Documentation Verification Report

Basic

📁 Source: Mathlib/Topology/MetricSpace/Basic.lean

Statistics

MetricCount
DefinitionsinstMetricSpace, replaceEDist, toMetricSpace, toMetricSpaceOfDist, comapMetricSpace, induced, ofT0PseudoMetricSpace, replaceDist, toEMetricSpace, instMetricSpace, metricSpaceMax, replaceEDist, replaceDist, metricSpace, instDist, instMetricSpace, metricSpace, comapMetricSpace, instMetricSpaceNNReal, instMetricSpaceULift, metricSpacePi
21
TheoremsreplaceEDist_eq, isClosedEmbedding_of_pairwise_le_dist, isClosed_of_pairwise_le_dist, isUniformEmbedding_bot_of_pairwise_le_dist, isUniformEmbedding_iff', secondCountable_of_countable_discretization, instT0Space, replaceDist_eq, isClosedEmbedding_coe, isEmbedding_coe, isUniformEmbedding_coe, replaceEDist_eq, replaceDist_eq, dist_mk
14
Total35

AddOpposite

Definitions

NameCategoryTheorems
instMetricSpace 📖CompOp

EMetricSpace

Definitions

NameCategoryTheorems
replaceEDist 📖CompOp
1 mathmath: replaceEDist_eq
toMetricSpace 📖CompOp
toMetricSpaceOfDist 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
replaceEDist_eq 📖mathematicalEDist.edist
PseudoEMetricSpace.toEDist
toPseudoEMetricSpace
replaceEDistext
EDist.ext

IsUniformEmbedding

Definitions

NameCategoryTheorems
comapMetricSpace 📖CompOp
1 mathmath: to_isometry

Metric

Theorems

NameKindAssumesProvesValidatesDepends On
isClosedEmbedding_of_pairwise_le_dist 📖mathematicalReal
Real.instLT
Real.instZero
Pairwise
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
Topology.IsClosedEmbedding
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
MetricSpace.toPseudoMetricSpace
isClosedEmbedding_of_spaced_out
MetricSpace.instT0Space
dist_mem_uniformity
isClosed_of_pairwise_le_dist 📖mathematicalReal
Real.instLT
Real.instZero
Set.Pairwise
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
MetricSpace.toPseudoMetricSpace
IsClosed
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
MetricSpace.toPseudoMetricSpace
isClosed_of_spaced_out
MetricSpace.instT0Space
dist_mem_uniformity
isUniformEmbedding_bot_of_pairwise_le_dist 📖mathematicalReal
Real.instLT
Real.instZero
Pairwise
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
IsUniformEmbedding
Bot.bot
UniformSpace
instBotUniformSpace
PseudoMetricSpace.toUniformSpace
isUniformEmbedding_of_spaced_out
dist_mem_uniformity
isUniformEmbedding_iff' 📖mathematicalIsUniformEmbedding
PseudoMetricSpace.toUniformSpace
MetricSpace.toPseudoMetricSpace
Real
Real.instLT
Real.instZero
Dist.dist
PseudoMetricSpace.toDist
isUniformEmbedding_iff_isUniformInducing
MetricSpace.instT0Space
isUniformInducing_iff
uniformContinuous_iff
secondCountable_of_countable_discretization 📖mathematicalEncodable
Real
Real.instLE
Dist.dist
PseudoMetricSpace.toDist
SecondCountableTopology
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
secondCountable_of_almost_dense_set
Set.countable_range
SetCoe.countable
Encodable.countable
Set.mem_range_self
Set.apply_rangeSplitting

MetricSpace

Definitions

NameCategoryTheorems
induced 📖CompOp
1 mathmath: isometry_induced
ofT0PseudoMetricSpace 📖CompOp
replaceDist 📖CompOp
1 mathmath: replaceDist_eq
toEMetricSpace 📖CompOp
524 mathmath: ContinuousLinearEquiv.antilipschitz, LipschitzWith.max_const, MemHolder.nsmul, Unitization.isometry_inr, NumberField.InfinitePlace.Completion.isometry_extensionEmbeddingOfIsReal, EuclideanGeometry.euclideanHausdorffMeasure_eq, ODE.FunSpace.exists_contractingWith_iterate_next, LightCondensed.isoFinYonedaComponents_hom_apply, EuclideanGeometry.side_angle_side, Real.dimH_of_mem_nhds, NonUnitalIsometricContinuousFunctionalCalculus.isometric, HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn, MeasureTheory.hausdorffMeasure_segment, ConvexOn.continuousOn_tfae, ConcaveOn.locallyLipschitz, MemHolder.holderWith, IsometryEquiv.toRealLinearIsometryEquiv_symm_apply, MemHolder.comp, SchwartzMap.pairing_apply_apply, ApproximatesLinearOn.lipschitz, ProperCone.innerDual_univ, ContinuousLinearMap.integral_apply, SchwartzMap.integral_clm_comp_lineDerivOp_right_eq_neg_left, NumberField.InfinitePlace.isometry_embedding, GromovHausdorff.isometry_optimalGHInjr, MeasureTheory.UniformIntegrable.uniformIntegrable_of_tendstoInMeasure, SchwartzMap.lineDerivOp_fourier_eq, LipschitzWith.lipschitzWith_compLp, LinearMap.adjoint_eq_toCLM_adjoint, LightCondSet.continuous_coinducingCoprod, NumberField.mixedEmbedding.fundamentalCone.abs_det_completeBasis_equivFunL_symm, Complex.lipschitz_equivRealProd, IsometryEquiv.toRealLinearIsometryEquiv_apply, LinearMap.isSelfAdjoint_toContinuousLinearMap_iff, LipschitzWith.completion_extension, SchwartzMap.bilinLeftCLM_apply, ODE.FunSpace.range_toContinuousMap, Delone.DeloneSet.isCover_coveringRadius, Real.ediam_eq, Real.dimH_lt_top, MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_enorm, CompHausLike.LocallyConstant.adjunction_counit, lintegral_fderiv_lineMap_eq_edist, Real.volume_closedEBall, MeasureTheory.isAddHaarMeasure_hausdorffMeasure, ContinuousLinearMap.isometry_mul_flip, Delone.DeloneSet.mapIsometry_trans, Set.infsep_zero_iff_subsingleton_of_finite, SchwartzMap.laplacianCLM_eq, UpperHalfPlane.isometry_real_vadd, ContDiff.lipschitzWith_of_hasCompactSupport, HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn, IsometryEquiv.midpoint_fixed, ContinuousMapZero.isometry_toContinuousMap, ContDiffOn.dimH_image_le, SchwartzMap.compCLMOfContinuousLinearEquiv_apply, HasFPowerSeriesWithinOnBall.isBigO_image_sub_image_sub_deriv_principal, GromovHausdorff.toGHSpace_eq_toGHSpace_iff_isometryEquiv, ConcaveOn.locallyLipschitzOn_interior, eHolderNorm_eq_zero, instIsAddHaarMeasureEuclideanSpaceRealFinHausdorffMeasureCast, Perfect.small_diam_splitting, LinearMap.adjoint_toContinuousLinearMap, MemHolder.coe_nnHolderNorm_eq_eHolderNorm, UpperHalfPlane.isometry_vertical_line, MeasureTheory.Integrable.edist_toL1_zero, Complex.isometry_conj, intervalIntegral.hasFDerivAt_integral_of_dominated_loc_of_lip, WithAbs.isometry_of_comp, SchwartzMap.fourier_lineDerivOp_eq, ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith, SchwartzMap.smulLeftCLM_compCLMOfContinuousLinearEquiv, SchwartzMap.smulLeftCLM_real_smul, Real.dimH_univ_eq_finrank, Finset.infsep_pos_iff_nontrivial, HasFiniteFPowerSeriesOnBall.cpolynomialOn, UnitAddTorus.mFourierCoeff_toLp, lipschitzWith_sup_right, Real.dimH_univ, MeasureTheory.tendsto_integral_approxOn_of_measurable, MeasureTheory.tendstoInMeasure_iff_tendsto_Lp, HasFPowerSeriesWithinOnBall.differentiableOn, DomAddAct.instIsIsometricVAddSubtypeAEEqFunMemAddAddSubgroupLp, CompHausLike.LocallyConstant.counitApp_app, Condensed.isoFinYonedaComponents_hom_apply, ContinuousLinearMap.antilipschitz_of_isEmbedding, hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall, NumberField.InfinitePlace.Completion.isometry_extensionEmbedding, MeasureTheory.convolution_precompR_apply, SchwartzMap.compSubConstCLM_apply, MeasureTheory.Integrable.norm_toL1, antilipschitzWith_mul_right, Real.ediam_Ioc, LinearMap.exists_antilipschitzWith, ConvexOn.locallyLipschitz, LipschitzWith.dist_left, CompHausLike.LocallyConstant.incl_of_counitAppApp, Delone.DeloneSet.mapIsometry_symm_apply_carrier, HasFPowerSeriesWithinOnBall.analyticOn, LipschitzWith.of_le_add, MeasureTheory.SimpleFunc.nnnorm_approxOn_le, lipschitzOnWith_cfc_fun_of_subset, SchwartzMap.smulLeftCLM_sum, Metric.isPreconnected_closedEBall, Real.volume_emetric_closedBall, IsLowerSet.cthickening, HasFPowerSeriesOnBall.differentiableOn, Metric.isPathConnected_closedEBall, LocallyLipschitz.const_max, MeasureTheory.UnifIntegrable.unifIntegrable_of_tendstoInMeasure, Isometry.completion_map, Real.dimH_univ_pi_fin, holderWith_zero_iff, LipschitzWith.projIcc, BoundedContinuousFunction.Lp_nnnorm_le, unitInterval.volume_uIcc, ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt, Real.dimH_ball_pi, Metric.toGlueL_isometry, IsCompactOperator.antilipschitz_of_not_hasEigenvalue, SchwartzMap.smulLeftCLM_fun_neg, lipschitzWith_thickenedIndicator, MeasureTheory.lintegral_edist_triangle, ClosedSubmodule.bot_orthogonal_eq_top, MeasureTheory.hausdorffMeasure_pi_real, SchwartzMap.pairing_apply, unitInterval.volume_uIoo, Set.Finite.infsep_pos_iff_nontrivial, StandardSubspace.IsSeparating, MeasureTheory.Measure.hausdorffMeasure_smul₀, MeasureTheory.tendsto_integral_approxOn_of_measurable_of_range_subset, MonotoneOn.locallyBoundedVariationOn, MeasureTheory.lintegral_edist_lt_top, BoundedVariationOn.sub_le, IsPicardLindelof.lipschitzOnWith, MeasureTheory.hasFDerivAt_convolution_right_with_param, MeasureTheory.hasFiniteIntegral_iff_edist, SchwartzMap.postcompCLM_apply, HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt, HasCompactSupport.hasFDerivAt_convolution_right, isometry_cfcₙHom, SchwartzMap.lineDerivOp_compCLMOfContinuousLinearEquiv, HasStrictFDerivAt.exists_lipschitzOnWith, ClosedSubmodule.top_orthogonal_eq_bot, Complex.antilipschitz_equivRealProd, Complex.isometry_ofReal, lipschitzWith_one_norm', LocallyLipschitz.max, MeasureTheory.SimpleFunc.norm_approxOn_y₀_le, CompHausLike.LocallyConstant.counit_app_hom_app, ContDiff.euclidean_dist, MeasureTheory.SimpleFunc.memLp_approxOn_range, DilationEquiv.mulRight_symm_apply, algebraMap_isometry, hasFDerivAt_integral_of_dominated_loc_of_lip', MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset, IsUpperSet.thickening', DilationEquiv.mulLeft_symm_apply, IsometryEquiv.map_midpoint, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed, SchwartzMap.compCLMOfAntilipschitz_apply, SchwartzMap.fourierInv_apply_eq, SchwartzMap.smulRightCLM_apply_apply, NNReal.lipschitzWith_sub, LightCondensed.isoFinYonedaComponents_inv_comp, EuclideanSpace.edist_single_same, lipschitzWith_circleMap, HasFPowerSeriesWithinOnBall.continuousOn, MeasureTheory.Lp.mem_boundedContinuousFunction_iff, LipschitzWith.min, DilationEquiv.mulLeft_apply, SchwartzMap.postcompCLM_postcompCLM, Real.lipschitzWith_sin, SchwartzMap.smulLeftCLM_const, Matrix.exp_sum_of_commute, Set.infsep_pos_iff_nontrivial_of_finite, MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm, lipschitzOnWith_cfc_fun, ContinuousLinearEquiv.dimH_image, MeasureTheory.hausdorffMeasure_prod_real, MemHolder.add, Orientation.areaForm'_apply, MeasureTheory.hausdorffMeasure_lineMap_image, AffineMap.lipschitzWith_of_finiteDimensional, ContinuousLinearMap.isClosed_range_iff_antilipschitz_of_injective, Convex.lipschitzOnWith_of_nnnorm_fderiv_le, LipschitzWith.min_const, ContinuousLinearMap.antilipschitz_antiLipschitzConstant_of_injective_of_isClosed_range, LightCondensed.isoLocallyConstantOfIsColimit_inv, Real.volume_pi_le_prod_diam, WithLp.volume_preserving_toLp, SchwartzMap.fourierMultiplierCLM_apply, LipschitzOnWith.extend_real, SchwartzMap.compSubConstCLM_zero, UniformSpace.Completion.coe_isometry, LipschitzWith.coordinate, HasFiniteFPowerSeriesOnBall.differentiableOn, LipschitzOnWith.extend_finite_dimension, Delone.DeloneSet.mapIsometry_symm, Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le, SchwartzMap.fourierInv_lineDerivOp_eq, LipschitzOnWith.extend_lp_infty, Metric.isPreconnected_eball, MeasureTheory.Lp.lipschitzWith_pos_part, PiNat.exists_lipschitz_retraction_of_isClosed, fourierCoeff_toLp, Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le, LinearMap.isPositive_toContinuousLinearMap_iff, HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn', Isometry.completion_extension, MemHolder.smul, LipschitzWith.const_min, SchwartzMap.smulLeftCLM_smulLeftCLM_apply, ContinuousLinearMap.isometry_iff_adjoint_comp_self, GromovHausdorff.ghDist_eq_hausdorffDist, ContinuousLinearMap.opNorm_extend_le, ApproximatesLinearOn.lipschitz_sub, lp.lipschitzWith_one_eval, HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal, Real.dimH_of_nonempty_interior, DilationEquiv.mulRight_apply, lipschitzOnWith_cfcₙ_fun, LocallyLipschitz.min_const, Delone.DeloneSet.mapIsometry_refl, BoundedContinuousFunction.mem_Lp, BoundedContinuousFunction.Lp_norm_le, Matrix.l2_opNNNorm_def, KuratowskiEmbedding.embeddingOfSubset_isometry, SchwartzMap.integral_pow_mul_iteratedFDeriv_le, CompHausLike.LocallyConstant.adjunction_left_triangle, MeasureTheory.euclideanHausdorffMeasure_homothety_image, RCLike.lipschitzWith_re, instIsRiemannianManifoldModelWithCornersSelfReal, MeasureTheory.euclideanHausdorffMeasure_homothety_preimage, SchwartzMap.smulLeftCLM_add, BoundedVariationOn.stieltjesFunctionRightLim_apply, Condensed.isoFinYonedaComponents_inv_comp, MeasureTheory.hausdorffMeasure_homothety_image, ContinuousMultilinearMap.integral_apply, UpperHalfPlane.instIsIsometricSMulSpecialLinearGroupFinOfNatNatReal, ApproximatesLinearOn.antilipschitz, Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt, IsUpperSet.thickening, CondensedSet.continuous_coinducingCoprod, EuclideanGeometry.similar_of_side_angle_side, GromovHausdorff.AuxGluingStruct.isom, Real.lipschitzWith_one_mulExpNegMulSq, ContinuousLinearMap.antilipschitz_of_injective_of_isClosed_range, HasFiniteFPowerSeriesOnBall.continuousOn, MeasureTheory.SimpleFunc.memLp_approxOn, memHolder_iff_holderWith, IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero, ContinuousLinearMap.isSelfAdjoint_toLinearMap_iff, EuclideanGeometry.measurePreserving_vaddConst, ContDiff.locallyLipschitz, HasFTaylorSeriesUpToOn.exists_lipschitzOnWith, MeasureTheory.MeasuredSets.lipschitzWith_measureReal, SchwartzMap.compSubConstCLM_comp, eHolderNorm_nsmul, LipschitzWith.dist_right, ConvexOn.locallyLipschitzOn_iff_continuousOn, HasFPowerSeriesOnBall.analyticOnNhd, Real.Convex.dimH_eq_finrank_vectorSpan, Set.Finite.infsep_zero_iff_subsingleton, variationOnFromTo.sub_self_monotoneOn, ClosedSubmodule.orthogonal_eq_top_iff, Real.volume_eball, LipschitzWith.completion_map, Real.volume_le_diam, MeasureTheory.L1.dist_def, Module.Basis.parallelepiped_eq_map, NumberField.InfinitePlace.isometry_embedding_of_isReal, FormalMultilinearSeries.analyticOnNhd, LipschitzOnWith.extend_pi, MeasureTheory.hausdorffMeasure_smul_right_image, ContDiffAt.exists_lipschitzOnWith_of_nnnorm_lt, LipschitzOnWith.coordinate, Real.dimH_univ_pi, LipschitzOnWith.iff_le_add_mul, Metric.isCompact_closure_iff_exists_finite_isCover, isConformalMap_complex_linear_conj, intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le, AbsoluteValue.Completion.isClosedEmbedding_extensionEmbedding_of_comp, SchwartzMap.tsupport_smulLeftCLM_subset, lipschitzWith_negPart, ContinuousMap.coe_toLp, Real.lipschitzWith_toNNReal, DomMulAct.instIsIsometricSMulSubtypeAEEqFunMemAddSubgroupLp, antilipschitz_of_bound_of_norm_one, MeasureTheory.SimpleFunc.integrable_approxOn_range, ContinuousLinearMap.intervalIntegral_apply, ConvexOn.locallyLipschitzOn_interior, SchwartzMap.smulLeftCLM_neg, LipschitzWith.of_le_add_mul, Delone.DeloneSet.mapBilipschitz_trans, lipschitzWith_of_nnnorm_fderiv_le, MemHolder.nnHolderNorm_nsmul, LipschitzWith.max, lipschitzWith_one_nnnorm, WithLp.volume_preserving_symm_measurableEquiv_toLp_prod, eventually_riemannianEDist_le_edist_extChartAt, ClosedSubmodule.inf_orthogonal_eq_bot, EuclideanGeometry.similar_of_angle_angle, MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz, SchwartzMap.pairing_continuous_left, lipschitzOnWith_cfcₙ_fun_of_subset, SchwartzMap.integralCLM_apply, ProbabilityTheory.condExp_ae_eq_integral_condDistrib', SchwartzMap.lineDerivOp_fourierInv_eq, HasFPowerSeriesOnBall.unique, AbsoluteValue.Completion.isometry_extensionEmbedding_of_comp, Metric.isometry_inl, LocallyLipschitz.max_const, LipschitzWith.iff_le_add_mul, Real.dimH_ball_pi_fin, EuclideanGeometry.triangle_congruent_iff_dist_eq, MemHolder.nnHolderNorm_smul, Metric.toInductiveLimit_isometry, Convex.lipschitz_gauge, MeasureTheory.tendstoInMeasure_of_tendsto_Lp, antilipschitzWith_mul_left, ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real, Real.ediam_Ioo, MemHolder.nnHolderNorm_add_le, SchwartzMap.compCLM_apply, TemperedDistribution.fourierMultiplierCLM_apply_apply, Dilation.ratio_comp, Isometry.isometry_mapRingHom, MeasureTheory.SimpleFunc.integrable_approxOn, ConvexOn.locallyLipschitzOn, Metric.toGlueR_isometry, Submodule.lipschitzWith_starProjection, Real.volume_pi_le_diam_pow, Finset.infsep_zero_iff_subsingleton, MeasureTheory.hausdorffMeasure_real, LipschitzOnWith.of_le_add, SchwartzMap.fderivCLM_fourier_eq, SchwartzMap.derivCLM_apply, Matrix.toLin_finTwoProd_toContinuousLinearMap, Metric.contractibleSpace_eball, ConcaveOn.locallyLipschitzOn, CompHausLike.LocallyConstant.instIsIsoFunctorTypeUnitSheafCoherentTopologyAdjunction, WithLp.volume_preserving_ofLp, Metric.isConnected_closedEBall, SchwartzMap.integral_clm_comp_deriv_right_eq_neg_left, GromovHausdorff.toGHSpace_lipschitz, NonUnitalStarAlgHom.isometry, IsometryEquiv.coeFn_toRealAffineIsometryEquiv, IsLowerSet.thickening, ContDiffOn.exists_lipschitzOnWith, LinearMap.isStarProjection_toContinuousLinearMap_iff, eHolderNorm_add_le, MeasureTheory.dist_integral_le_lintegral_edist, dimH_orthogonalProjection_le, ProbabilityTheory.iIndepFun.condExp_natural_ae_eq_of_lt, Convex.lipschitzWith_gauge, ConcaveOn.exists_lipschitzOnWith_of_isBounded, MeasureTheory.lintegral_enorm_eq_lintegral_edist, ContinuousLinearEquiv.dimH_univ, HasFPowerSeriesWithinOnBall.unique, MeasureTheory.SimpleFunc.norm_approxOn_zero_le, hasFDerivAt_integral_of_dominated_of_fderiv_le, EuclideanGeometry.angle_side_angle, RCLike.lipschitzWith_ofReal, Module.Basis.parallelepiped_map, LocallyLipschitz.const_min, ConvexOn.exists_lipschitzOnWith_of_isBounded, Metric.lipschitz_infDist_pt, MeasureTheory.hausdorffMeasure_orthogonalProjection_le, Real.dimH_segment, EuclideanGeometry.angle_angle_side, Submodule.lipschitzWith_orthogonalProjection, EuclideanGeometry.side_side_side, lipschitzWith_of_nnnorm_deriv_le, Real.lipschitzWith_cos, Metric.lipschitz_infDist, NormedSpace.continuousOn_exp, ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection, MeasureTheory.hausdorffMeasure_affineSegment, TemperedDistribution.derivCLM_toTemperedDistributionCLM_eq, Metric.eball_contractible, IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm, HolderWith.nnholderNorm_le, MonotoneOn.eVariationOn_le, MeasureTheory.ProbabilityMeasure.testAgainstNN_lipschitz, Metric.isPathConnected_eball, Dilation.mulRight_toFun, MeasureTheory.BoundedContinuousFunction.inner_toLp, tendsto_measure_cthickening_of_isCompact, Metric.contractibleSpace_closedEBall, ContDiff.dimH_range_le, NNReal.isometry_coe, SchwartzMap.tsupport_derivCLM_subset, HasFPowerSeriesOnBall.continuousOn, ContinuousLinearEquiv.dimH_preimage, ConcaveOn.continuousOn_tfae, lp.isometry_single, CompHausLike.LocallyConstant.unit_app, Delone.DeloneSet.isSeparated_packingRadius, LipschitzOnWith.of_le_add_mul', Metric.lipschitz_infDist_set, Unitization.antilipschitzWith_addEquiv, ClosedSubmodule.orthogonal_disjoint, Metric.Sigma.isometry_mk, IsUpperSet.cthickening', SchwartzMap.smulLeftCLM_sub, SchwartzMap.smulLeftCLM_apply_apply, Real.hausdorffMeasure_of_finrank_lt, SchwartzMap.fourier_fderivCLM_eq, MeasureTheory.lintegral_norm_eq_lintegral_edist, Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt, MeasureTheory.tendsto_setToFun_approxOn_of_measurable, MemHolder.nnHolderNorm_eq_zero, gelfandTransform_isometry, CompHausLike.LocallyConstant.incl_comap, MeasureTheory.isometry_lpMeasSubgroupToLpTrim, InnerProductSpace.euclideanHausdorffMeasure_eq_volume, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed_symm, Module.Basis.opNNNorm_le, LinearMap.antilipschitz_of_comap_nhds_le, SchwartzMap.smulLeftCLM_ofReal, IsPicardLindelof.exists_forall_mem_closedBall_eq_hasDerivWithinAt_lipschitzOnWith, IsLowerSet.thickening', lipschitzWith_one_nnnorm', KuratowskiEmbedding.exists_isometric_embedding, Metric.isConnected_eball, Complex.edist_of_re_eq, EuclideanSpace.euclideanHausdorffMeasure_eq_volume, Complex.exists_cthickening_tendstoUniformlyOn, Module.Basis.opNorm_le, isConformalMap_complex_linear, unitInterval.volume_uIoc, lipschitzWith_max, SchwartzMap.smulLeftCLM_apply, MeasureTheory.Measure.euclideanHausdorffMeasure_smul₀, Metric.isometry_inr, MeasureTheory.isAddHaarMeasure_euclideanHausdorffMeasure, lipschitzWith_lineMap, Dilation.mulLeft_toFun, SchwartzMap.smulLeftCLM_compL_smulLeftCLM, MeasureTheory.SimpleFunc.tendsto_approxOn_L1_enorm, Unitization.lipschitzWith_addEquiv, Complex.edist_of_im_eq, TemperedDistribution.derivCLM_apply_apply, NormedSpace.exp_sum_of_commute, IsometricContinuousFunctionalCalculus.isometric, MeasureTheory.hausdorffMeasure_homothety_preimage, MeasureTheory.ContinuousMap.inner_toLp, lipschitzWith_posPart, SchwartzMap.integral_clm_comp_laplacian_right_eq_left, HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn', NumberField.InfinitePlace.LiesOver.isometry_algebraMap, Matrix.l2_opNorm_def, isometry_induced, MeasureTheory.tendsto_integral_norm_approxOn_sub, AffineMap.antilipschitzWith_of_finiteDimensional, AbsoluteValue.Completion.extensionEmbedding_dist_eq_of_comp, MeasureTheory.L1.edist_def, antilipschitzWith_lineMap, LinearMap.injective_iff_antilipschitz, UpperHalfPlane.isometry_pos_mul, ContDiffOn.locallyLipschitzOn, SchwartzMap.fourier_convolution, eHolderNorm_smul, ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz, EuclideanSpace.edist_eq, ApproximatesLinearOn.lipschitzOnWith, LipschitzWith.const_max, UniformSpace.Completion.edist_eq, FormalMultilinearSeries.continuousOn, ContDiffAt.exists_lipschitzOnWith, ContDiffWithinAt.exists_lipschitzOnWith, lipschitzWith_min, MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp, MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite, AffineSubspace.euclideanHausdorffMeasure_coe_image, IsLowerSet.cthickening', Convex.lipschitzOnWith_of_nnnorm_deriv_le, GromovHausdorff.isometry_optimalGHInjl, AbsoluteValue.Completion.extensionEmbedding_of_comp_coe, Metric.lipschitz_infNndist_pt, MeasureTheory.Integrable.edist_toL1_toL1, WithLp.unitization_isometry_inr, MeasureTheory.L1.setToL1_lipschitz, GromovHausdorff.eq_toGHSpace_iff, LipschitzOnWith.of_le_add_mul, MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_eLpNorm, IsUniformEmbedding.to_isometry, MeasureTheory.Lp.isometry_compMeasurePreserving, SchwartzMap.smulLeftCLM_smul, MemHolder.smul_iff, ODE.FunSpace.lipschitzWith, ConcaveOn.locallyLipschitzOn_iff_continuousOn, LocallyLipschitz.min, Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le, MeasureTheory.Measure.euclideanHausdorffMeasure_def, LipschitzWith.of_le_add_mul', RCLike.lipschitzWith_im, ContinuousMap.idealOpensGI_choice, HasCompactSupport.hasFDerivAt_convolution_left, TemperedDistribution.smulLeftCLM_apply_apply, MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_eLpNorm, BoundedContinuousFunction.lintegral_le_edist_mul, Real.ediam_Icc, CompHausLike.LocallyConstant.adjunction_unit, IsUpperSet.cthickening, ConvexOn.lipschitzOnWith_of_abs_le, HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt, StarAlgEquiv.isometry, Delone.DeloneSet.mapIsometry_apply_carrier, hasFDerivAt_integral_of_dominated_loc_of_lip, Convex.lipschitzOnWith_of_nnnorm_derivWithin_le, Complex.isometry_intCast, kuratowskiEmbedding.isometry, Real.ediam_Ico, ContinuousLinearMap.antilipschitz_of_forall_le_inner_map, TensorProduct.edist_tmul_le, LipschitzWith.dist, isometry_cfcHom, Real.volume_emetric_ball, lipschitzWith_one_norm, ConcaveOn.lipschitzOnWith_of_abs_le, IsCoercive.antilipschitz, MeasureTheory.edist_integral_le_lintegral_edist

Theorems

NameKindAssumesProvesValidatesDepends On
instT0Space 📖mathematicalT0Space
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
toPseudoMetricSpace
eq_of_dist_eq_zero
Metric.inseparable_iff
replaceDist_eq 📖mathematicalDist.dist
PseudoMetricSpace.toDist
toPseudoMetricSpace
replaceDistext
Dist.ext

MulOpposite

Definitions

NameCategoryTheorems
instMetricSpace 📖CompOp

NNReal

Theorems

NameKindAssumesProvesValidatesDepends On
isClosedEmbedding_coe 📖mathematicalTopology.IsClosedEmbedding
NNReal
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
instPseudoMetricSpaceNNReal
Real.pseudoMetricSpace
toReal
IsClosed.isClosedEmbedding_subtypeVal
isClosed_Ici
instClosedIciTopology
OrderTopology.to_orderClosedTopology
instOrderTopologyReal
isEmbedding_coe 📖mathematicalTopology.IsEmbedding
NNReal
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
instPseudoMetricSpaceNNReal
Real.pseudoMetricSpace
toReal
IsUniformEmbedding.isEmbedding
isUniformEmbedding_coe
isUniformEmbedding_coe 📖mathematicalIsUniformEmbedding
NNReal
Real
PseudoMetricSpace.toUniformSpace
instPseudoMetricSpaceNNReal
Real.pseudoMetricSpace
toReal
isUniformEmbedding_subtype_val

Prod

Definitions

NameCategoryTheorems
metricSpaceMax 📖CompOp

PseudoEMetricSpace

Definitions

NameCategoryTheorems
replaceEDist 📖CompOp
1 mathmath: replaceEDist_eq

Theorems

NameKindAssumesProvesValidatesDepends On
replaceEDist_eq 📖mathematicalEDist.edist
toEDist
replaceEDistext
EDist.ext

PseudoMetricSpace

Definitions

NameCategoryTheorems
replaceDist 📖CompOp
1 mathmath: replaceDist_eq

Theorems

NameKindAssumesProvesValidatesDepends On
replaceDist_eq 📖mathematicalDist.dist
toDist
replaceDistext
Dist.ext

Real

Definitions

NameCategoryTheorems
metricSpace 📖CompOp
247 mathmath: range_cfcₙ_nnreal, LipschitzWith.max_const, NumberField.InfinitePlace.Completion.isometry_extensionEmbeddingOfIsReal, CFC.abs_eq_cfc_norm, ConvexOn.continuousOn_tfae, ConcaveOn.locallyLipschitz, CFC.sqrt_eq_one_iff', CFC.posPart_one, nnnorm_cfcₙ_nnreal_le_iff, Matrix.IsHermitian.det_abs, CFC.posPart_natCast, CFC.tendsto_cfc_rpow_sub_one_log, Matrix.IsHermitian.cfc_eq, CFC.sq_sqrt, Unitization.real_cfcₙ_eq_cfc_inr, NumberField.mixedEmbedding.fundamentalCone.abs_det_completeBasis_equivFunL_symm, Complex.lipschitz_equivRealProd, CFC.nnnorm_sqrt, ODE.FunSpace.range_toContinuousMap, ediam_eq, CFC.sqrt_rpow, volume_closedEBall, cfc_complex_eq_real, CFC.negPart_algebraMap, cfcₙHom_nnreal_eq_restrict, ConcaveOn.locallyLipschitzOn_interior, instIsAddHaarMeasureEuclideanSpaceRealFinHausdorffMeasureCast, Continuous.cfc_nnreal', CFC.abs_coe_unitary, UpperHalfPlane.isometry_vertical_line, IsUnit.cfcSqrt, CFC.continuousOn_log, ContinuousOn.cfc_nnreal', IsGreatest.nnnorm_cfc_nnreal, CFC.continuousOn_sqrt, UnitAddTorus.mFourierCoeff_toLp, dimH_univ, Matrix.PosSemidef.det_sqrt, Commute.cfc_real, ediam_Ioc, Matrix.IsHermitian.charpoly_cfc_eq, ConvexOn.locallyLipschitz, nnnorm_cfcₙ_nnreal_lt, CFC.sqrt_eq_rpow, LipschitzWith.dist_left, LipschitzWith.of_le_add, cfc_real_eq_nnreal, volume_emetric_closedBall, LocallyLipschitz.const_max, dimH_univ_pi_fin, LipschitzWith.projIcc, unitInterval.volume_uIcc, CFC.posPart_algebraMap_nnreal, dimH_ball_pi, MeasureTheory.hausdorffMeasure_pi_real, unitInterval.volume_uIoo, MonotoneOn.locallyBoundedVariationOn, CFC.sqrt_eq_one_iff, BoundedVariationOn.sub_le, CStarAlgebra.norm_posPart_le, IsGreatest.nnnorm_cfcₙ_nnreal, Complex.antilipschitz_equivRealProd, CFC.continuous_abs, Complex.isometry_ofReal, continuousOn_cfc_nnreal, lipschitzWith_one_norm', LocallyLipschitz.max, cfc_nnreal_eq_real, cfcₙ_real_eq_complex, cfcₙ_comp_re, cfc_comp_im, CFC.abs_natCast, CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt, lipschitzWith_circleMap, LipschitzWith.min, cfcₙ_comp_im, lipschitzWith_sin, Continuous.cfc_nnreal_of_mem_nhdsSet, MeasureTheory.hausdorffMeasure_prod_real, IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus, MeasureTheory.hausdorffMeasure_lineMap_image, Continuous.cfcₙ_nnreal, LipschitzWith.min_const, cfcHom_real_eq_restrict, volume_pi_le_prod_diam, ContinuousAt.cfcₙ_nnreal, LipschitzOnWith.extend_real, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, CFC.abs_algebraMap, LipschitzWith.coordinate, MeasureTheory.Lp.lipschitzWith_pos_part, IsStrictlyPositive.nnrpow, fourierCoeff_toLp, CFC.monotoneOn_one_sub_one_add_inv_real, LipschitzWith.const_min, ContinuousWithinAt.cfcₙ_nnreal, CFC.continuousOn_nnrpow, range_cfcₙ_nnreal_eq_image_cfcₙ_real, Filter.Tendsto.cfcₙ_nnreal, LocallyLipschitz.min_const, IsUnit.cfcNNRpow, ContinuousOn.cfc_nnreal, nnnorm_cfcₙ_nnreal_lt_iff, RCLike.lipschitzWith_re, nnnorm_cfc_nnreal_lt, IsStrictlyPositive.sqrt, cfcₙ_real_eq_nnreal, cfc_comp_re, nnnorm_cfc_nnreal_le, IsSelfAdjoint.norm_eq_max_norm_posPart_negPart, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Continuous.cfc_nnreal, nnnorm_cfc_nnreal_le_iff, CFC.sqrt_sq, lipschitzWith_one_mulExpNegMulSq, CFC.rpow_sqrt_nnreal, MeasureTheory.MeasuredSets.lipschitzWith_measureReal, LipschitzWith.dist_right, ConvexOn.locallyLipschitzOn_iff_continuousOn, IsSelfAdjoint.instIsometricContinuousFunctionalCalculus, cfcHom_nnreal_eq_restrict, variationOnFromTo.sub_self_monotoneOn, volume_eball, ContinuousOn.cfcₙ_nnreal', volume_le_diam, Matrix.IsHermitian.instContinuousFunctionalCalculusIsClosedEmbedding, range_cfc_nnreal, NumberField.InfinitePlace.isometry_embedding_of_isReal, LipschitzOnWith.extend_pi, MeasureTheory.hausdorffMeasure_smul_right_image, LipschitzOnWith.coordinate, CFC.abs_ofNat, dimH_univ_pi, LipschitzOnWith.iff_le_add_mul, CFC.negPart_one, Matrix.PosSemidef.inv_sqrt, Matrix.IsHermitian.cfcHom_eq_cfcAux, cfcₙHom_real_eq_restrict, CFC.abs_algebraMap_nnreal, lipschitzWith_toNNReal, ConvexOn.locallyLipschitzOn_interior, LipschitzWith.of_le_add_mul, CFC.nnnorm_rpow, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, LipschitzWith.max, CStarAlgebra.isStrictlyPositive_iff_isUnit_sqrt_and_eq_sqrt_mul_sqrt, cfc_comp_norm, CFC.abs_intCast, apply_le_nnnorm_cfc_nnreal, range_cfc_nnreal_eq_image_cfc_real, LocallyLipschitz.max_const, LipschitzWith.iff_le_add_mul, dimH_ball_pi_fin, Convex.lipschitz_gauge, ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real, ediam_Ioo, CFC.exists_measure_nnrpow_eq_integral_cfcₙ_rpowIntegrand₀₁, CFC.spectrum_abs, ConvexOn.locallyLipschitzOn, instContinuousMapUniqueHom, volume_pi_le_diam_pow, MeasureTheory.hausdorffMeasure_real, LipschitzOnWith.of_le_add, CFC.abs_of_mem_unitary, ConcaveOn.locallyLipschitzOn, CFC.sqrt_eq_cfc, Filter.Tendsto.cfc_nnreal, IsSelfAdjoint.instContinuousFunctionalCalculus, Convex.lipschitzWith_gauge, ConcaveOn.exists_lipschitzOnWith_of_isBounded, CFC.abs_eq_cfcₙ_norm, continuousOn_cfcₙ_nnreal, CFC.abs_sq, RCLike.lipschitzWith_ofReal, LocallyLipschitz.const_min, ConvexOn.exists_lipschitzOnWith_of_isBounded, Metric.lipschitz_infDist_pt, CStarAlgebra.isStrictlyPositive_TFAE, lipschitzWith_cos, Metric.lipschitz_infDist, CStarAlgebra.norm_negPart_le, cfcₙ_nnreal_eq_real, CFC.sqrt_algebraMap, CFC.posPart_algebraMap, CFC.sqrt_one, CFC.nnrpow_eq_cfcₙ_real, MonotoneOn.eVariationOn_le, CFC.negPart_def, NNReal.isometry_coe, CFC.sqrt_rpow_nnreal, nnnorm_cfc_nnreal_lt_iff, ConcaveOn.continuousOn_tfae, LipschitzOnWith.of_le_add_mul', Metric.lipschitz_infDist_set, cfc_real_eq_complex, CFC.norm_rpow, MonotoneOn.nnnorm_cfc, ContinuousOn.cfcₙ_nnreal, CFC.posPart_def, CFC.nnrpow_eq_rpow, CFC.cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one, continuousOn_cfc_nnreal_setProd, CFC.abs_one, Complex.edist_of_re_eq, EuclideanSpace.euclideanHausdorffMeasure_eq_volume, CFC.isUnit_nnrpow_iff, unitInterval.volume_uIoc, lipschitzWith_max, CFC.monotoneOn_cfcₙ_rpowIntegrand₀₁, Complex.edist_of_im_eq, ContinuousAt.cfc_nnreal, ContinuousWithinAt.cfc_nnreal, IsSelfAdjoint.instNonUnitalIsometricContinuousFunctionalCalculus, nnnorm_cfcₙ_nnreal_le, CFC.sqrt_eq_real_sqrt, apply_le_nnnorm_cfcₙ_nnreal, continuousOn_cfcₙ_nnreal_setProd, CFC.real_exp_eq_normedSpace_exp, CFC.norm_sqrt, CFC.continuousOn_rpow, LipschitzWith.const_max, MonotoneOn.nnnorm_cfcₙ, lipschitzWith_min, CFC.nnnorm_nnrpow, Matrix.PosDef.posDef_sqrt, CFC.norm_nnrpow, Commute.cfcₙ_real, LipschitzOnWith.of_le_add_mul, cfcₙ_complex_eq_real, Matrix.IsHermitian.instContinuousFunctionalCalculus, ODE.FunSpace.lipschitzWith, ConcaveOn.locallyLipschitzOn_iff_continuousOn, LocallyLipschitz.min, MeasureTheory.Measure.euclideanHausdorffMeasure_def, LipschitzWith.of_le_add_mul', RCLike.lipschitzWith_im, CFC.rpow_eq_cfc_real, ediam_Icc, ConvexOn.lipschitzOnWith_of_abs_le, Continuous.cfcₙ_nnreal', CFC.isUnit_sqrt_iff, ediam_Ico, LipschitzWith.dist, volume_emetric_ball, lipschitzWith_one_norm, CFC.rpow_sqrt, ConcaveOn.lipschitzOnWith_of_abs_le

SeparationQuotient

Definitions

NameCategoryTheorems
instDist 📖CompOp
1 mathmath: dist_mk
instMetricSpace 📖CompOp
1 mathmath: instIsBoundedSMulSeparationQuotient

Theorems

NameKindAssumesProvesValidatesDepends On
dist_mk 📖mathematicalDist.dist
SeparationQuotient
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
instDist
PseudoMetricSpace.toDist

Subtype

Definitions

NameCategoryTheorems
metricSpace 📖CompOp
28 mathmath: Affine.Simplex.altitudeFoot_restrict, EuclideanGeometry.Cospherical.subtype_val_iff, EuclideanGeometry.Cospherical.inclusion_iff, EuclideanGeometry.Cospherical.inclusion, Affine.Simplex.circumradius_restrict, Affine.Simplex.ninePointCircle_restrict, EuclideanGeometry.Sphere.coe_secondInter, Affine.Simplex.altitude_restrict_eq_comap_subtype, Affine.Simplex.ExcenterExists.touchpointWeights_restrict, EuclideanGeometry.orthogonalProjection_subtype, Affine.Simplex.ExcenterExists.excenter_restrict, Affine.Simplex.map_altitude_restrict, Affine.Simplex.exradius_restrict, Affine.Simplex.circumcenter_restrict, Affine.Simplex.height_restrict, GromovHausdorff.eq_toGHSpace, Affine.Simplex.incenter_restrict, Affine.Simplex.inradius_restrict, Affine.Simplex.orthogonalProjectionSpan_restrict, Affine.Simplex.excenterWeights_restrict, Affine.Simplex.mongePoint_restrict, Affine.Simplex.eulerPoint_restrict, Affine.Simplex.ExcenterExists.touchpoint_restrict, Affine.Simplex.excenterWeightsUnnorm_restrict, IsConvexMetricSpace.of_convex, AffineSubspace.angle_coe, EuclideanGeometry.reflection_subtype, Affine.Simplex.excenterExists_restrict

Topology.IsEmbedding

Definitions

NameCategoryTheorems
comapMetricSpace 📖CompOp

(root)

Definitions

NameCategoryTheorems
instMetricSpaceNNReal 📖CompOp
76 mathmath: range_cfcₙ_nnreal, CFC.monotoneOn_one_sub_one_add_inv, nnnorm_cfcₙ_nnreal_le_iff, Commute.cfcₙ_nnreal, cfcₙHom_nnreal_eq_restrict, Continuous.cfc_nnreal', CFC.rpow_def, ContinuousOn.cfc_nnreal', IsGreatest.nnnorm_cfc_nnreal, NNReal.instContinuousMap.UniqueHom, nnnorm_cfcₙ_nnreal_lt, cfc_real_eq_nnreal, lipschitzWith_thickenedIndicator, IsGreatest.nnnorm_cfcₙ_nnreal, continuousOn_cfc_nnreal, cfc_nnreal_eq_real, CFC.nnrpow_def, NNReal.lipschitzWith_sub, Nonneg.instNonUnitalIsometricContinuousFunctionalCalculus, Continuous.cfc_nnreal_of_mem_nhdsSet, Continuous.cfcₙ_nnreal, ContinuousAt.cfcₙ_nnreal, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, cfcₙ_tsub, CFC.rpow_neg_one_eq_cfc_inv, ContinuousWithinAt.cfcₙ_nnreal, range_cfcₙ_nnreal_eq_image_cfcₙ_real, Filter.Tendsto.cfcₙ_nnreal, ContinuousOn.cfc_nnreal, nnnorm_cfcₙ_nnreal_lt_iff, nnnorm_cfc_nnreal_lt, cfcₙ_real_eq_nnreal, nnnorm_cfc_nnreal_le, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, cfc_nnreal_le_iff, Continuous.cfc_nnreal, nnnorm_cfc_nnreal_le_iff, Nonneg.instContinuousFunctionalCalculus, cfcHom_nnreal_eq_restrict, ContinuousOn.cfcₙ_nnreal', range_cfc_nnreal, Unitization.nnreal_cfcₙ_eq_cfc_inr, Real.lipschitzWith_toNNReal, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, lipschitzWith_one_nnnorm, apply_le_nnnorm_cfc_nnreal, MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz, range_cfc_nnreal_eq_image_cfc_real, range_cfcₙ_nnreal_subset, Nonneg.instNonUnitalContinuousFunctionalCalculus, CFC.sqrt_eq_cfc, Filter.Tendsto.cfc_nnreal, norm_cfcₙ_one_sub_one_add_inv_lt_one, continuousOn_cfcₙ_nnreal, cfcₙ_nnreal_eq_real, cfc_tsub, MeasureTheory.ProbabilityMeasure.testAgainstNN_lipschitz, NNReal.isometry_coe, nnnorm_cfc_nnreal_lt_iff, Commute.cfc_nnreal, MonotoneOn.nnnorm_cfc, ContinuousOn.cfcₙ_nnreal, NNReal.instContinuousMapZero.UniqueHom, continuousOn_cfc_nnreal_setProd, lipschitzWith_one_nnnorm', ContinuousAt.cfc_nnreal, ContinuousWithinAt.cfc_nnreal, nnnorm_cfcₙ_nnreal_le, apply_le_nnnorm_cfcₙ_nnreal, continuousOn_cfcₙ_nnreal_setProd, MonotoneOn.nnnorm_cfcₙ, range_cfc_nnreal_subset, Metric.lipschitz_infNndist_pt, BoundedContinuousFunction.lintegral_le_edist_mul, Continuous.cfcₙ_nnreal', Nonneg.instIsometricContinuousFunctionalCalculus
instMetricSpaceULift 📖CompOp
metricSpacePi 📖CompOp

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