpace.liftPropWithinAt_iff'
toPartialEquiv 📖CompOp
351 mathmath: Unitary.openPartialHomeomorph_source, NumberField.mixedEmbedding.integral_comp_polarSpaceCoord_symm, Homeomorph.toOpenPartialHomeomorph_target, Trivialization.pullback_source, bijOn, SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source, NumberField.mixedEmbedding.lintegral_comp_polarCoord_symm, ApproximatesLinearOn.toPartialHomeomorph_target, UniqueMDiffOn.uniqueMDiffOn_preimage, isOpen_extend_preimage, subtypeRestr_symm_trans_subtypeRestr, restr_inter_source, restr_target, Trivialization.domExtend_source, symm_toPartialEquiv, Complex.integral_comp_pi_polarCoord_symm, IsOpen.trivializationDiscrete_target, ofSet_toPartialEquiv, symm_image_target_eq_source, Manifold.LiftSourceTargetPropertyAt.mem_domChart_source, transHomeomorph_source, IsImage.preimage_eq', contMDiffOn_chart, chart_target_mem_nhds, Manifold.LocalPresentationAt.mem_codChart_source, Trivialization.codExtend_source, image_trans_source, ImplicitFunctionData.map_pt_mem_toOpenPartialHomeomorph_target, ImplicitFunctionData.pt_mem_toPartialHomeomorph_source, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_source, EqOnSource.source_eq, ofSet_source, mem_chart_source, ofSet_trans', restr_source_inter, mdifferentiableOn_extChartAt, transHomeomorph_target, Manifold.IsImmersionAt.mem_domChart_source, ext_iff, IsLocalHomeomorphOn.OpenPartialHomeomorph.isLocalHomeomorphOn, EqOnSource.eqOn, refl_partialEquiv, VectorBundleCore.mem_localTriv_target, trans_source', preimage_interior, coe_coe, Trivialization.pullback_target, stereographic'_source, contMDiffOn_chart_symm, Trivialization.contMDiffOn_symm_trans, chart_source_mem_nhds, Homeomorph.toOpenPartialHomeomorphOfImageEq_target, symm_mapsTo, NumberField.mixedEmbedding.polarSpaceCoord_target, open_source, Trivialization.eqOn, FiberBundleCore.mk_mem_localTrivAt_source, NumberField.mixedEmbedding.fundamentalCone.expMap_target, FiberBundleCore.mem_localTrivAt_source, self_trans_symm, SmoothBumpFunction.exists_r_pos_lt_subset_ball, Trivialization.continuousOn, source_inter_preimage_inv_preimage, ofContinuousOpen_toPartialEquiv, ImplicitFunctionData.map_pt_mem_toPartialHomeomorph_target, isLocalStructomorphWithinAt_source_iff, lift_openEmbedding_source, Manifold.LocalPresentationAt.mem_domChart_source, Complex.integral_comp_polarCoord_symm, HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_target, ofContinuousOpenRestrict_toPartialEquiv, pi_target, FiberBundleCore.localTrivAsPartialEquiv_target, ImplicitFunctionData.pt_mem_toOpenPartialHomeomorph_source, toHomeomorphSourceTarget_apply_coe, restr_toPartialEquiv, restrOpen_source, HasStrictFDerivAt.mem_toOpenPartialHomeomorph_source, iUnion_source_chartAt, Trivialization.target_eq, lintegral_comp_pi_polarCoord_symm, ContDiffAt.image_mem_toOpenPartialHomeomorph_target, polarCoord_target, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target, Trivialization.restrictPreimage'_source, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_target, Bundle.Trivial.trivialization_target, toFun_eq_coe, SmoothBumpFunction.support_eq_inter_preimage, trans_source'', Manifold.IsImmersionAtOfComplement.mem_domChart_source, pi_source, Manifold.IsImmersionAt.mem_codChart_source, toPartialEquiv_injective, target_inter_inv_preimage_preimage, prod_target, NumberField.mixedEmbedding.polarCoordReal_source, stereographic'_target, isOpen_image_source_inter, inv_image_trans_target, leftInvOn, Trivialization.prod_symm_apply, Manifold.LocalPresentationAt.source_subset_preimage_source, restr_source', measurableSet_pi_polarCoord_target, FiberwiseLinear.source_trans_PartialHomeomorph, integral_comp_polarCoord_symm, extend_target, VectorBundleCore.mem_trivChange_source, EqOnSource.target_eq, SmoothBumpFunction.tsupport_subset_chartAt_source, symm_image_target_inter_eq, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_source, SmoothBumpFunction.eqOn_source, contMDiffOn_extChartAt, symm_trans_self, isOpenEmbedding_restrict, IsImage.iff_preimage_eq', preimage_frontier, Trivialization.preimage_subset_source, Trivialization.proj_surjOn_baseSet, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_target, image_source_eq_target, isOpen_extChartAt_preimage, univUnitBall_target, MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul, FiberBundleCore.mem_localTriv_source, VectorBundleCore.mem_source_at, refl_source, restrContDiff_target, Trivialization.mem_target, prod_toPartialEquiv, extChartAt_source, FiberBundleCore.mem_localTrivAt_target, TangentBundle.trivializationAt_source, Trivialization.coe_mem_source, image_source_inter_eq, ext_coord_change_source, Trivialization.codExtend'_target, univUnitBall_source, Manifold.IsImmersionAtOfComplement.source_subset_preimage_source, isLocalStructomorphOn_contDiffGroupoid_iff, ChartedSpace.mem_chart_source, SmoothBumpCovering.mem_chartAt_ind_source, unitBallBall_target, contMDiffOn_of_mem_contDiffGroupoid, trans_toPartialEquiv, FiberBundleCore.mem_localTriv_target, symm_target, pi_toPartialEquiv, extend_image_source_inter, Trivialization.mem_source, FiberBundle.extChartAt, StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_groupoid, restrContDiff_source, Complex.lintegral_comp_polarCoord_symm, Trivialization.source_inter_preimage_target_inter, NumberField.mixedEmbedding.polarCoord_source, Homeomorph.toOpenPartialHomeomorphOfImageEq_toPartialEquiv, mem_achart_source, StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_maximalAtlas, continuousOn_invFun, Unitary.openPartialHomeomorph_target, Manifold.IsImmersionAt.source_subset_preimage_source, IsImage.mapsTo, surjOn, AddCircle.openPartialHomeomorphCoe_source, contDiffOn_restrContDiff_source, IsImage.isOpen_iff, Trivialization.proj_clift, tangentBundle_model_space_chartAt, TangentBundle.mem_chart_target_iff, FiberwiseLinear.target_trans_openPartialHomeomorph, TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_target, isImage_source_target, ChartedSpace.isOpen_iff, NumberField.mixedEmbedding.polarCoord_target, extend_target', Homeomorph.toOpenPartialHomeomorphOfImageEq_source, IsImage.symm_mapsTo, SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source_of_isClosed, ContDiffAt.image_mem_toPartialHomeomorph_target, Trivialization.symm_trans_target_eq, Complex.polarCoord_source, image_source_inter_eq', NumberField.mixedEmbedding.polarSpaceCoord_target', trans_source, IsImage.leftInvOn_piecewise, IsOpen.trivializationDiscrete_source, stereographic_target, trans_target', secondCountableTopology_source, mdifferentiableOn_atlas, restr_toPartialEquiv', Trivialization.restrictPreimage'_target, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_target, isOpen_inter_preimage, symm_trans_restr, invFun_eq_coe, FiberBundleCore.localTrivAsPartialEquiv_source, StructureGroupoid.LocalInvariantProp.liftPropOn_chart, IsImage.iff_symm_preimage_eq, EqOnSource.symm_eqOn_target, TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_source, continuousOn, stereographic_source, hom_trivializationAt_source, IsImage.toPartialEquiv, restrOpen_toPartialEquiv, Trivialization.domExtend_target, Trivialization.prod_target, SmoothBumpFunction.coe_def, tangentBundleCore_baseSet, FiberBundle.mem_trivializationAt_proj_source, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff, extend_source, unitBallBall_source, Manifold.IsImmersionAtOfComplement.mem_codChart_source, Topology.IsQuotientMap.trivializationOfVAddDisjoint_source, refl_target, contDiffOn_restrContDiff_target, lift_openEmbedding_target, restr_source, FiberwiseLinear.target_trans_PartialHomeomorph, ApproximatesLinearOn.closedBall_subset_target, tangentBundleCore_localTriv_baseSet, Trivialization.restrictPreimage_source, IsImage.image_eq, preimage_closure, injOn, subtypeRestr_symm_eqOn, NumberField.mixedEmbedding.lintegral_comp_polarCoordReal_symm, NumberField.mixedEmbedding.fundamentalCone.expMap_source, Topology.IsQuotientMap.trivializationOfSMulDisjoint_target, HasStrictFDerivAt.image_mem_toPartialHomeomorph_target, Trivialization.continuousOn_proj, injOn_pi_polarCoord_symm, trans_of_set', Manifold.LiftSourceTargetPropertyAt.mem_codChart_source, subtypeRestr_target_subset, const_target, Homeomorph.transOpenPartialHomeomorph_target, rightInvOn, TangentBundle.trivializationAt_baseSet, IsImage.symm_image_eq, lintegral_comp_polarCoord_symm, HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_source, subtypeRestr_symm_eqOn_of_le, contMDiffOn_of_mem_maximalAtlas, ApproximatesLinearOn.toOpenPartialHomeomorph_target, Trivialization.sourceHomeomorphBaseSetProd_symm_apply, NumberField.mixedEmbedding.polarCoordReal_symm_target_ae_eq_univ, Trivialization.symm_trans_source_eq, lift_openEmbedding_symm_target, continuousOn_toFun, ContMDiffFiberwiseLinear.locality_aux₁, polarCoord_source_ae_eq_univ, Homeomorph.transOpenPartialHomeomorph_source, VectorBundleCore.mem_localTriv_source, ContDiffAt.mem_toPartialHomeomorph_source, contMDiffOn_extend_symm, symm_source, invOn, IsImage.iff_symm_preimage_eq', Trivialization.mk_mem_target, map_source'', extChartAt_target, Trivialization.contMDiffOn_symm, StructureGroupoid.LocalInvariantProp.liftPropOn_symm_of_mem_maximalAtlas, mapsTo, prod_source, continuousOn_symm, NumberField.mixedEmbedding.polarSpaceCoord_source, Trivialization.codExtend_target, Complex.lintegral_comp_pi_polarCoord_symm, ofSet_target, Topology.IsQuotientMap.trivializationOfSMulDisjoint_source, source_preimage_target, NumberField.mixedEmbedding.polarCoord_target_eq_polarCoordReal_target, IsImage.preimage_eq, pi_polarCoord_symm_target_ae_eq_univ, ball_subset_univBall_target, extChartAt_comp, ApproximatesLinearOn.toPartialHomeomorph_source, HasStrictFDerivAt.mem_implicitToPartialHomeomorph_source, Homeomorph.toOpenPartialHomeomorph_source, HasStrictFDerivAt.mem_implicitToPartialHomeomorph_target, Trivialization.sourceHomeomorphBaseSetProd_apply, Real.expPartialHomeomorph_source, lift_openEmbedding_symm_source, SmoothBumpFunction.support_subset_source, isOpen_inter_preimage_symm, Real.expPartialHomeomorph_target, open_target, ContDiffAt.mem_toOpenPartialHomeomorph_source, Topology.IsQuotientMap.trivializationOfVAddDisjoint_target, TangentBundle.chartAt_toPartialEquiv, IsImage.symm_preimage_eq', ModelWithCorners.uniqueDiffOn_preimage_source, StructureGroupoid.LocalInvariantProp.liftPropOn_chart_symm, NumberField.mixedEmbedding.polarCoordReal_target, coe_coe_symm, extend_coord_change_source, Complex.polarCoord_target, contMDiffOn_symm_of_mem_maximalAtlas, univBall_target, Trivialization.frontier_preimage, mem_chart_target, subtypeRestr_source, Trivialization.codExtend'_source, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_source, ApproximatesLinearOn.toOpenPartialHomeomorph_source, Units.chartAt_source, NumberField.mixedEmbedding.integral_comp_polarCoord_symm, FiberwiseLinear.source_trans_openPartialHomeomorph, eqOnSource_iff, IsImage.symm_preimage_eq, trans_target, HasStrictFDerivAt.image_mem_toOpenPartialHomeomorph_target, const_source, NumberField.mixedEmbedding.fundamentalCone.expMap_single_source, Trivialization.contMDiffOn, mdifferentiableOn_atlas_symm, NumberField.mixedEmbedding.integral_comp_polarCoordReal_symm, hom_trivializationAt_target, Bundle.Trivial.trivialization_source, integral_comp_pi_polarCoord_symm, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_source, polarCoord_source, Trivialization.prod_source, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_source, StructureGroupoid.restriction_mem_maximalAtlas_subtype, HasStrictFDerivAt.mem_toPartialHomeomorph_source, AddCircle.openPartialHomeomorphCoe_target, FiberBundleCore.localTrivAsPartialEquiv_symm, FiberBundleCore.localTrivAsPartialEquiv_trans, SmoothBumpCovering.mem_chartAt_source_of_eq_one, trans_target'', NumberField.mixedEmbedding.fundamentalCone.expMap_single_target, source_inter_preimage_target_inter, toHomeomorphSourceTarget_symm_apply_coe, TangentBundle.mem_chart_source_iff, Manifold.LiftSourceTargetPropertyAt.source_subset_preimage_source, TangentBundle.trivializationAt_target, mem_groupoid_of_pregroupoid, IsImage.iff_preimage_eq, FiberBundleCore.mem_trivChange_source, univBall_source, Trivialization.restrictPreimage_target, NumberField.mixedEmbedding.lintegral_comp_polarSpaceCoord_symm, Trivialization.source_eq

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn 📖mathematicalSet.BijOn
toFun'
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
Set.InvOn.bijOn
invOn
mapsTo
symm_mapsTo
coe_coe 📖mathematicalPartialEquiv.toFun
toPartialEquiv
toFun'
coe_coe_symm 📖mathematicalPartialEquiv.toFun
PartialEquiv.symm
toPartialEquiv
toFun'
symm
continuousOn 📖mathematicalContinuousOn
toFun'
PartialEquiv.source
toPartialEquiv
continuousOn_toFun
continuousOn_invFun 📖mathematicalContinuousOn
PartialEquiv.invFun
toPartialEquiv
PartialEquiv.target
continuousOn_symm 📖mathematicalContinuousOn
toFun'
symm
PartialEquiv.target
toPartialEquiv
continuousOn_invFun
continuousOn_toFun 📖mathematicalContinuousOn
PartialEquiv.toFun
toPartialEquiv
PartialEquiv.source
eq_symm_apply 📖mathematicalSet
Set.instMembership
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
toFun'
symm
PartialEquiv.eq_symm_apply
ext 📖toFun'
symm
PartialEquiv.source
toPartialEquiv
toPartialEquiv_injective
PartialEquiv.ext
ext_iff 📖mathematicaltoFun'
symm
PartialEquiv.source
toPartialEquiv
ext
injOn 📖mathematicalSet.InjOn
toFun'
PartialEquiv.source
toPartialEquiv
Set.LeftInvOn.injOn
leftInvOn
invFun_eq_coe 📖mathematicalPartialEquiv.invFun
toPartialEquiv
toFun'
symm
invOn 📖mathematicalSet.InvOn
toFun'
symm
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
leftInvOn
rightInvOn
leftInvOn 📖mathematicalSet.LeftInvOn
toFun'
symm
PartialEquiv.source
toPartialEquiv
left_inv
left_inv 📖mathematicalSet
Set.instMembership
PartialEquiv.source
toPartialEquiv
toFun'
symm
PartialEquiv.left_inv'
map_source 📖mathematicalSet
Set.instMembership
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
toFun'
PartialEquiv.map_source'
map_source'' 📖mathematicalSet
Set.instHasSubset
Set.image
toFun'
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
Set.mem_of_eq_of_mem
PartialEquiv.map_source'
map_target 📖mathematicalSet
Set.instMembership
PartialEquiv.target
toPartialEquiv
PartialEquiv.source
toFun'
symm
PartialEquiv.map_target'
mapsTo 📖mathematicalSet.MapsTo
toFun'
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
map_source
mk_coe 📖mathematicalIsOpen
PartialEquiv.source
PartialEquiv.target
ContinuousOn
PartialEquiv.toFun
PartialEquiv.invFun
toFun'
mk_coe_symm 📖mathematicalIsOpen
PartialEquiv.source
PartialEquiv.target
ContinuousOn
PartialEquiv.toFun
PartialEquiv.invFun
toFun'
symm
PartialEquiv.symm
open_source 📖mathematicalIsOpen
PartialEquiv.source
toPartialEquiv
open_target 📖mathematicalIsOpen
PartialEquiv.target
toPartialEquiv
replaceEquiv_eq_self 📖mathematicaltoPartialEquivreplaceEquiv
rightInvOn 📖mathematicalSet.RightInvOn
toFun'
symm
PartialEquiv.target
toPartialEquiv
right_inv
right_inv 📖mathematicalSet
Set.instMembership
PartialEquiv.target
toPartialEquiv
toFun'
symm
PartialEquiv.right_inv'
surjOn 📖mathematicalSet.SurjOn
toFun'
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
Set.BijOn.surjOn
bijOn
symm_bijective 📖mathematicalFunction.Bijective
OpenPartialHomeomorph
symm
Function.bijective_iff_has_inverse
symm_symm
symm_mapsTo 📖mathematicalSet.MapsTo
toFun'
symm
PartialEquiv.target
toPartialEquiv
PartialEquiv.source
mapsTo
symm_source 📖mathematicalPartialEquiv.source
toPartialEquiv
symm
PartialEquiv.target
symm_symm 📖mathematicalsymm
symm_target 📖mathematicalPartialEquiv.target
toPartialEquiv
symm
PartialEquiv.source
symm_toPartialEquiv 📖mathematicaltoPartialEquiv
symm
PartialEquiv.symm
toFun_eq_coe 📖mathematicalPartialEquiv.toFun
toPartialEquiv
toFun'
toPartialEquiv_injective 📖mathematicalOpenPartialHomeomorph
PartialEquiv
toPartialEquiv

OpenPartialHomeomorph.Simps

Definitions

NameCategoryTheorems
apply 📖CompOp
symm_apply 📖CompOp

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