Basic

📁 Source: Mathlib/Analysis/SpecialFunctions/Integrability/Basic.lean

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Theorems integrableOn_Ioo_cpow_iff, integrableOn_Ioo_rpow_iff, intervalIntegrable_const, intervalIntegrable_cos, intervalIntegrable_cpow, intervalIntegrable_cpow', intervalIntegrable_exp, intervalIntegrable_id, intervalIntegrable_inv, intervalIntegrable_inv_one_add_sq, intervalIntegrable_log, intervalIntegrable_log', intervalIntegrable_one_div, intervalIntegrable_one_div_one_add_sq, intervalIntegrable_pow, intervalIntegrable_rpow, intervalIntegrable_rpow', intervalIntegrable_sin, intervalIntegrable_zpow	19
Total	19

intervalIntegral

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
integrableOn_Ioo_cpow_iff 📖	mathematical	Real Real.instLT Real.instZero	MeasureTheory.IntegrableOn Complex MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing Complex.instNormedField SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing Complex.instPow Complex.ofReal Set.Ioo Real.instPreorder MeasureTheory.MeasureSpace.volume Real.instNeg Real.instOne Complex.re	—	MeasureTheory.integrableOn_congr_fun Complex.norm_cpow_eq_rpow_re_of_pos measurableSet_Ioo BorelSpace.opensMeasurable Real.borelSpace HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid MeasureTheory.Integrable.norm integrableOn_Ioo_rpow_iff intervalIntegrable_iff_integrableOn_Ioo_of_le PseudoEMetricSpace.pseudoMetrizableSpace Real.noAtoms_volume LT.lt.le enorm_ne_top intervalIntegrable_cpow'
integrableOn_Ioo_rpow_iff 📖	mathematical	Real Real.instLT Real.instZero	MeasureTheory.IntegrableOn MeasureTheory.MeasureSpace.toMeasurableSpace Real.measureSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing Real.instPow Set.Ioo Real.instPreorder MeasureTheory.MeasureSpace.volume Real.instNeg Real.instOne	—	Mathlib.Tactic.Contrapose.contrapose₁ lt_min zero_lt_one Real.instZeroLEOneClass NeZero.charZero_one RCLike.charZero_rclike MeasureTheory.IntegrableOn.mono Set.Ioo_subset_Ioo le_rfl min_le_right MeasureTheory.Integrable.mono' Measurable.aestronglyMeasurable PseudoEMetricSpace.pseudoMetrizableSpace instSecondCountableTopologyReal BorelSpace.opensMeasurable Real.borelSpace MeasurableInv.measurable_inv ContinuousInv₀.measurableInv T5Space.toT1Space T6Space.toT5Space instT6SpaceOfMetrizableSpace EMetricSpace.metrizableSpace IsTopologicalDivisionRing.toContinuousInv₀ instIsTopologicalDivisionRingReal Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.ae_restrict_mem measurableSet_Ioo HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid Filter.univ_mem' norm_inv abs_of_nonneg IsOrderedAddMonoid.toAddLeftMono le_of_lt Real.rpow_neg_one Real.rpow_le_rpow_left_iff_of_base_lt_one lt_of_lt_of_le min_le_left intervalIntegrable_iff_integrableOn_Ioo_of_le Real.noAtoms_volume LT.lt.le enorm_ne_top LT.lt.ne intervalIntegrable_rpow'
intervalIntegrable_const 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup	—	Continuous.intervalIntegrable continuous_const
intervalIntegrable_cos 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.cos	—	Continuous.intervalIntegrable Real.continuous_cos
intervalIntegrable_cpow 📖	mathematical	Real Real.instLE Real.instZero Complex.re Set Set.instMembership Set.uIcc Real.lattice	IntervalIntegrable Complex UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing Complex.instNormedField NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Complex.instNormedAddCommGroup Complex.instPow Complex.ofReal	—	ContinuousOn.intervalIntegrable continuousOn_of_forall_continuousAt Complex.continuousAt_ofReal_cpow_const lt_or_eq_of_le Continuous.intervalIntegrable Complex.continuous_ofReal_cpow_const IntervalIntegrable.intervalIntegrable_norm_iff Measurable.aestronglyMeasurable PseudoEMetricSpace.pseudoMetrizableSpace UniformSpace.secondCountable_of_separable EMetric.instIsCountablyGeneratedUniformity TopologicalSpace.SecondCountableTopology.to_separableSpace secondCountable_of_proper Complex.instProperSpace BorelSpace.opensMeasurable Complex.borelSpace measurable_of_continuousOn_compl_singleton Real.borelSpace T5Space.toT1Space T6Space.toT5Space instT6SpaceOfMetrizableSpace EMetricSpace.metrizableSpace le_or_gt intervalIntegrable_const enorm_ne_top intervalIntegrable_iff_integrableOn_Ioc_of_le MeasureTheory.IntegrableOn.congr_fun Complex.norm_cpow_eq_rpow_re_of_pos Real.rpow_zero measurableSet_Ioc instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid IntervalIntegrable.symm LT.lt.le Set.Ioo_union_right MeasureTheory.integrableOn_union MeasureTheory.integrableOn_singleton instMeasurableSingletonClassOfMeasurableEq instMeasurableEqOfSecondCountableTopologyOfT2Space instSecondCountableTopologyReal TopologicalSpace.t2Space_of_metrizableSpace enorm_norm MeasureTheory.IsFiniteMeasureOnCompacts.lt_top_of_isCompact isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure isCompact_singleton Complex.ofReal_cpow_of_nonpos norm_mul NormedDivisionRing.toNormMulClass Complex.ofReal_neg neg_pos IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono AddLeftCancelSemigroup.toIsLeftCancelAdd IsOrderedAddMonoid.toAddLeftMono one_mul MeasureTheory.integrableOn_const_iff LE.le.trans_lt MeasureTheory.measure_mono MeasureTheory.Measure.instOuterMeasureClass Set.Ioo_subset_Icc_self CompactIccSpace.isCompact_Icc ConditionallyCompleteLinearOrder.toCompactIccSpace instOrderTopologyReal measurableSet_Ioo IntervalIntegrable.trans
intervalIntegrable_cpow' 📖	mathematical	Real Real.instLT Real.instNeg Real.instOne Complex.re	IntervalIntegrable Complex UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing Complex.instNormedField NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Complex.instNormedAddCommGroup Complex.instPow Complex.ofReal MeasureTheory.MeasureSpace.volume Real.measureSpace	—	IntervalIntegrable.intervalIntegrable_norm_iff ContinuousOn.aestronglyMeasurable secondCountableTopologyEither_of_right UniformSpace.secondCountable_of_separable EMetric.instIsCountablyGeneratedUniformity TopologicalSpace.SecondCountableTopology.to_separableSpace secondCountable_of_proper Complex.instProperSpace BorelSpace.opensMeasurable Real.borelSpace PseudoEMetricSpace.pseudoMetrizableSpace continuousOn_of_forall_continuousAt ContinuousAt.comp continuousAt_cpow_const Complex.ofReal_re Set.uIoc_of_le Continuous.continuousAt Complex.continuous_ofReal measurableSet_uIoc instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid intervalIntegrable_iff MeasureTheory.IntegrableOn.congr_fun intervalIntegrable_rpow' Complex.norm_cpow_eq_rpow_re_of_pos le_total IntervalIntegrable.iff_comp_neg enorm_ne_top neg_zero IntervalIntegrable.const_mul le_of_not_gt Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.cast_zero Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Linarith.add_lt_of_le_of_neg Real.instIsStrictOrderedRing Mathlib.Tactic.Linarith.sub_nonpos_of_le Real.instIsOrderedRing Mathlib.Tactic.Linarith.sub_neg_of_lt Mathlib.Meta.NormNum.IsNat.to_raw_eq Mathlib.Tactic.Linarith.add_neg Complex.ofReal_cpow_of_nonpos mul_comm Complex.ofReal_neg neg_neg measurableSet_Ioc IntervalIntegrable.trans IntervalIntegrable.symm
intervalIntegrable_exp 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.exp	—	Continuous.intervalIntegrable Real.continuous_exp
intervalIntegrable_id 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup	—	Continuous.intervalIntegrable continuous_id
intervalIntegrable_inv 📖	mathematical	ContinuousOn Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Set.uIcc Real.lattice	IntervalIntegrable NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.instInv	—	one_div intervalIntegrable_one_div
intervalIntegrable_inv_one_add_sq 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.instInv Real.instAdd Real.instOne Monoid.toNatPow Real.instMonoid	—	—
intervalIntegrable_log 📖	mathematical	Real Set Set.instMembership Set.uIcc Real.lattice Real.instZero	IntervalIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.log	—	IntervalIntegrable.log continuousOn_id
intervalIntegrable_log' 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.log MeasureTheory.MeasureSpace.volume Real.measureSpace	—	intervalIntegrable_of_even Real.log_neg_eq_log enorm_ne_top IntervalIntegrable.trans PseudoEMetricSpace.pseudoMetrizableSpace neg_neg IntervalIntegrable.neg intervalIntegrable_deriv_of_nonneg ContinuousOn.neg IsTopologicalRing.toContinuousNeg instIsTopologicalRingReal ContinuousOn.sub IsTopologicalAddGroup.to_continuousSub instIsTopologicalAddGroupReal Continuous.continuousOn Real.continuous_mul_log continuous_id IsModuleTopology.toContinuousSMul IsTopologicalSemiring.toIsModuleTopology IsTopologicalRing.toIsTopologicalSemiring HasDerivAt.congr_simp neg_sub IsBoundedSMul.continuousSMul NormedSpace.toIsBoundedSMul sub_add_cancel_right HasDerivAt.sub hasDerivAt_id Real.hasDerivAt_mul_log LT.lt.ne inf_of_le_left Mathlib.Meta.NormNum.isNat_le_true Real.instIsOrderedRing Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Nat.cast_one IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid Real.log_nonpos_iff LT.lt.le sup_of_le_right ContinuousOn.intervalIntegrable Real.locallyFinite_volume ContinuousOn.mono Real.continuousOn_log Set.notMem_uIcc_of_lt zero_lt_one Real.instZeroLEOneClass NeZero.charZero_one RCLike.charZero_rclike
intervalIntegrable_one_div 📖	mathematical	ContinuousOn Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Set.uIcc Real.lattice	IntervalIntegrable NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne	—	ContinuousOn.intervalIntegrable ContinuousOn.div IsTopologicalDivisionRing.toContinuousInv₀ instIsTopologicalDivisionRingReal IsTopologicalSemiring.toContinuousMul IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal continuousOn_const
intervalIntegrable_one_div_one_add_sq 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne Real.instAdd Monoid.toNatPow Real.instMonoid	—	Continuous.intervalIntegrable Continuous.div₀ IsTopologicalDivisionRing.toContinuousInv₀ instIsTopologicalDivisionRingReal IsTopologicalSemiring.toContinuousMul IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal continuous_const Continuous.fun_add IsTopologicalSemiring.toContinuousAdd Continuous.pow continuous_id' Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.cast_zero Nat.cast_zero Mathlib.Tactic.Ring.pow_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.pow_add Mathlib.Tactic.Ring.single_pow Mathlib.Tactic.Ring.mul_pow Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.one_pow Mathlib.Tactic.Ring.pow_zero Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Tactic.Linarith.lt_of_lt_of_eq Mathlib.Tactic.Linarith.add_lt_of_neg_of_le Real.instIsStrictOrderedRing neg_neg_of_pos Real.instIsOrderedAddMonoid Mathlib.Tactic.Linarith.zero_lt_one Mathlib.Tactic.Linarith.sub_nonpos_of_le Real.instIsOrderedRing sq_nonneg AddGroup.existsAddOfLE IsOrderedRing.toPosMulMono IsOrderedAddMonoid.toAddLeftMono sub_eq_zero_of_eq
intervalIntegrable_pow 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Monoid.toNatPow Real.instMonoid	—	Continuous.intervalIntegrable continuous_pow IsTopologicalSemiring.toContinuousMul IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal
intervalIntegrable_rpow 📖	mathematical	Real Real.instLE Real.instZero Set Set.instMembership Set.uIcc Real.lattice	IntervalIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.instPow	—	ContinuousOn.intervalIntegrable ContinuousOn.rpow_const continuousOn_id
intervalIntegrable_rpow' 📖	mathematical	Real Real.instLT Real.instNeg Real.instOne	IntervalIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.instPow MeasureTheory.MeasureSpace.volume Real.measureSpace	—	intervalIntegrable_iff PseudoEMetricSpace.pseudoMetrizableSpace Set.uIoc_of_le IsModuleTopology.toContinuousSMul IsTopologicalSemiring.toIsModuleTopology IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal IsBoundedSMul.continuousSMul NormedSpace.toIsBoundedSMul add_sub_cancel_right mul_div_cancel_left₀ EuclideanDomain.toMulDivCancelClass Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_gt Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.cast_zero Nat.cast_zero Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Tactic.Linarith.lt_of_lt_of_eq Mathlib.Tactic.Linarith.sub_neg_of_lt Real.instIsOrderedRing sub_eq_zero_of_eq HasDerivAt.div_const Real.hasDerivAt_rpow_const LT.lt.ne' integrableOn_deriv_of_nonneg ContinuousOn.div_const IsTopologicalSemiring.toContinuousMul ContinuousOn.rpow_const continuousOn_id le_of_not_gt Mathlib.Tactic.Linarith.add_neg Real.instIsStrictOrderedRing Real.rpow_nonneg LT.lt.le le_total IntervalIntegrable.iff_comp_neg enorm_ne_top neg_zero IntervalIntegrable.smul Real.isBoundedSMul Mathlib.Tactic.Linarith.add_lt_of_le_of_neg Mathlib.Tactic.Linarith.sub_nonpos_of_le MeasureTheory.IntegrableOn.congr_fun Real.rpow_def_of_pos mul_comm Real.rpow_def_of_neg lt_of_not_ge Mathlib.Meta.NormNum.IsNat.to_raw_eq Real.log_neg_eq_log measurableSet_Ioc BorelSpace.opensMeasurable Real.borelSpace instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid IntervalIntegrable.trans IntervalIntegrable.symm
intervalIntegrable_sin 📖	mathematical	—	IntervalIntegrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup Real.sin	—	Continuous.intervalIntegrable Real.continuous_sin
intervalIntegrable_zpow 📖	mathematical	Real Set Set.instMembership Set.uIcc Real.lattice Real.instZero	IntervalIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup Real.normedAddCommGroup DivInvMonoid.toZPow Real.instDivInvMonoid	—	ContinuousOn.intervalIntegrable ContinuousOn.zpow₀ IsTopologicalDivisionRing.toContinuousInv₀ instIsTopologicalDivisionRingReal IsTopologicalSemiring.toContinuousMul IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal continuousOn_id

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