Documentation Verification Report

Basic

📁 Source: Mathlib/Algebra/GroupWithZero/Units/Basic.lean

Statistics

MetricCount
DefinitionstoDivisionCommMonoid, inverse, mk0, commGroupWithZeroOfIsUnitOrEqZero, groupWithZeroOfIsUnitOrEqZero
5
Theoremseq_zero_or_unit, noZeroDivisors, mk0, mul_left_eq_zero, mul_right_eq_zero, ne_zero, ringInverse, isUnit, eq_mul_inverse_iff_mul_eq, inverse_eq_inv, inverse_eq_inv', inverse_mul_cancel, inverse_mul_cancel_left, inverse_mul_cancel_right, inverse_mul_eq_iff_eq_mul, inverse_non_unit, inverse_of_isUnit, inverse_one, inverse_unit, inverse_zero, mul_inverse_cancel, mul_inverse_cancel_left, mul_inverse_cancel_right, exists0, exists0', exists_iff_ne_zero, inv_mul', mk0_inj, mk0_mul, mk0_one, mk0_val, mul_inv', mul_left_eq_zero, mul_right_eq_zero, ne_zero, val_mk0, div_div_cancel_left', div_div_cancel₀, div_div_div_cancel_left', div_div_div_cancel_right₀, div_eq_div_iff, div_eq_div_iff_div_eq_div', div_eq_div_of_div_eq_div, div_eq_iff, div_eq_iff_mul_eq, div_eq_of_eq_mul, div_eq_one_iff_eq, div_eq_zero_iff, div_helper, div_left_inj', div_mul_cancel_left₀, div_mul_cancel_of_imp, div_mul_cancel_right₀, div_mul_cancel₀, div_mul_div_cancel₀, div_mul_div_cancel₀', div_ne_zero, div_ne_zero_iff, div_self, div_self_eq_one₀, divp_mk0, eq_div_iff, eq_div_iff_mul_eq, eq_div_of_mul_eq, eq_inv_mul_iff_mul_eq₀, eq_mul_inv_iff_mul_eq₀, eq_mul_of_inv_mul_eq₀, eq_mul_of_mul_inv_eq₀, eq_zero_of_zpow_eq_zero, inv_mul_eq_iff_eq_mul₀, inv_mul_eq_inv_mul_iff, inv_mul_eq_one₀, inv_pow_sub_of_lt, inv_pow_sub₀, isUnit_iff_ne_zero, isUnit_ringInverse, isUnit_zero_iff, mul_div_cancel_left_of_imp, mul_div_cancel_of_imp, mul_div_cancel_of_imp', mul_div_cancel₀, mul_div_mul_left, mul_div_mul_right, mul_eq_mul_of_div_eq_div, mul_eq_of_eq_inv_mul₀, mul_eq_of_eq_mul_inv₀, mul_eq_one_iff_eq_inv₀, mul_eq_one_iff_inv_eq₀, mul_inv_eq_iff_eq_mul₀, mul_inv_eq_mul_inv_iff, mul_inv_eq_one₀, mul_mul_div, mul_one_div_cancel, not_isUnit_zero, one_div_mul_cancel, pow_sub_of_lt, pow_sub₀, zpow_eq_zero_iff, zpow_natCast_sub_natCast₀, zpow_natCast_sub_one₀, zpow_ne_zero, zpow_ne_zero_iff, zpow_neg_mul_zpow_self, zpow_one_sub_natCast₀, zpow_sub₀
105
Total110

CommGroupWithZero

Definitions

NameCategoryTheorems
toDivisionCommMonoid 📖CompOp
440 mathmath: ordinaryHypergeometric_sum_eq, Polynomial.coe_normUnit_of_ne_zero, monovaryOn_inv₀, coe_normUnit, ValuationSubring.valuation_lt_one_iff, HasStrictDerivAt.to_localInverse, tendsto_inv_atTop_zero, RatFunc.num_div, Valuation.isEquiv_iff_val_sub_one_lt_one, mul_inv_eq_mul_inv_iff, NormedSpace.expSeries_eq_ofScalars, inv_atBot₀, Lagrange.eval_interpolate_not_at_node', Valuation.mem_maximalIdeal_iff, Lagrange.nodalWeight_eq_eval_nodal_erase_inv, AdjoinRoot.minpoly_root, round_neg_two_inv, Algebra.normalizedTrace_eq_of_finiteDimensional, Int.log_inv, Meromorphic.fun_inv, Valued.toNormedField.norm_le_one_iff, Polynomial.div_def, Asymptotics.SuperpolynomialDecay.param_inv_mul, ValuationSubring.mem_or_inv_mem, OpenPartialHomeomorph.hasDerivAt_symm, spectrum.inv₀_mem_iff, norm_smul_inv_norm', Lagrange.leadingCoeff_basis, Int.ceil_div_ceil_inv_sub_one, Quaternion.coe_inv, round_two_inv, Valuation.val_le_one_or_val_inv_lt_one, minpoly.eq_of_irreducible, HasStrictDerivAt.of_local_left_inverse, inv_atTop₀, ValuativeRel.inv_srel_one, IsStrictOrderedRing.toContinuousInv₀, ValuationSubring.ofPrime_valuation_eq_one_iff_mem_primeCompl, NormedSpace.exp_series_hasSum_exp', Lagrange.eval_interpolate_not_at_node, ValuationSubring.valuation_unit, DiffContOnCl.inv, AddCircle.homeomorphAddCircle_apply_mk, Valuation.IsEquiv.valuedCompletion_le_one_iff, Subring.center.coe_inv, sub_inv_antitoneOn_Ioi, two_inv_lt_one, difference_quotients_converge_uniformly, AntivaryOn.inv_left₀, Valuation.isNontrivial_iff_exists_lt_one, Valuation.Integers.valuation_irreducible_lt_one, hasFDerivWithinAt_inv, Ring.choose_eq_smul, norm_smul_inv_norm, inv_pow_le_inv_pow_of_le, inv_pow_anti, Polynomial.roots_degree_eq_one, deriv_fun_inv'', Asymptotics.isTheta_inv, Set.inv_Iio₀, Set.inv_Ioo_0_right, inv_le_inv_of_neg, inv_pow_strictAnti, antivary_inv₀, Valuation.Integers.map_le_one, HasFPowerSeriesOnBall.hasSum_iteratedFDeriv, lt_inv_mul_iff₀', MeromorphicAt.inv_iff, inv_strictAntiOn, inv_lt_zero', Mathlib.Tactic.FieldSimp.NF.eval_inv, HasLineDerivAt.tendsto_slope_zero_right, ContDiffAt.inv, Hyperreal.infinitesimal_inv_of_infinite, le_inv_of_neg, antivaryOn_inv_left₀, Derivation.leibniz_div, div_mul_cancel_left₀, NormedSpace.exp_eq_tsum, Valuation.leIdeal_map_algebraMap_eq_leSubmodule_min, Nonneg.inv_mk, Valuation.val_lt_one_iff, Valuation.val_le_one_or_val_inv_le_one, EulerProduct.eulerProduct_completely_multiplicative, Quaternion.inv_def, Localization.subalgebra.mem_range_mapToFractionRing_iff_ofField, Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval, IsAlgebraic.inv, Hyperreal.inv_omega, Valuation.one_lt_val_iff, Hyperreal.infinite_iff_infinitesimal_inv, abs_inv, ValuationSubring.valuation_lt_one_or_eq_one, ValuationRing.iff_isInteger_or_isInteger, Int.clog_of_right_le_one, MeromorphicOn.divisor_inv, Finset.lubell_yamamoto_meshalkin_inequality_sum_inv_choose, Valuation.val_eq_one_iff, inv_pow_lt_inv_pow_of_lt, NormedSpace.expSeries_sum_eq, IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mk, NormedSpace.smul_mem_polar, div_helper, FractionalIdeal.one_div_spanSingleton, monovary_inv_right₀, ValuativeRel.one_rel_inv, IntermediateField.inv_mem', TrivSqZeroExt.eq_smul_exp_of_ne_zero, Derivation.leibniz_inv, RingPreordering.inv_mem, Antivary.inv_right₀, inv_mul_eq_inv_mul_iff, MvPowerSeries.constantCoeff_inv, Valuation.mem_unitGroup_iff, derivWithin_inv, FractionalIdeal.count_ne_zero, RCLike.normSq_inv, ValuationSubring.valuation_le_one, RatFunc.numDenom_div, MvPowerSeries.coeff_inv, geom_sum_lt, sub_inv_antitoneOn_Icc_left, NormedSpace.norm_expSeries_summable_of_mem_ball', le_mul_inv_iff₀', tendsto_inv_nhdsGT_zero, HasLineDerivAt.tendsto_slope_zero, HasStrictDerivAt.to_local_left_inverse, ordinaryHypergeometricSeries_apply_eq', differentiableAt_inv_iff, MonovaryOn.inv_left₀, ValuationSubring.primeSpectrumOrderEquiv_apply_coe_carrier, Hyperreal.infinitePos_iff_infinitesimal_inv_pos, mul_inv_lt_of_lt_mul₀, Valuation.isEquiv_iff_val_le_one, traceForm_dualSubmodule_adjoin, Antivary.inv_left₀, Polynomial.roots_C_mul_X_add_C, AddCircle.equivAddCircle_apply_mk, meromorphicNFAt_inv, Valuation.mem_valuationSubring_iff, HasLineDerivAt.tendsto_slope_zero_left, ValuationRing.isFractionRing_iff, EulerProduct.prod_filter_prime_geometric_eq_tsum_factoredNumbers, Valuation.Integers.isIntegral_iff_v_le_one, antivary_inv_right₀, deriv_inv, Valuation.isEquiv_iff_val_eq_one, tendsto_inv_atBot_nhdsLT_zero, Algebra.norm_inv, NormedSpace.expSeries_summable', ValuationSubring.valuation_le_one_iff, HasDerivAt.tendsto_slope_zero_left, ValuationSubring.mem_unitGroup_iff, Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero, EulerProduct.eulerProduct_completely_multiplicative_tprod, Quaternion.normSq_inv, meromorphicOrderAt_inv, Algebra.normalizedTrace_eq_of_fininteDimensional, inv_nhdsGT_zero, FractionalIdeal.mul_generator_self_inv, RCLike.inv_im, FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul, MonovaryOn.inv_right₀, NormedSpace.expSeries_apply_eq, MvPowerSeries.C_inv, ContDiffWithinAt.inv, tendsto_inv_atTop_nhdsGT_zero, hasDerivAt_inv, le_inv_mul_iff₀', tendsto_inv_atBot_zero, PowerSeries.constantCoeff_inv, spectrum.inv₀_mem_inv_iff, MeromorphicOn.inv_iff, Hyperreal.inv_epsilon, inv_lt_inv_of_neg, EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric, sum_Ioo_inv_sq_le, meromorphicTrailingCoeffAt_inv, EulerProduct.prod_primesBelow_geometric_eq_tsum_smoothNumbers, CancelDenoms.inv_subst, ordinaryHypergeometric_eq_tsum, gauge_smul_left_of_nonneg, meromorphicNFOn_inv, GenContFract.convs_succ, LieAlgebra.IsKilling.rootSpace_one_div_two_smul, cfc_inv, antivaryOn_inv₀, InnerProductSpace.gramSchmidtOrthonormalBasis_apply_of_orthogonal, Field.isSeparable_inv, contDiffAt_inv, Mathlib.Tactic.Ring.inv_zero, Valuation.Integers.valuation_unit, EulerProduct.eulerProduct_completely_multiplicative_hasProd, Hyperreal.IsSt.inv, Monovary.inv_right₀, Antivary.inv₀, lt_mul_inv_iff₀', Height.mulHeight₁_inv, Hyperreal.coe_inv, NormedField.continuousAt_inv, ValuativeRel.inv_rel_one, Hyperreal.infinitesimal_neg_iff_infiniteNeg_inv, MeromorphicOn.inv, hasDerivAt_iff_tendsto_slope_zero, Mathlib.Tactic.FieldSimp.list_prod_zpow', EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric, spectrum.inv₀_mem, ordinaryHypergeometricSeries_apply_eq, ValuativeRel.one_apply_posSubmonoid, Matrix.SpecialLinearGroup.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero, hasStrictFDerivAt_inv, IsFractionRing.associated_num_den_inv, fderiv_inv, inv_sub_inv, Valuation.extendToLocalization_mk', Monovary.inv_left₀, Nonneg.coe_inv, slope_fun_def, Height.logHeight₁_inv, deriv_inv'', selfAdjoint.val_inv, HasFDerivAt.lim, cfc_inv_id, NormedSpace.norm_expSeries_summable', ValuativeRel.one_vlt_inv, SimpleGraph.FarFromTriangleFree.lt_half, NormedSpace.expSeries_apply_eq', contDiffOn_inv, Valued.toNormedField.one_lt_norm_iff, Valuation.IsNontrivial_iff_exists_one_lt, RatFunc.denom_div, val_inv_cfcUnits, Valuation.Integers.eq_algebraMap_or_inv_eq_algebraMap, Monovary.inv₀, Valuation.map_inv, inv_antitoneOn_Icc_right, stdSimplex.barycenter_apply, ContinuousLinearEquiv.det_coe_symm, Polynomial.div_C, MeromorphicOn.divisor_fun_inv, inv_nhdsLT_zero, Polynomial.leadingCoeff_preHilbertPoly, Hyperreal.infinitesimal_iff_infinite_inv, ValuationSubring.mem_or_inv_mem', instContMDiffInv₀ModelWithCornersSelf, tendsto_algebraMap_inverse_atTop_nhds_zero_nat, spectrum.map_inv, derivWithin_inv', MulChar.inv_apply_eq_inv', inv_mul_lt_of_lt_mul₀, Positive.coe_inv, Valuation.IsNontrivial.exists_one_lt, HasDerivWithinAt.inv, RCLike.inv_pos_of_pos, ValuationSubring.inv_mem_nonunits_iff, logDeriv_inv, MonovaryOn.inv₀, Asymptotics.IsTheta.inv, NumberField.linearDisjoint_of_isGalois_isCoprime_discr, Valuation.IsEquiv.val_sub_one_lt_one_iff, IsStrictOrderedRing.toHasContinuousInv₀, hasLineDerivAt_iff_tendsto_slope_zero, Algebra.normalizedTrace_minpoly, ValuationSubring.mem_nonunits_iff_or, Polynomial.coeff_inv_units, monovaryOn_inv_left₀, iter_deriv_inv, PowerSeries.coeff_inv, ContDiff.inv, RCLike.inv_def, Algebra.normalizedTrace_eq_of_finiteDimensional_apply, ValuationSubring.mem_principalUnitGroup_iff, fderivWithin_inv, FractionalIdeal.div_spanSingleton, RatFunc.num_div_dvd', inv_smul_lt_iff_of_neg, iter_deriv_inv', Asymptotics.IsLittleOTVS.tendsto_inv_smul, Algebra.normalizedTrace_def, Units.continuousOn_inv₀_spectrum, IsAbsoluteValue.abv_inv, IntermediateField.coe_inv, Hyperreal.infiniteNeg_iff_infinitesimal_inv_neg, tendsto_inv_nhdsLT_zero, Seminorm.smul_closedBall_preimage, RCLike.inv_pos, IsLocalMin.inv, HahnSeries.inv_def, Valuation.Integer.not_isUnit_iff_valuation_lt_one, Valuation.map_eq_one_of_forall_lt, smul_inv_le_iff_of_neg, MulChar.inv_apply', Asymptotics.SuperpolynomialDecay.inv_param_mul, Filter.Tendsto.inv_tendsto_nhdsLT_zero, EulerProduct.one_sub_inv_eq_geometric_of_summable_norm, inv_mul_lt_iff₀', Polynomial.roots_C_mul_X_sub_C, ValuativeRel.one_vle_inv, smul_inv_lt_iff_of_neg, Asymptotics.IsEquivalent.inv, Asymptotics.isLittleOTVS_iff_tendsto_inv_smul, Seminorm.smul_ball_preimage, normalize_eq_one, lt_inv_of_neg, hasStrictDerivAt_inv, GenContFract.IntFractPair.succ_nth_stream_b_le_nth_stream_fr_inv, FractionalIdeal.exists_eq_spanSingleton_mul, AdjoinRoot.minpoly_powerBasis_gen, divMonoidWithZeroHom_apply, ValuationSubring.mem_nonunits_iff, HasDerivAt.tendsto_slope_zero, sub_one_div_inv_le_two, Filter.Tendsto.inv_tendsto_nhdsGT_zero, iter_deriv_inv_linear_sub, RCLike.conj_inv, IsAlgebraic.inv_iff, monovary_inv₀, AntivaryOn.inv_right₀, Valuation.integer.v_irreducible_lt_one, sum_Ioc_inv_sq_le_sub, PowerSeries.C_inv, sub_inv_antitoneOn_Icc_right, Hyperreal.st_inv, Valuation.IsNontrivial.exists_lt_one, Valuation.Integers.isUnit_iff_valuation_eq_one, ContDiffOn.inv, LieAlgebra.IsKilling.traceForm_coroot, ValuativeRel.inv_vle_one, toPrincipalIdeal_def, inv_antitoneOn_Ioi, MeromorphicAt.inv, MeromorphicOn.fun_inv, spectrum.inv₀_mem_inv, Meromorphic.inv, HasDerivAt.fun_inv, ValuativeRel.inv_vlt_one, PowerSeries.inv_eq_inv_aux, mul_inv_lt_iff₀', deriv_inv', LinearEquiv.det_coe_symm, Filter.Tendsto.inv_tendsto_atTop, RCLike.inv_eq_conj, iter_deriv_inv_linear, ValuationRing.isInteger_or_isInteger, monovary_inv_left₀, Nat.inv_pos_of_nat, gauge_smul_left, Derivation.leibniz_div_const, IntermediateField.inv_mem, Valuation.one_le_val_iff, ValuationSubring.valuation_eq_one_iff, inv_le_of_neg, IsFractionRing.associated_den_num_inv, Algebra.normalizedTrace_eq_of_fininteDimensional_apply, NormedSpace.expSeries_hasSum_exp_of_mem_ball', Polynomial.div_C_mul, Hyperreal.infinitesimal_pos_iff_infinitePos_inv, Int.log_of_right_le_one, div_div_cancel_left', AntivaryOn.inv₀, GenContFract.sub_convs_eq, inv_smul_le_iff_of_neg, homothety_inv_two, IsLocalization.AtPrime.equivQuotMaximalIdeal_symm_apply_mk, ValuativeRel.one_srel_inv, Valuation.isEquiv_iff_val_lt_one, HahnSeries.inv_single, Valuation.IsUniformizer.val_lt_one, hasDerivWithinAt_inv, OpenPartialHomeomorph.hasStrictDerivAt_symm, Valuation.mem_integer_iff, MeromorphicNFAt.inv, algebraMap.coe_inv, ValuationSubring.top_coe_inv, tendsto_algebraMap_inv_atTop_nhds_zero_nat, derivWithin_fun_inv', Subalgebra.inv_mem_of_algebraic, spectrum.resolvent_isBigO_inv, NormedSpace.expSeries_summable_of_mem_ball', Int.neg_log_inv_eq_clog, AddCircle.equivAddCircle_symm_apply_mk, RatFunc.single_inv, RCLike.ofReal_inv, PerfectClosure.mk_inv, antivaryOn_inv_right₀, inv_lt_of_neg, PowerSeries.smul_inv, Valued.toNormedField.norm_lt_one_iff, two_mul_le_add_mul_sq, Polynomial.mod_def, Valuation.Integers.one_of_isUnit, Int.neg_clog_inv_eq_log, hasFDerivAt_inv, inv_nonpos', HahnSeries.cardSupp_inv_le, ValueDistribution.logCounting_inv, MeromorphicAt.fun_inv, MvPowerSeries.smul_inv, UniformSpace.Completion.hatInv_extends, NumberField.IsCMField.complexConj_torsion, sub_inv_antitoneOn_Iio, Valuation.val_le_one_iff, Transcendental.linearIndependent_sub_inv, CauSeq.lim_inv, Valued.toNormedField.one_le_norm_iff, ArchimedeanClass.stdPart_inv, slope_def_module, inv_antitoneOn_Icc_left, Nat.cast_inv_le_one, inv_eq_of_aeval_divX_ne_zero, AddChar.inv_apply_eq_conj, antivary_inv_left₀, HasDerivAt.of_local_left_inverse, Polynomial.coeff_preHilbertPoly_self, Filter.IsBoundedUnder.isLittleO_sub_self_inv, monovaryOn_inv_right₀, Int.clog_inv, ArchimedeanClass.mk_inv, RCLike.inv_I, HasDerivAt.inv, inv_antitoneOn_Iio, Mathlib.Tactic.FieldSimp.NF.inv_eq_eval, HasDerivWithinAt.fun_inv, Filter.Tendsto.inv_tendsto_atBot, inv_eq_of_root_of_coeff_zero_ne_zero, HasDerivAt.tendsto_slope_zero_right, FractionalIdeal.spanSingleton_inv, AddCircle.homeomorphAddCircle_symm_apply_mk, inv_mul_le_iff₀', inv_add_inv, RCLike.inv_re, CompletableTopField.nice, WithZeroTopology.instContinuousInv₀, NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat, Lagrange.nodalWeight_eq_eval_derivative_nodal, Padic.valuation_inv, UniformSpace.Completion.coe_inv, Lagrange.eval_basis_not_at_node, mul_inv_le_iff₀', NormedSpace.exp_eq_ofScalarsSum

GroupWithZero

Theorems

NameKindAssumesProvesValidatesDepends On
eq_zero_or_unit 📖mathematical—MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
toMonoidWithZero
Units.val
MonoidWithZero.toMonoid
—em
noZeroDivisors 📖mathematical—NoZeroDivisors
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
toMonoidWithZero
MulZeroClass.toZero
—Mathlib.Tactic.Contrapose.contrapose₁
Units.ne_zero
toNontrivial

IsUnit

Theorems

NameKindAssumesProvesValidatesDepends On
mk0 📖mathematical—IsUnit
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—Units.isUnit
mul_left_eq_zero 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
MulZeroClass.toZero
—Units.mul_left_eq_zero
mul_right_eq_zero 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
MulZeroClass.toZero
—Units.mul_right_eq_zero
ne_zero 📖—IsUnit
MonoidWithZero.toMonoid
——Units.ne_zero
ringInverse 📖mathematicalIsUnit
MonoidWithZero.toMonoid
Ring.inverse—Ring.inverse_unit

Ne

Theorems

NameKindAssumesProvesValidatesDepends On
isUnit 📖mathematical—IsUnit
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—isUnit_iff_ne_zero

Ring

Definitions

NameCategoryTheorems
inverse 📖CompOp
85 mathmath: DividedPowers.RatAlgebra.dpow_eq_inv_fact_smul, NormedRing.inverse_one_sub_norm, tsum_coe_mul_geometric_of_norm_lt_one', mul_inverse_cancel_right, cfc_map_div, analyticOnNhd_inverse, inverse_eq_inv', NormedRing.inverse_add_norm_diff_nth_order, DividedPowers.OfInvertibleFactorial.dpow_apply, tsum_choose_mul_geometric_of_norm_lt_one', hasSum_coe_mul_geometric_of_norm_lt_one', inverse_mul_eq_iff_eq_mul, inverse_mul_cancel, DifferentiableWithinAt.inverse, MulChar.inv_apply_eq_inv, inverse_sub_inverse, NormedRing.inverse_one_sub_nth_order, inverse_invertible, spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul, MonoidWithZero.coe_inverse, mul_inverse_rev', hasFDerivAt_ringInverse, DifferentiableAt.inverse, ContinuousLinearMap.ringInverse_equiv, inverse_eq_inv, fderiv_inverse, Matrix.inv_def, Differentiable.inverse, inverse_add_inverse, mul_inverse_cancel, Nat.castChoose_eq, DividedPowers.RatAlgebra.dpow_apply, inverse_pow_mul_eq_iff_eq_mul, Commute.ringInverse_ringInverse, inverse_star, Matrix.nonsing_inv_eq_ringInverse, cfc_inv, NormedRing.inverse_add, NormedRing.inverse_one_sub, NormedRing.inverse_add_norm, NormedRing.inverse_add_norm_diff_second_order, inverse_non_unit, hasSum_geom_series_inverse, hasSum_choose_mul_geometric_of_norm_lt_one', inverse_unit, hasFPowerSeriesOnBall_inverse_one_sub, hasStrictFDerivAt_ringInverse, LinearMap.equivariantProjection_apply, differentiableOn_inverse, ContinuousLinearMap.ringInverse_eq_inverse, spectrum.differentiableOn_inverse_one_sub_smul, DividedPowers.OfInvertibleFactorial.dpow_eq_of_mem, inverse_one, differentiableWithinAt_inverse, DifferentiableOn.inverse, inverse_mul_cancel_right, DividedPowers.RatAlgebra.dpow_eq_of_mem, IsUnit.ringInverse, NormedRing.inverse_add_norm_diff_first_order, Matrix.charpoly_inv, ContinuousLinearMap.inverse_eq_ringInverse, mul_inverse_rev, NormedRing.inverse_continuousAt, inverse_mul_cancel_left, mul_inverse_cancel_left, contDiffAt_ringInverse, analyticAt_inverse_one_sub, Polynomial.newtonMap_apply, isUnit_ringInverse, NormedRing.inverse_one_sub_nth_order', NormedRing.inverse_add_nth_order, MonoidWithZero.inverse_apply, differentiableAt_inverse, Matrix.inv_subsingleton, PadicInt.coe_dpow_eq, inverse_of_isUnit, inverse_exp, MulChar.inv_apply, analyticAt_inverse, geom_series_eq_inverse, Matrix.inv_diagonal, inverse_zero, eq_mul_inverse_iff_mul_eq, Matrix.det_nonsing_inv, inverse_pow

Theorems

NameKindAssumesProvesValidatesDepends On
eq_mul_inverse_iff_mul_eq 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—inverse_mul_cancel_right
mul_inverse_cancel_right
inverse_eq_inv 📖mathematical—inverse
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—eq_or_ne
inverse_non_unit
inv_zero
inverse_unit
inverse_eq_inv' 📖mathematical—inverse
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—inverse_eq_inv
inverse_mul_cancel 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
MulOne.toOne
MulOneClass.toMulOne
MulZeroOneClass.toMulOneClass
—inverse_unit
Units.inv_mul
inverse_mul_cancel_left 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—mul_assoc
inverse_mul_cancel
one_mul
inverse_mul_cancel_right 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—mul_assoc
inverse_mul_cancel
mul_one
inverse_mul_eq_iff_eq_mul 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—mul_inverse_cancel_left
inverse_mul_cancel_left
inverse_non_unit 📖mathematicalIsUnit
MonoidWithZero.toMonoid
inverse
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
——
inverse_of_isUnit 📖mathematicalIsUnit
MonoidWithZero.toMonoid
inverse
Units.val
Units
Units.instInv
IsUnit.unit
——
inverse_one 📖mathematical—inverse
MulOne.toOne
MulOneClass.toMulOne
MulZeroOneClass.toMulOneClass
MonoidWithZero.toMulZeroOneClass
—inverse_unit
inverse_unit 📖mathematical—inverse
Units.val
MonoidWithZero.toMonoid
Units
Units.instInv
—inverse.eq_1
Units.isUnit
IsUnit.unit_of_val_units
inverse_zero 📖mathematical—inverse
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
—Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
inverse_non_unit
not_isUnit_zero
mul_inverse_cancel 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
MulOne.toOne
MulOneClass.toMulOne
MulZeroOneClass.toMulOneClass
—inverse_unit
Units.mul_inv
mul_inverse_cancel_left 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—mul_assoc
mul_inverse_cancel
one_mul
mul_inverse_cancel_right 📖mathematicalIsUnit
MonoidWithZero.toMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
inverse
—mul_assoc
mul_inverse_cancel
mul_one

Units

Definitions

NameCategoryTheorems
mk0 📖CompOp—

Theorems

NameKindAssumesProvesValidatesDepends On
exists0 📖————ne_zero
GroupWithZero.toNontrivial
exists0' 📖mathematical—val
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
ne_zero
GroupWithZero.toNontrivial
—ne_zero
GroupWithZero.toNontrivial
exists0
exists_iff_ne_zero 📖mathematical—val
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
——
inv_mul' 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
val
MonoidWithZero.toMonoid
InvOneClass.toOne
—inv_mul_cancel₀
ne_zero
GroupWithZero.toNontrivial
mk0_inj 📖————mul_inv_cancel₀
inv_mul_cancel₀
ext
mk0_mul 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
Units
MonoidWithZero.toMonoid
instMul
MulZeroClass.toZero
mul_ne_zero_iff
GroupWithZero.noZeroDivisors
—ext
mul_ne_zero_iff
GroupWithZero.noZeroDivisors
mk0_one 📖mathematicalInvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
one_ne_zero
NeZero.one
GroupWithZero.toNontrivial
Units
MonoidWithZero.toMonoid
instOne
—ext
mk0_val 📖mathematical—val
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—ext
mul_inv' 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
val
MonoidWithZero.toMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
InvOneClass.toOne
—mul_inv_cancel₀
ne_zero
GroupWithZero.toNontrivial
mul_left_eq_zero 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
val
MonoidWithZero.toMonoid
MulZeroClass.toZero
—mul_inv_cancel_right
mul_eq_zero_of_left
mul_right_eq_zero 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
val
MonoidWithZero.toMonoid
MulZeroClass.toZero
—inv_mul_cancel_left
mul_eq_zero_of_right
ne_zero 📖————left_ne_zero_of_mul_eq_one
mul_inv
val_mk0 📖mathematical—val
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
——

(root)

Definitions

NameCategoryTheorems
commGroupWithZeroOfIsUnitOrEqZero 📖CompOp—
groupWithZeroOfIsUnitOrEqZero 📖CompOp—

Theorems

NameKindAssumesProvesValidatesDepends On
div_div_cancel_left' 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
—IsUnit.div_div_cancel_left
Ne.isUnit
div_div_cancel₀ 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
—IsUnit.div_div_cancel
Ne.isUnit
div_div_div_cancel_left' 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
—div_div_div_eq
mul_comm
mul_div_mul_right
div_div_div_cancel_right₀ 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—div_div_eq_mul_div
div_mul_cancel₀
div_eq_div_iff 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.div_eq_div_iff
Ne.isUnit
div_eq_div_iff_div_eq_div' 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
—mul_left_inj'
IsCancelMulZero.toIsRightCancelMulZero
instIsCancelMulZero
div_mul_cancel₀
mul_comm
div_mul_eq_mul_div
mul_div_assoc
div_eq_div_of_div_eq_div 📖—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
——div_ne_zero
div_zero
div_eq_div_iff_div_eq_div'
div_eq_iff 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.div_eq_iff
Ne.isUnit
div_eq_iff_mul_eq 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.div_eq_iff
Ne.isUnit
div_eq_of_eq_mul 📖mathematicalMulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—IsUnit.div_eq_of_eq_mul
Ne.isUnit
div_eq_one_iff_eq 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.div_eq_one_iff_eq
Ne.isUnit
div_eq_zero_iff 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—div_eq_mul_inv
GroupWithZero.noZeroDivisors
div_helper 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
—div_mul_eq_mul_div
one_mul
div_mul_cancel_left₀
one_div
div_left_inj' 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—Ne.isUnit
div_mul_cancel_left₀ 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
—IsUnit.div_mul_cancel_left
Ne.isUnit
div_mul_cancel_of_imp 📖mathematicalMulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
MulZeroClass.toMul
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—eq_or_ne
div_zero
MulZeroClass.mul_zero
IsUnit.div_mul_cancel
div_mul_cancel_right₀ 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.div_mul_cancel_right
Ne.isUnit
div_mul_cancel₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—IsUnit.div_mul_cancel
div_mul_div_cancel₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—mul_div_assoc
div_mul_cancel₀
div_mul_div_cancel₀' 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—mul_comm
div_mul_div_cancel₀
div_ne_zero 📖————div_eq_mul_inv
mul_ne_zero
GroupWithZero.noZeroDivisors
inv_ne_zero
div_ne_zero_iff 📖————Iff.not
div_eq_zero_iff
div_self 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.div_self
Ne.isUnit
div_self_eq_one₀ 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—Mathlib.Tactic.Contrapose.contrapose₃
div_zero
NeZero.one
GroupWithZero.toNontrivial
div_self
divp_mk0 📖mathematical—divp
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—divp_eq_div
eq_div_iff 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.eq_div_iff
Ne.isUnit
eq_div_iff_mul_eq 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.eq_div_iff
Ne.isUnit
eq_div_of_mul_eq 📖mathematicalMulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—IsUnit.eq_div_of_mul_eq
Ne.isUnit
eq_inv_mul_iff_mul_eq₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.eq_inv_mul_iff_mul_eq
Ne.isUnit
eq_mul_inv_iff_mul_eq₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.eq_mul_inv_iff_mul_eq
Ne.isUnit
eq_mul_of_inv_mul_eq₀ 📖—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
——mul_eq_of_eq_inv_mul₀
eq_mul_of_mul_inv_eq₀ 📖—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
——mul_eq_of_eq_mul_inv₀
eq_zero_of_zpow_eq_zero 📖—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
——not_imp_not
zpow_ne_zero
inv_mul_eq_iff_eq_mul₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.inv_mul_eq_iff_eq_mul
Ne.isUnit
inv_mul_eq_inv_mul_iff 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
—IsUnit.inv_mul_eq_inv_mul_iff
Ne.isUnit
inv_mul_eq_one₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
InvOneClass.toOne
—IsUnit.inv_mul_eq_one
Ne.isUnit
inv_pow_sub_of_lt 📖mathematical—Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
—pow_sub_of_lt
inv_pow
inv_inv
inv_pow_sub₀ 📖mathematical—Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
—pow_sub₀
inv_ne_zero
inv_pow
inv_inv
isUnit_iff_ne_zero 📖mathematical—IsUnit
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—Units.exists_iff_ne_zero
isUnit_ringInverse 📖mathematical—IsUnit
MonoidWithZero.toMonoid
Ring.inverse
—subsingleton_or_nontrivial
Lean.Meta.FastSubsingleton.elim
Mathlib.Tactic.Contrapose.contrapose₁
Ring.inverse_non_unit
not_isUnit_zero
IsUnit.ringInverse
isUnit_zero_iff 📖mathematical—IsUnit
MonoidWithZero.toMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
MulOne.toOne
MulOneClass.toMulOne
MulZeroOneClass.toMulOneClass
—MulZeroClass.zero_mul
isUnit_of_subsingleton
subsingleton_of_zero_eq_one
mul_div_cancel_left_of_imp 📖mathematicalMulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
—mul_comm
mul_div_cancel_of_imp
mul_div_cancel_of_imp 📖mathematicalMulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
—eq_or_ne
MulZeroClass.mul_zero
div_zero
IsUnit.mul_div_cancel_right
mul_div_cancel_of_imp' 📖mathematicalMulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
MulZeroClass.toMul
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—mul_comm
div_mul_cancel_of_imp
mul_div_cancel₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
—IsUnit.mul_div_cancel
Ne.isUnit
mul_div_mul_left 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.mul_div_mul_left
Ne.isUnit
mul_div_mul_right 📖mathematical—DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—IsUnit.mul_div_mul_right
Ne.isUnit
mul_eq_mul_of_div_eq_div 📖mathematicalDivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
CommGroupWithZero.toGroupWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—mul_one
div_self
mul_comm_div
div_mul_eq_mul_div
div_mul_cancel₀
mul_eq_of_eq_inv_mul₀ 📖—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
——eq_inv_mul_iff_mul_eq₀
inv_zero
MulZeroClass.zero_mul
mul_eq_of_eq_mul_inv₀ 📖—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
——eq_mul_inv_iff_mul_eq₀
inv_zero
MulZeroClass.mul_zero
mul_eq_one_iff_eq_inv₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
InvOneClass.toInv
—IsUnit.mul_eq_one_iff_eq_inv
Ne.isUnit
mul_eq_one_iff_inv_eq₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
InvOneClass.toInv
—IsUnit.mul_eq_one_iff_inv_eq
Ne.isUnit
mul_inv_eq_iff_eq_mul₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.mul_inv_eq_iff_eq_mul
Ne.isUnit
mul_inv_eq_mul_inv_iff 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
—IsUnit.mul_inv_eq_mul_inv_iff
Ne.isUnit
mul_inv_eq_one₀ 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
InvOneClass.toOne
—IsUnit.mul_inv_eq_one
Ne.isUnit
mul_mul_div 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.mul_mul_div
Ne.isUnit
mul_one_div_cancel 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.mul_one_div_cancel
Ne.isUnit
not_isUnit_zero 📖mathematical—IsUnit
MonoidWithZero.toMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
—isUnit_zero_iff
zero_ne_one
NeZero.one
one_div_mul_cancel 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—IsUnit.one_div_mul_cancel
Ne.isUnit
pow_sub_of_lt 📖mathematical—Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—eq_or_ne
zero_pow
MulZeroClass.zero_mul
pow_sub₀
pow_sub₀ 📖mathematical—Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—pow_add
div_eq_mul_inv
eq_div_of_mul_eq
pow_ne_zero
isReduced_of_noZeroDivisors
GroupWithZero.noZeroDivisors
zpow_eq_zero_iff 📖mathematical—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
—eq_zero_of_zpow_eq_zero
zero_zpow
zpow_natCast_sub_natCast₀ 📖mathematical—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
DivInvMonoid.toDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—zpow_natCast
zpow_sub₀
zpow_natCast_sub_one₀ 📖mathematical—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
DivInvMonoid.toDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—zpow_natCast
zpow_one
zpow_sub₀
zpow_ne_zero 📖————zpow_natCast
pow_ne_zero
isReduced_of_noZeroDivisors
GroupWithZero.noZeroDivisors
zpow_negSucc
inv_ne_zero
zpow_ne_zero_iff 📖————Iff.ne
zpow_eq_zero_iff
zpow_neg_mul_zpow_self 📖mathematical—MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
—zpow_neg
inv_mul_cancel₀
zpow_ne_zero
zpow_one_sub_natCast₀ 📖mathematical—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
DivInvMonoid.toDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
GroupWithZero.toMonoidWithZero
—zpow_one
zpow_natCast
zpow_sub₀
zpow_sub₀ 📖mathematical—DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
DivInvMonoid.toDiv
—zpow_add₀
zpow_neg
div_eq_mul_inv

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