Documentation Verification Report

NNReal

πŸ“ Source: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean

Statistics

MetricCount
DefinitionsinstPowReal, orderIsoRpow, rpow, evalNNRealRPow, proveIsNatNNRealRPowIsNNRat, evalENNRealRpow, evalNNRealRpow, instPowReal, orderIsoRpow, rpow, rpowMonoidHom
11
Theoremsadd_rpow_le_two_rpow_mul_rpow_add_rpow, coe_mul_rpow, coe_rpow_def, coe_rpow_of_ne_zero, coe_rpow_of_nonneg, div_rpow_of_nonneg, inv_rpow, le_rpow_inv_iff, le_rpow_self_of_one_le, lt_rpow_inv_iff, max_rpow, monotone_rpow_of_nonneg, mul_rpow_eq_ite, mul_rpow_of_ne_top, mul_rpow_of_ne_zero, mul_rpow_of_nonneg, ofReal_rpow_of_nonneg, ofReal_rpow_of_pos, one_le_rpow, one_le_rpow_of_pos_of_le_one_of_neg, one_lt_rpow, one_lt_rpow_of_pos_of_lt_one_of_neg, one_rpow, orderIsoRpow_apply, orderIsoRpow_symm_apply, pow_rpow_inv_natCast, prod_coe_rpow, prod_rpow_of_ne_top, prod_rpow_of_nonneg, rpow_add, rpow_add_of_add_pos, rpow_add_of_nonneg, rpow_eq_pow, rpow_eq_top_iff, rpow_eq_top_iff_of_pos, rpow_eq_top_of_nonneg, rpow_eq_zero_iff, rpow_eq_zero_iff_of_pos, rpow_intCast, rpow_intCast_mul, rpow_inv_le_iff, rpow_inv_lt_iff, rpow_inv_natCast_pow, rpow_inv_rpow, rpow_le_one, rpow_le_one_of_one_le_of_neg, rpow_le_rpow, rpow_le_rpow_iff, rpow_le_rpow_of_exponent_ge, rpow_le_rpow_of_exponent_le, rpow_le_self_of_le_one, rpow_left_bijective, rpow_left_injective, rpow_left_surjective, rpow_lt_one, rpow_lt_one_of_one_lt_of_neg, rpow_lt_rpow, rpow_lt_rpow_iff, rpow_lt_rpow_of_exponent_gt, rpow_lt_rpow_of_exponent_lt, rpow_lt_top_iff_of_pos, rpow_lt_top_of_nonneg, rpow_mul, rpow_mul_intCast, rpow_mul_natCast, rpow_natCast, rpow_natCast_mul, rpow_ne_top_of_ne_zero, rpow_ne_top_of_nonneg, rpow_ne_top_of_nonneg', rpow_neg, rpow_neg_one, rpow_ofNNReal, rpow_ofNat, rpow_one, rpow_pos, rpow_pos_of_nonneg, rpow_rpow_inv, rpow_sub, rpow_two, rpow_zero, rpow_zero_pos, strictMono_rpow_of_pos, toNNReal_rpow, toReal_rpow, top_rpow_def, top_rpow_of_neg, top_rpow_of_pos, zero_rpow_def, zero_rpow_mul_self, zero_rpow_of_neg, zero_rpow_of_pos, nnreal_rpow_eq_inv_nnreal_pow, nnreal_rpow_isNNRat, nnreal_rpow_eq_nnreal_pow, nnreal_rpow_isNNRat, nnreal_rpow_isRat_eq_inv_nnreal_rpow, coe_rpow, div_rpow, eq_rpow_inv_iff, finset_prod_rpow, inv_rpow, le_rpow_inv_iff, le_rpow_inv_iff_of_neg, le_rpow_inv_iff_of_pos, list_prod_map_rpow, list_prod_map_rpow', lt_rpow_inv_iff, lt_rpow_inv_iff_of_neg, lt_rpow_inv_iff_of_pos, monotone_rpow_of_nonneg, mul_rpow, multiset_prod_map_rpow, one_le_rpow, one_le_rpow_of_pos_of_le_one_of_nonpos, one_lt_rpow, one_lt_rpow_of_pos_of_lt_one_of_neg, one_rpow, orderIsoRpow_apply, orderIsoRpow_symm_eq, pow_rpow_inv_natCast, rpowMonoidHom_apply, rpow_add, rpow_add', rpow_add_intCast, rpow_add_intCast', rpow_add_natCast, rpow_add_natCast', rpow_add_of_nonneg, rpow_add_one, rpow_add_one', rpow_eq_pow, rpow_eq_rpow_iff, rpow_eq_zero, rpow_eq_zero_iff, rpow_intCast, rpow_intCast_mul, rpow_inv_eq_iff, rpow_inv_le_iff, rpow_inv_le_iff_of_neg, rpow_inv_le_iff_of_pos, rpow_inv_lt_iff, rpow_inv_lt_iff_of_neg, rpow_inv_lt_iff_of_pos, rpow_inv_natCast_pow, rpow_inv_rpow, rpow_inv_rpow_self, rpow_le_one, rpow_le_one_of_one_le_of_nonpos, rpow_le_rpow, rpow_le_rpow_iff, rpow_le_rpow_iff_of_neg, rpow_le_rpow_of_exponent_ge, rpow_le_rpow_of_exponent_le, rpow_le_rpow_of_nonpos, rpow_le_self_of_le_one, rpow_left_bijective, rpow_left_injective, rpow_left_surjective, rpow_lt_one, rpow_lt_one_of_one_lt_of_neg, rpow_lt_rpow, rpow_lt_rpow_iff, rpow_lt_rpow_iff_of_neg, rpow_lt_rpow_of_exponent_gt, rpow_lt_rpow_of_exponent_lt, rpow_lt_rpow_of_neg, rpow_mul, rpow_mul_intCast, rpow_mul_natCast, rpow_natCast, rpow_natCast_mul, rpow_neg, rpow_neg_one, rpow_ofNat, rpow_of_add_eq, rpow_one, rpow_one_add', rpow_one_sub', rpow_pos, rpow_rpow_inv, rpow_self_rpow_inv, rpow_sub, rpow_sub', rpow_sub_intCast, rpow_sub_intCast', rpow_sub_natCast, rpow_sub_natCast', rpow_sub_one, rpow_sub_one', rpow_two, rpow_zero, rpow_zero_pos, sqrt_eq_rpow, strictMono_rpow_of_pos, zero_rpow, enorm_rpow_of_nonneg, finset_prod_rpow, list_prod_map_rpow, list_prod_map_rpow', multiset_prod_map_rpow, nnnorm_rpow_of_nonneg, toNNReal_rpow_of_nonneg
203
Total214

ENNReal

Definitions

NameCategoryTheorems
instPowReal πŸ“–CompOp
241 mathmath: ofReal_rpow_of_pos, one_rpow, MeasureTheory.Lp.mul_meas_ge_le_pow_enorm', MeasureTheory.lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top, max_rpow, rpow_lt_top_iff_of_pos, MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv, MeasureTheory.eLpNorm'_const', AntilipschitzWith.hausdorffMeasure_preimage_le, rpow_mul, rpow_lt_rpow_iff, rpow_eq_zero_iff, rpow_intCast_mul, zero_rpow_of_pos, rpow_left_injective, rpow_add, lintegral_mul_prod_norm_pow_le, HolderWith.edist_le_of_le, rpow_add_le_add_rpow, monotone_rpow_of_nonneg, MeasureTheory.le_eLpNorm_of_bddBelow, rpow_inv_rpow, rpow_le_rpow_of_exponent_ge, rpow_le_self_of_le_one, MeasureTheory.lintegral_prod_lintegral_pow_le, one_lt_rpow, rpow_left_surjective, lintegral_rpow_funMulInvSnorm_eq_one, MeasureTheory.eLpNorm_le_of_ae_nnnorm_bound, tendsto_rpow_atTop_of_one_lt_base, MeasureTheory.mul_meas_ge_le_pow_eLpNorm', tendsto_const_mul_rpow_nhds_zero_of_pos, hasMeasurablePow, lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow, rpow_add_of_add_pos, HolderOnWith.ediam_image_le_of_subset_of_le, enorm_fderiv_norm_rpow_le, inv_rpow, MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum, spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div, mul_rpow_of_nonneg, rpow_ofNNReal, lintegral_prod_norm_pow_le, pow_rpow_inv_natCast, Metric.Snowflaking.image_toSnowflaking_emetricClosedBall, rpow_sub, lintegral_mul_norm_pow_le, top_rpow_def, rpow_pos, prod_rpow_of_ne_top, MeasureTheory.eLpNorm_indicator_sub_le_of_dist_bdd, rpow_one, rpow_natCast, lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add, Metric.Snowflaking.image_ofSnowflaking_eball, Metric.Snowflaking.preimage_toSnowflaking_emetricClosedBall, mul_rpow_of_ne_zero, Metric.Snowflaking.image_toSnowflaking_eball, rpow_arith_mean_le_arith_mean_rpow, PiLp.edist_eq_sum, MeasureTheory.MemLp.integrable_enorm_rpow', rpow_inv_le_iff, eventually_pow_one_div_le, Metric.Snowflaking.ediam_image_ofSnowflaking, rpow_zero_pos, ofReal_rpow_of_nonneg, MeasureTheory.lintegral_mul_prod_lintegral_pow_le, WithLp.prod_edist_eq_of_L2, coe_rpow_of_nonneg, rpow_le_rpow_of_exponent_le, spectrum.gelfand_formula, MeasureTheory.enorm_indicatorConstLp_le, prod_coe_rpow, lt_rpow_inv_iff, HolderWith.ediam_image_le, mul_rpow_eq_ite, rpow_two, tendsto_rpow_at_top, rpow_mul_natCast, div_rpow_of_nonneg, MeasureTheory.eLpNorm'_smul_measure, MeasureTheory.eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ, MeasureTheory.eLpNorm_smul_measure_of_ne_top, rpow_le_one_of_one_le_of_neg, rpow_natCast_mul, MeasureTheory.L2.integral_inner_eq_sq_eLpNorm, spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius, rpow_eq_zero_iff_of_pos, Metric.Snowflaking.image_ofSnowflaking_closedEBall, ProbabilityTheory.IsAEKolmogorovProcess.kolmogorovCondition, MeasureTheory.lintegral_rpow_enorm_eq_rpow_eLpNorm', rpow_eq_top_iff, ProbabilityTheory.IsKolmogorovProcess.kolmogorovCondition, rpow_rpow_inv, top_rpow_of_pos, le_rpow_inv_iff, rpow_add_le_mul_rpow_add_rpow, funMulInvSnorm_rpow, coe_rpow_of_ne_zero, rpow_lt_rpow_of_exponent_lt, MeasureTheory.ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd, Metric.Snowflaking.ediam_preimage_ofSnowflaking, EReal.ENNReal.rpow_eq_exp_mul_log, rpow_lt_one_of_one_lt_of_neg, AntilipschitzWith.le_hausdorffMeasure_image, lintegral_Lp_add_le, MeasureTheory.eLpNorm_indicator_const_le, MeasureTheory.integrable_enorm_rpow_iff, rpow_intCast, toNNReal_rpow, MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf, MeasureTheory.eLpNorm_le_of_ae_enorm_bound, Metric.Snowflaking.preimage_toSnowflaking_eball, MeasureTheory.eLpNorm_const', MeasureTheory.GridLines.T_empty, spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius, rpow_add_rpow_le, rpow_lt_rpow, zero_rpow_def, MeasureTheory.MemLp.enorm_rpow, MeasureTheory.eLpNorm_nnreal_eq_lintegral, Metric.Snowflaking.preimage_toSnowflaking_closedEBall, MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum, coe_mul_rpow, mul_rpow_of_ne_top, Lp_add_le, rpow_pos_of_nonneg, prod_rpow_of_nonneg, lintegral_mul_le_Lp_mul_Lq, zero_rpow_of_neg, inner_le_weight_mul_Lp_of_nonneg, MeasureTheory.eLpNorm_norm_rpow, tendsto_rpow_atTop_of_base_lt_one, MeasureTheory.eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ, coe_rpow_def, Metric.Snowflaking.edist_toSnowflaking_toSnowflaking, rpow_le_rpow, MeasureTheory.eLpNorm'_add_le_of_le_one, log_rpow, one_le_rpow_of_pos_of_le_one_of_neg, MeasureTheory.SimpleFunc.eLpNorm'_eq, zero_rpow_mul_self, rpow_zero, HolderOnWith.ediam_image_le_of_le, rpow_ofNat, MeasureTheory.meas_ge_le_mul_pow_eLpNorm_enorm, rpow_left_bijective, tendsto_rpow_atBot_of_one_lt_base, MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux, EReal.exp_mul, Real.enorm_rpow_of_nonneg, MeasureTheory.eLpNorm_eq_lintegral_rpow_enorm_toReal, rpow_lt_one, MeasureTheory.eLpNorm_indicator_const', Metric.Snowflaking.image_ofSnowflaking_emetricClosedBall, Metric.Snowflaking.edist_ofSnowflaking_ofSnowflaking, Metric.Snowflaking.preimage_ofSnowflaking_eball, MeasureTheory.Measure.hausdorffMeasure_apply, add_rpow_le_rpow_add, Metric.Snowflaking.image_toSnowflaking_closedEBall, rpow_eq_top_iff_of_pos, MeasureTheory.pow_mul_meas_ge_le_eLpNorm, MeasureTheory.eLpNorm_const, LipschitzOnWith.hausdorffMeasure_image_le, one_lt_rpow_of_pos_of_lt_one_of_neg, MeasureTheory.eLpNorm_le_eLpNorm_mul_rpow_measure_univ, toReal_rpow, rpow_neg_one, HolderOnWith.edist_le, MeasureTheory.eLpNorm_eq_lintegral_rpow_enorm, rpow_mul_intCast, HolderOnWith.ediam_image_inter_le_of_le, MeasureTheory.eLpNorm_indicator_constβ‚€, rpow_le_one, lintegral_Lp_add_le_of_le_one, Metric.Snowflaking.edist_def, rpow_le_rpow_iff, Metric.Snowflaking.preimage_ofSnowflaking_emetricBall, MeasureTheory.Lp.meas_ge_le_mul_pow_enorm, LipschitzWith.hausdorffMeasure_image_le, MeasureTheory.memLp_enorm_rpow_iff, HolderWith.edist_le, orderIsoRpow_apply, rpow_inv_lt_iff, MeasureTheory.lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top, HolderOnWith.ediam_image_inter_le, MeasureTheory.eLpNorm'_eq_lintegral_enorm, continuous_rpow_const, rpow_add_of_nonneg, rpow_neg, Metric.Snowflaking.ediam_preimage_toSnowflaking, Metric.Snowflaking.preimage_ofSnowflaking_emetricClosedBall, MeasureTheory.eLpNorm_sub_le_of_dist_bdd, rpow_eq_pow, MeasureTheory.GridLines.T_univ, MeasureTheory.eLpNorm_enorm_rpow, MeasureTheory.MemLp.integrable_enorm_rpow, MeasureTheory.eLpNorm_smul_measure_of_ne_zero, MeasureTheory.MemLp.enorm_rpow_div, fun_eq_funMulInvSnorm_mul_eLpNorm, strictMono_rpow_of_pos, add_rpow_le_two_rpow_mul_rpow_add_rpow, rpow_inv_natCast_pow, MeasureTheory.eLpNorm_nnreal_pow_eq_lintegral, le_rpow_self_of_one_le, HolderOnWith.ediam_image_le, HolderOnWith.edist_le_of_le, rpow_lt_top_of_nonneg, young_inequality, inner_le_Lp_mul_Lq, MeasureTheory.eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top, tendsto_rpow_atBot_of_base_lt_one, MeasureTheory.eLpNorm'_norm_rpow, HolderOnWith.hausdorffMeasure_image_le, MeasureTheory.mul_meas_ge_le_pow_eLpNorm, Metric.Snowflaking.image_ofSnowflaking_emetricBall, MeasureTheory.eLpNorm'_const, rpow_lt_rpow_of_exponent_gt, spectrum.spectralRadius_le_pow_nnnorm_pow_one_div, EuclideanSpace.edist_eq, MeasureTheory.Lp.pow_mul_meas_ge_le_enorm, top_rpow_of_neg, rpow_arith_mean_le_arith_mean2_rpow, Metric.Snowflaking.preimage_ofSnowflaking_closedEBall, Filter.Tendsto.ennrpow_const, HolderOnWith.ediam_image_le_of_subset, Metric.Snowflaking.preimage_toSnowflaking_emetricBall, WithLp.prod_edist_eq_add, lintegral_Lp_mul_le_Lq_mul_Lr, one_le_rpow, Metric.Snowflaking.image_toSnowflaking_emetricBall, Metric.Snowflaking.ediam_image_toSnowflaking, MeasureTheory.eLpNorm'_enorm_rpow, MeasureTheory.eLpNorm_le_of_ae_bound, rpow_add_rpow_le_add, NNReal.lintegral_mul_le_Lp_mul_Lq, PiLp.edist_eq_of_L2, MeasureTheory.eLpNorm_indicator_const, MeasureTheory.Lp.mul_meas_ge_le_pow_enorm, rpow_sum_le_const_mul_sum_rpow, lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top
orderIsoRpow πŸ“–CompOp
2 mathmath: orderIsoRpow_symm_apply, orderIsoRpow_apply
rpow πŸ“–CompOp
1 mathmath: rpow_eq_pow

Theorems

NameKindAssumesProvesValidatesDepends On
add_rpow_le_two_rpow_mul_rpow_add_rpow πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPowReal
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
Distrib.toMul
instOfNatAtLeastTwo
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Nat.instAtLeastTwoHAddOfNat
β€”Nat.instAtLeastTwoHAddOfNat
two_mul
rpow_le_rpow
add_le_add
IsOrderedAddMonoid.toAddLeftMono
instIsOrderedAddMonoid
covariant_swap_add_of_covariant_add
mul_rpow_of_nonneg
max_rpow
mul_le_mul_right
IsOrderedMonoid.toMulLeftMono
instIsOrderedMonoid
max_le_add_of_nonneg
instCanonicallyOrderedAdd
coe_mul_rpow πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
ofNNReal
β€”mul_rpow_of_ne_top
coe_ne_top
coe_rpow_def πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
ofNNReal
NNReal
instZeroNNReal
Real.instLT
Real.instZero
LinearOrder.toDecidableEq
NNReal.instLinearOrder
Real.decidableLT
Top.top
instTopENNReal
NNReal.instPowReal
β€”β€”
coe_rpow_of_ne_zero πŸ“–mathematicalβ€”ofNNReal
NNReal
Real
NNReal.instPowReal
ENNReal
instPowReal
β€”some_eq_coe
coe_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ofNNReal
NNReal
NNReal.instPowReal
ENNReal
instPowReal
β€”le_iff_eq_or_lt
NNReal.rpow_zero
rpow_zero
NNReal.zero_rpow
ne_of_gt
zero_rpow_of_pos
coe_rpow_of_ne_zero
div_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
DivInvMonoid.toDiv
instDivInvMonoid
β€”div_eq_mul_inv
mul_rpow_of_nonneg
inv_rpow
inv_rpow πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
instInv
β€”eq_or_ne
rpow_zero
inv_one
Ne.lt_or_gt
inv_zero
top_rpow_of_neg
zero_rpow_of_neg
inv_top
top_rpow_of_pos
zero_rpow_of_pos
eq_inv_of_mul_eq_one_left
mul_rpow_of_ne_zero
inv_ne_zero
inv_mul_cancel
one_rpow
le_rpow_inv_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPowReal
Real.instInv
β€”rpow_one
mul_inv_cancelβ‚€
LT.lt.ne'
rpow_mul
rpow_le_rpow_iff
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
le_rpow_self_of_one_le πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLE
Real.instOne
instPowRealβ€”rpow_one
rpow_le_rpow_of_exponent_le
lt_rpow_inv_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instPowReal
Real.instInv
β€”rpow_one
mul_inv_cancelβ‚€
ne_of_lt
rpow_mul
rpow_lt_rpow_iff
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
max_rpow πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
instMax
β€”le_total
max_eq_right
rpow_le_rpow
max_eq_left
monotone_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
Monotone
ENNReal
PartialOrder.toPreorder
instPartialOrder
instPowReal
β€”LE.le.eq_or_lt
rpow_zero
StrictMono.monotone
strictMono_rpow_of_pos
mul_rpow_eq_ite πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
instZeroENNReal
Top.top
instTopENNReal
Real.instLT
Real.instZero
LinearOrder.toDecidableEq
instLinearOrder
Real.decidableLT
β€”eq_or_ne
rpow_zero
mul_one
Ne.lt_or_gt
MulZeroClass.zero_mul
zero_rpow_of_neg
zero_rpow_of_pos
LT.lt.not_gt
top_rpow_of_pos
top_mul
bot_unique
top_unique
mul_top
top_rpow_of_neg
MulZeroClass.mul_zero
coe_eq_zero
Nat.cast_zero
coe_rpow_of_ne_zero
mul_ne_zero
NNReal.instNoZeroDivisors
NNReal.mul_rpow
mul_comm
le_of_not_ge
mul_rpow_of_ne_top πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”mul_rpow_eq_ite
mul_rpow_of_ne_zero πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”mul_rpow_eq_ite
mul_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”mul_rpow_eq_ite
LE.le.not_gt
ofReal_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
ofReal
Real.instPow
β€”rpow_zero
Real.rpow_zero
ofReal_one
lt_of_le_of_ne
ofReal_zero
zero_rpow_of_pos
Real.zero_rpow
LT.lt.ne
ofReal_rpow_of_pos
LE.le.lt_of_ne
ofReal_rpow_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
ofReal
Real.instPow
β€”coe_rpow_of_ne_zero
Real.toNNReal_rpow_of_nonneg
LT.lt.le
one_le_rpow πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”top_rpow_of_pos
coe_rpow_of_nonneg
le_of_lt
NNReal.one_le_rpow
one_le_rpow_of_pos_of_le_one_of_neg πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
Preorder.toLE
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_le_of_lt
coe_lt_top
coe_rpow_of_ne_zero
ne_of_gt
NNReal.one_le_rpow_of_pos_of_le_one_of_nonpos
le_of_lt
one_lt_rpow πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”top_rpow_of_pos
coe_rpow_of_nonneg
le_of_lt
NNReal.one_lt_rpow
one_lt_rpow_of_pos_of_lt_one_of_neg πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_lt_of_le
le_top
coe_rpow_of_ne_zero
ne_of_gt
NNReal.one_lt_rpow_of_pos_of_lt_one_of_neg
one_rpow πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
β€”coe_one
coe_rpow_of_ne_zero
one_ne_zero
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
NNReal.instIsStrictOrderedRing
NNReal.one_rpow
orderIsoRpow_apply πŸ“–mathematicalReal
Real.instLT
Real.instZero
DFunLike.coe
RelIso
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
RelIso.instFunLike
orderIsoRpow
instPowReal
β€”β€”
orderIsoRpow_symm_apply πŸ“–mathematicalReal
Real.instLT
Real.instZero
OrderIso.symm
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
orderIsoRpow
DivInvMonoid.toDiv
Real.instDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
DivisionSemiring.toGroupWithZero
Semifield.toDivisionSemiring
Field.toSemifield
Real.instField
Preorder.toLT
Real.partialOrder
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
GroupWithZero.toDivInvMonoid
one_div_pos
PosMulReflectLE.toPosMulReflectLT
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Real.normedCommRing
PosMulStrictMono.toPosMulReflectLE
MulZeroClass.toMul
Real.linearOrder
IsStrictOrderedRing.toPosMulStrictMono
Real.semiring
Real.instIsStrictOrderedRing
β€”one_div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
strictMono_rpow_of_pos
one_div_one_div
StrictMono.orderIsoOfRightInverse.congr_simp
pow_rpow_inv_natCast πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
Real.instInv
Real.instNatCast
β€”rpow_natCast
rpow_mul
mul_inv_cancelβ‚€
ne_of_gt
Nat.cast_pos'
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
Real.instZeroLEOneClass
NeZero.charZero_one
RCLike.charZero_rclike
lt_of_le_of_ne'
zero_le
Nat.instCanonicallyOrderedAdd
rpow_one
prod_coe_rpow πŸ“–mathematicalβ€”Finset.prod
ENNReal
CommSemiring.toCommMonoid
instCommSemiring
Real
instPowReal
ofNNReal
NNReal
instCommSemiringNNReal
β€”Finset.induction
one_rpow
Finset.prod_insert
prod_rpow_of_ne_top πŸ“–mathematicalβ€”Finset.prod
ENNReal
CommSemiring.toCommMonoid
instCommSemiring
Real
instPowReal
β€”Finset.induction
one_rpow
Finset.mem_insert_of_mem
Finset.prod_insert
mul_rpow_of_ne_top
Finset.mem_insert_self
prod_ne_top
prod_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
Finset.prod
ENNReal
CommSemiring.toCommMonoid
instCommSemiring
instPowReal
β€”Finset.induction
one_rpow
Finset.prod_insert
mul_rpow_of_nonneg
rpow_add πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instAdd
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”coe_rpow_of_ne_zero
NNReal.rpow_add
rpow_add_of_add_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
Real.instAdd
ENNReal
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”eq_or_ne
zero_rpow_of_pos
MulZeroClass.zero_mul
lt_of_not_ge
Mathlib.Tactic.Linarith.lt_irrefl
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.sub_congr
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Ring.atom_pf
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.sub_pf
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.neg_zero
Mathlib.Tactic.Ring.add_pf_add_zero
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.Linarith.add_lt_of_neg_of_le
Real.instIsStrictOrderedRing
Mathlib.Tactic.Linarith.sub_neg_of_lt
Real.instIsOrderedRing
Mathlib.Tactic.Linarith.sub_nonpos_of_le
le_of_not_gt
MulZeroClass.mul_zero
rpow_add
rpow_add_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
Real.instAdd
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
β€”LE.le.eq_or_lt
rpow_zero
one_mul
zero_add
mul_one
add_zero
top_rpow_of_pos
add_pos
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
mul_top
add_nonneg
NNReal.rpow_add_of_nonneg
rpow_eq_pow πŸ“–mathematicalβ€”rpow
ENNReal
Real
instPowReal
β€”β€”
rpow_eq_top_iff πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Top.top
instTopENNReal
instZeroENNReal
Real.instLT
Real.instZero
β€”lt_trichotomy
top_rpow_of_neg
rpow_zero
top_rpow_of_pos
zero_rpow_of_neg
zero_rpow_of_pos
coe_rpow_of_ne_zero
rpow_eq_top_iff_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
Top.top
instTopENNReal
β€”asymm
instAsymmLt
rpow_eq_top_of_nonneg πŸ“–β€”Real
Real.instLE
Real.instZero
ENNReal
instPowReal
Top.top
instTopENNReal
β€”β€”LE.le.not_gt
rpow_eq_zero_iff πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
instZeroENNReal
Real.instLT
Real.instZero
Top.top
instTopENNReal
β€”lt_trichotomy
top_rpow_of_neg
rpow_zero
NeZero.charZero_one
instCharZero
top_rpow_of_pos
zero_rpow_of_neg
zero_rpow_of_pos
coe_rpow_of_ne_zero
rpow_eq_zero_iff_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
instZeroENNReal
β€”LT.lt.not_gt
rpow_intCast πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instIntCast
DivInvMonoid.toZPow
instDivInvMonoid
β€”Int.cast_natCast
rpow_natCast
zpow_natCast
Int.cast_negSucc
rpow_neg
zpow_negSucc
rpow_intCast_mul πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instMul
Real.instIntCast
DivInvMonoid.toZPow
instDivInvMonoid
β€”rpow_mul
rpow_intCast
rpow_inv_le_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPowReal
Real.instInv
β€”rpow_one
mul_inv_cancelβ‚€
LT.lt.ne
rpow_mul
rpow_le_rpow_iff
inv_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
rpow_inv_lt_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instPowReal
Real.instInv
β€”le_rpow_inv_iff
rpow_inv_natCast_pow πŸ“–mathematicalβ€”ENNReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
Real
instPowReal
Real.instInv
Real.instNatCast
β€”rpow_natCast
rpow_mul
inv_mul_cancelβ‚€
ne_of_gt
Nat.cast_pos'
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
Real.instZeroLEOneClass
NeZero.charZero_one
RCLike.charZero_rclike
lt_of_le_of_ne'
zero_le
Nat.instCanonicallyOrderedAdd
rpow_one
rpow_inv_rpow πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instInv
β€”rpow_mul
inv_mul_cancelβ‚€
rpow_one
rpow_le_one πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_le_of_lt
coe_lt_top
coe_rpow_of_nonneg
NNReal.rpow_le_one
rpow_le_one_of_one_le_of_neg πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”top_rpow_of_neg
instCanonicallyOrderedAdd
coe_rpow_of_ne_zero
ne_of_gt
lt_of_lt_of_le
zero_lt_one
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
NNReal.instIsStrictOrderedRing
NNReal.rpow_le_one_of_one_le_of_nonpos
le_of_lt
rpow_le_rpow πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
Real
Real.instLE
Real.instZero
instPowRealβ€”monotone_rpow_of_nonneg
rpow_le_rpow_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPowReal
β€”StrictMono.le_iff_le
strictMono_rpow_of_pos
rpow_le_rpow_of_exponent_ge πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLE
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_le_of_lt
coe_lt_top
lt_trichotomy
zero_rpow_of_neg
rpow_zero
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.sub_pf
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.neg_zero
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Ring.add_pf_add_zero
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.neg_congr
Mathlib.Tactic.Linarith.lt_of_lt_of_eq
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
neg_eq_zero
sub_eq_zero_of_eq
zero_rpow_of_pos
instCanonicallyOrderedAdd
Mathlib.Tactic.Linarith.add_neg
NeZero.charZero_one
instCharZero
Mathlib.Tactic.Linarith.le_of_le_of_eq
coe_rpow_of_ne_zero
NNReal.rpow_le_rpow_of_exponent_ge
bot_lt_iff_ne_bot
coe_le_one_iff
rpow_le_rpow_of_exponent_le πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLE
instPowRealβ€”lt_trichotomy
top_rpow_of_neg
rpow_zero
instCanonicallyOrderedAdd
top_rpow_of_pos
NeZero.charZero_one
instCharZero
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.sub_pf
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.neg_zero
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.neg_congr
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Ring.add_pf_add_zero
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.Linarith.add_lt_of_le_of_neg
Real.instIsStrictOrderedRing
Mathlib.Tactic.Linarith.le_of_le_of_eq
Mathlib.Tactic.Linarith.sub_nonpos_of_le
Real.instIsOrderedRing
neg_eq_zero
sub_eq_zero_of_eq
Mathlib.Tactic.Linarith.sub_neg_of_lt
Mathlib.Tactic.Linarith.add_neg
Mathlib.Tactic.Linarith.lt_of_lt_of_eq
coe_rpow_of_ne_zero
ne_of_gt
lt_of_lt_of_le
zero_lt_one
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
IsStrictOrderedRing.toCharZero
NNReal.instIsStrictOrderedRing
NNReal.rpow_le_rpow_of_exponent_le
rpow_le_self_of_le_one πŸ“–mathematicalENNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLE
Real.instOne
instPowRealβ€”rpow_one
rpow_le_rpow_of_exponent_ge
rpow_left_bijective πŸ“–mathematicalβ€”Function.Bijective
ENNReal
Real
instPowReal
β€”rpow_left_injective
rpow_left_surjective
rpow_left_injective πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
β€”rpow_rpow_inv
rpow_left_surjective πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
β€”rpow_inv_rpow
rpow_lt_one πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_lt_of_le
le_top
coe_rpow_of_nonneg
le_of_lt
NNReal.rpow_lt_one
rpow_lt_one_of_one_lt_of_neg πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”top_rpow_of_neg
IsOrderedRing.toZeroLEOneClass
instIsOrderedRing
NeZero.charZero_one
instCharZero
coe_rpow_of_ne_zero
ne_of_gt
lt_trans
zero_lt_one
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
IsStrictOrderedRing.toCharZero
NNReal.instIsStrictOrderedRing
NNReal.rpow_lt_one_of_one_lt_of_neg
rpow_lt_rpow πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
Real
Real.instLT
Real.instZero
instPowRealβ€”strictMono_rpow_of_pos
rpow_lt_rpow_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instPowReal
β€”StrictMono.lt_iff_lt
strictMono_rpow_of_pos
rpow_lt_rpow_of_exponent_gt πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
instPowRealβ€”CanLift.prf
canLift
ne_of_lt
lt_of_lt_of_le
le_top
coe_rpow_of_ne_zero
ne_of_gt
NNReal.rpow_lt_rpow_of_exponent_gt
rpow_lt_rpow_of_exponent_lt πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Real
Real.instLT
instPowRealβ€”CanLift.prf
canLift
coe_rpow_of_ne_zero
ne_of_gt
lt_trans
zero_lt_one
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
NNReal.instIsStrictOrderedRing
one_lt_coe_iff
NNReal.rpow_lt_rpow_of_exponent_lt
rpow_lt_top_iff_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instPowReal
Top.top
instTopENNReal
β€”rpow_eq_top_iff_of_pos
rpow_lt_top_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instPowReal
Top.top
instTopENNReal
β€”lt_top_iff_ne_top
rpow_ne_top_of_nonneg
rpow_mul πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instMul
β€”lt_trichotomy
top_rpow_of_pos
AddGroup.existsAddOfLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
contravariant_swap_add_of_contravariant_add
contravariant_lt_of_covariant_le
top_rpow_of_neg
zero_rpow_of_neg
MulZeroClass.mul_zero
rpow_zero
zero_rpow_of_pos
MulZeroClass.zero_mul
one_rpow
IsStrictOrderedRing.toPosMulStrictMono
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
NNReal.rpow_mul
rpow_mul_intCast πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instMul
Real.instIntCast
DivInvMonoid.toZPow
instDivInvMonoid
β€”rpow_mul
rpow_intCast
rpow_mul_natCast πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instMul
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
β€”rpow_mul
rpow_natCast
rpow_natCast πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
β€”CharP.cast_eq_zero
CharP.ofCharZero
RCLike.charZero_rclike
rpow_zero
pow_zero
Nat.cast_add
Nat.cast_one
top_rpow_of_pos
Nat.cast_add_one_pos
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
Real.instZeroLEOneClass
NeZero.charZero_one
top_pow
coe_rpow_of_nonneg
Nat.cast_nonneg
Real.instIsOrderedRing
NNReal.rpow_natCast
rpow_natCast_mul πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instMul
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
β€”rpow_mul
rpow_natCast
rpow_ne_top_of_ne_zero πŸ“–β€”β€”β€”β€”β€”
rpow_ne_top_of_nonneg πŸ“–β€”Real
Real.instLE
Real.instZero
β€”β€”rpow_eq_top_of_nonneg
rpow_ne_top_of_nonneg' πŸ“–β€”ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
β€”β€”LT.lt.ne'
rpow_neg πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instNeg
instInv
β€”lt_trichotomy
top_rpow_of_pos
neg_pos
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
top_rpow_of_neg
inv_zero
neg_zero
rpow_zero
inv_one
inv_top
zero_rpow_of_pos
zero_rpow_of_neg
coe_rpow_of_ne_zero
NNReal.rpow_neg
coe_inv
rpow_neg_one πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instNeg
Real.instOne
instInv
β€”rpow_neg
rpow_one
rpow_ofNNReal πŸ“–mathematicalReal
Real.instLE
Real.instZero
ENNReal
instPowReal
ofNNReal
NNReal
NNReal.instPowReal
β€”coe_rpow_of_nonneg
rpow_eq_pow
rpow_ofNat πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
β€”rpow_natCast
rpow_one πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instOne
β€”zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
RCLike.charZero_rclike
NNReal.rpow_one
LE.le.not_gt
zero_le_one
rpow_pos πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
Real
instPowReal
β€”lt_or_ge
rpow_pos_of_nonneg
le_of_lt
neg_neg
rpow_neg
inv_pos
rpow_ne_top_of_nonneg
Right.nonneg_neg_iff
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
rpow_pos_of_nonneg πŸ“–mathematicalENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”rpow_zero
IsOrderedRing.toZeroLEOneClass
instIsOrderedRing
NeZero.charZero_one
instCharZero
lt_of_le_of_ne
zero_rpow_of_pos
rpow_lt_rpow
rpow_rpow_inv πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instInv
β€”rpow_mul
mul_inv_cancelβ‚€
rpow_one
rpow_sub πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instSub
DivInvMonoid.toDiv
instDivInvMonoid
β€”sub_eq_add_neg
rpow_add
rpow_neg
div_eq_mul_inv
rpow_two πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
instCommSemiring
β€”rpow_ofNat
Nat.instAtLeastTwoHAddOfNat
rpow_zero πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Real.instZero
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
β€”NNReal.rpow_zero
rpow_zero_pos πŸ“–mathematicalβ€”ENNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrder
instZeroENNReal
Real
instPowReal
Real.instZero
β€”rpow_zero
one_pos
IsOrderedRing.toZeroLEOneClass
instIsOrderedRing
NeZero.charZero_one
instCharZero
strictMono_rpow_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
StrictMono
ENNReal
PartialOrder.toPreorder
instPartialOrder
instPowReal
β€”CanLift.prf
canLift
ne_top_of_lt
eq_or_ne
coe_rpow_of_nonneg
LT.lt.le
top_rpow_of_pos
NNReal.rpow_lt_rpow
coe_lt_coe
toNNReal_rpow πŸ“–mathematicalβ€”toNNReal
ENNReal
Real
instPowReal
NNReal
NNReal.instPowReal
β€”lt_trichotomy
top_rpow_of_neg
NNReal.zero_rpow
zero_rpow_of_neg
coe_rpow_of_ne_zero
rpow_zero
NNReal.rpow_zero
top_rpow_of_pos
coe_rpow_of_nonneg
le_of_lt
toReal_rpow πŸ“–mathematicalβ€”Real
Real.instPow
toReal
ENNReal
instPowReal
β€”toReal.eq_1
NNReal.coe_rpow
toNNReal_rpow
top_rpow_def πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
Top.top
instTopENNReal
Real.instLT
Real.instZero
Real.decidableLT
Real.decidableEq
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
instZeroENNReal
β€”β€”
top_rpow_of_neg πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
Top.top
instTopENNReal
instZeroENNReal
β€”asymm
instAsymmLt
ne_of_lt
top_rpow_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
Top.top
instTopENNReal
β€”β€”
zero_rpow_def πŸ“–mathematicalβ€”ENNReal
Real
instPowReal
instZeroENNReal
Real.instLT
Real.instZero
Real.decidableLT
Real.decidableEq
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
instAddCommMonoidWithOneENNReal
Top.top
instTopENNReal
β€”lt_trichotomy
zero_rpow_of_pos
rpow_zero
zero_rpow_of_neg
asymm
instAsymmLt
zero_rpow_mul_self πŸ“–mathematicalβ€”ENNReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instCommSemiring
Real
instPowReal
instZeroENNReal
β€”zero_rpow_def
MulZeroClass.zero_mul
one_mul
top_mul_top
zero_rpow_of_neg πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
instZeroENNReal
Top.top
instTopENNReal
β€”coe_zero
some_eq_coe
zero_rpow_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
ENNReal
instPowReal
instZeroENNReal
β€”coe_zero
some_eq_coe
asymm
instAsymmLt
NNReal.zero_rpow
ne_of_gt

Mathlib.Meta.NormNum

Definitions

NameCategoryTheorems
evalNNRealRPow πŸ“–CompOpβ€”
proveIsNatNNRealRPowIsNNRat πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
nnreal_rpow_isRat_eq_inv_nnreal_rpow πŸ“–mathematicalIsRat
Real
Real.instRing
NNReal
NNReal.instPowReal
NNReal.instInv
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instNatCast
β€”NNReal.inv_rpow
NNReal.rpow_neg
IsRat.neg_to_eq

Mathlib.Meta.NormNum.IsInt

Theorems

NameKindAssumesProvesValidatesDepends On
nnreal_rpow_eq_inv_nnreal_pow πŸ“–mathematicalMathlib.Meta.NormNum.IsInt
Real
Real.instRing
NNReal
NNReal.instPowReal
NNReal.instInv
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”out
NNReal.rpow_intCast
zpow_neg
zpow_natCast

Mathlib.Meta.NormNum.IsNNRat

Theorems

NameKindAssumesProvesValidatesDepends On
nnreal_rpow_isNNRat πŸ“–β€”Mathlib.Meta.NormNum.IsNNRat
NNReal
instSemiringNNReal
Mathlib.Meta.NormNum.IsNat
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Real
NNReal.instPowReal
AddMonoidWithOne.toNatCast
β€”β€”of_raw
Mathlib.Meta.NormNum.IsNat.out
den_nz
to_eq
NNReal.div_rpow

Mathlib.Meta.NormNum.IsNat

Theorems

NameKindAssumesProvesValidatesDepends On
nnreal_rpow_eq_nnreal_pow πŸ“–mathematicalMathlib.Meta.NormNum.IsNat
Real
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Real.instRing
NNReal
NNReal.instPowReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”out
NNReal.rpow_natCast
nnreal_rpow_isNNRat πŸ“–mathematicalMathlib.Meta.NormNum.IsNat
NNReal
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
instSemiringNNReal
Mathlib.Meta.NormNum.IsNNRat
Real
Real.semiring
Monoid.toNatPow
Nat.instMonoid
NNReal.instPowRealβ€”Nat.cast_zero
RCLike.charZero_rclike
Mathlib.Meta.NormNum.IsNNRat.den_nz
Mathlib.Meta.NormNum.IsNNRat.to_eq
div_eq_mul_inv
NNReal.rpow_natCast_mul
Nat.cast_pow
NNReal.pow_rpow_inv_natCast
ne_of_gt
lt_of_le_of_ne'
zero_le
Nat.instCanonicallyOrderedAdd

Mathlib.Meta.Positivity

Definitions

NameCategoryTheorems
evalENNRealRpow πŸ“–CompOpβ€”
evalNNRealRpow πŸ“–CompOpβ€”

NNReal

Definitions

NameCategoryTheorems
instPowReal πŸ“–CompOp
187 mathmath: PiLp.lipschitzWith_toLp, PiLp.nndist_eq_sum, rpow_sub_intCast, rpow_rpow_inv, rpowMonoidHom_apply, HolderOnWith.interpolate, zero_rpow, rpow_lt_one, HolderWith.of_le, HolderOnWith.comp, MeasureTheory.Lp.nnnorm_le_of_ae_bound, rpow_add_rpow_le_add, rpow_intCast_mul, pow_rpow_inv_natCast, WithLp.prod_nnnorm_eq_add, MeasureTheory.eLpNorm_smul_measure_of_ne_top', orderIsoRpow_apply, rpow_natCast, rpow_lt_rpow_iff_of_neg, sqrt_eq_rpow, rpow_mul_intCast, young_inequality, CFC.rpow_def, ENNReal.rpow_ofNNReal, rpow_inv_lt_iff_of_pos, CFC.rpow_algebraMap, eventually_pow_one_div_le, rpow_zero_pos, Mathlib.Meta.NormNum.IsNat.nnreal_rpow_isNNRat, rpow_lt_rpow_of_neg, HolderOnWith.nndist_le_of_le, coe_rpow, rpow_add_intCast', one_le_rpow, rpow_le_one_of_one_le_of_nonpos, HolderWith.holderWith_zero_of_bounded, rpow_lt_rpow, rpow_eq_pow, PiLp.nnnorm_toLp_one, arith_mean_le_rpow_mean, isGreatest_Lp, rpow_inv_rpow, MeasureTheory.lpNorm_smul_measure_of_ne_zero, rpow_sub_natCast, rpow_sub, rpow_sub', rpow_neg_one, rpow_le_rpow, rpow_sum_le_const_mul_sum_rpow, BoundedContinuousFunction.Lp_nnnorm_le, eq_rpow_inv_iff, rpow_le_rpow_of_exponent_ge, rpow_mul, PiLp.nnnorm_toLp_const', rpow_add_natCast', ENNReal.coe_rpow_of_nonneg, MeasureTheory.Measure.hausdorffMeasure_smulβ‚€, rpow_lt_rpow_of_exponent_gt, rpow_lt_rpow_of_exponent_lt, FormalMultilinearSeries.radius_inv_eq_limsup, inv_rpow, continuousAt_rpow, rpow_ofNat, rpow_arith_mean_le_arith_mean_rpow, one_lt_rpow_of_pos_of_lt_one_of_neg, nnrpow_def, add_rpow_le_rpow_add, rpow_two, strictMono_rpow_of_pos, lt_rpow_inv_iff, rpow_natCast_mul, geom_mean_le_arith_mean4_weighted, ENNReal.coe_rpow_of_ne_zero, tendsto_rpow_atTop, rpow_inv_eq_iff, rpow_le_one, rpow_add_natCast, geom_mean_le_arith_mean2_weighted, WithLp.prod_nndist_eq_add, rpow_inv_le_iff_of_pos, MeasureTheory.eLpNormLESNormFDerivOneConst_def, rpow_le_self_of_le_one, rpow_lt_rpow_iff, list_prod_map_rpow', rpow_inv_le_iff, Mathlib.Meta.NormNum.IsNat.nnreal_rpow_eq_nnreal_pow, WithLp.prod_lipschitzWith_toLp, rpow_eq_zero, one_le_rpow_of_pos_of_le_one_of_nonpos, rpow_add', ENNReal.toNNReal_rpow, HolderOnWith.holderOnWith_zero_of_bounded, HolderWith.comp_holderOnWith, rpow_add, one_lt_rpow, le_rpow_inv_iff, MeasureTheory.hausdorffMeasure_homothety_image, rpow_inv_rpow_self, rpow_add_rpow_le, PiLp.antilipschitzWith_ofLp, rpow_sub_one, rpow_inv_le_iff_of_neg, rpow_one_sub', rpow_intCast, continuousOn_rpow_const, rpow_left_injective, MeasureTheory.nnnorm_indicatorConstLp_le, rpow_add_le_mul_rpow_add_rpow, rpow_le_rpow_of_nonpos, rpow_add_intCast, le_rpow_inv_iff_of_neg, rpow_eq_rpow_iff, continuousAt_rpow_const, lt_rpow_inv_iff_of_neg, rpow_add_one, rpow_lt_one_of_one_lt_of_neg, inner_le_weight_mul_Lp, rpow_add_one', lt_rpow_inv_iff_of_pos, rpow_le_rpow_iff_of_neg, geom_mean_le_arith_mean_weighted, CFC.nnnorm_rpow, nnnorm_fderiv_norm_rpow_le, ENNReal.coe_rpow_def, summable_one_div_rpow, rpow_neg, rpow_add_le_add_rpow, rpow_sub_one', rpow_sub_natCast', HolderOnWith.comp_holderWith, continuous_rpow_const, finset_prod_rpow, rpow_inv_lt_iff, summable_rpow_inv, rpow_eq_zero_iff, rpow_arith_mean_le_arith_mean2_rpow, monotone_rpow_of_nonneg, hasMeasurablePow, Lp_add_le, rpow_one_add', continuousOn_rpow_const_compl_zero, HolderOnWith.of_le, geom_mean_le_arith_mean3_weighted, rpow_zero, PiLp.nnnorm_eq_sum, one_rpow, rpow_le_rpow_of_exponent_le, Matrix.frobenius_nnnorm_def, Real.nnnorm_rpow_of_nonneg, HolderOnWith.nndist_le, rpow_mul_natCast, MeasureTheory.lpNorm_smul_measure_of_ne_top, div_rpow, rpow_inv_natCast_pow, rpow_add_of_nonneg, rpow_self_rpow_inv, rpow_one, Real.toNNReal_rpow_of_nonneg, rpow_left_surjective, rpow_of_add_eq, MeasureTheory.hausdorffMeasure_homothety_preimage, Filter.Tendsto.nnrpow, list_prod_map_rpow, PiLp.nnnorm_toLp_const, mul_rpow, MeasureTheory.eLpNorm_smul_measure_of_ne_zero', multiset_prod_map_rpow, rpow_le_rpow_iff, strictConcaveOn_rpow, le_rpow_inv_iff_of_pos, CFC.nnnorm_nnrpow, FormalMultilinearSeries.radius_eq_liminf, HolderWith.interpolate, HolderWith.nndist_le, young_inequality_real, Mathlib.Meta.NormNum.nnreal_rpow_isRat_eq_inv_nnreal_rpow, HolderWith.comp, inner_le_Lp_mul_Lq, rpow_inv_lt_iff_of_neg, rpow_sub_intCast', Mathlib.Meta.NormNum.IsInt.nnreal_rpow_eq_inv_nnreal_pow, rpow_pos, summable_rpow, concaveOn_rpow, WithLp.prod_antilipschitzWith_ofLp, HolderWith.nndist_le_of_le, rpow_left_bijective
orderIsoRpow πŸ“–CompOp
2 mathmath: orderIsoRpow_apply, orderIsoRpow_symm_eq
rpow πŸ“–CompOp
1 mathmath: rpow_eq_pow
rpowMonoidHom πŸ“–CompOp
1 mathmath: rpowMonoidHom_apply

Theorems

NameKindAssumesProvesValidatesDepends On
coe_rpow πŸ“–mathematicalβ€”toReal
NNReal
Real
instPowReal
Real.instPow
β€”β€”
div_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instDiv
β€”eq
Real.div_rpow
eq_rpow_inv_iff πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instInv
β€”rpow_eq_rpow_iff
one_div
rpow_self_rpow_inv
finset_prod_rpow πŸ“–mathematicalβ€”Finset.prod
NNReal
CommSemiring.toCommMonoid
instCommSemiringNNReal
Real
instPowReal
β€”multiset_prod_map_rpow
inv_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instInv
β€”eq
Real.inv_rpow
le_rpow_inv_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
Real.instInv
β€”rpow_le_rpow_iff
one_div
rpow_self_rpow_inv
LT.lt.ne'
le_rpow_inv_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
Preorder.toLE
instPowReal
Real.instInv
β€”Real.le_rpow_inv_iff_of_neg
le_rpow_inv_iff_of_pos πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowReal
Real.instInv
β€”Real.le_rpow_inv_iff_of_pos
list_prod_map_rpow πŸ“–mathematicalβ€”NNReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
instOneNNReal
Real
instPowReal
β€”List.prod_hom
MonoidHom.instMonoidHomClass
list_prod_map_rpow' πŸ“–mathematicalβ€”NNReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
instOneNNReal
Real
instPowReal
β€”list_prod_map_rpow
lt_rpow_inv_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
Real.instInv
β€”rpow_inv_le_iff
lt_rpow_inv_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowReal
Real.instInv
β€”Real.lt_rpow_inv_iff_of_neg
lt_rpow_inv_iff_of_pos πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
Preorder.toLT
instPowReal
Real.instInv
β€”Real.lt_rpow_inv_iff_of_pos
monotone_rpow_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
Monotone
NNReal
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
β€”LE.le.eq_or_lt
rpow_zero
StrictMono.monotone
strictMono_rpow_of_pos
mul_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”eq
Real.mul_rpow
multiset_prod_map_rpow πŸ“–mathematicalβ€”Multiset.prod
NNReal
CommSemiring.toCommMonoid
instCommSemiringNNReal
Multiset.map
Real
instPowReal
β€”Multiset.prod_hom'
MonoidHom.instMonoidHomClass
one_le_rpow πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.one_le_rpow
one_le_rpow_of_pos_of_le_one_of_nonpos πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Preorder.toLE
instOneNNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.one_le_rpow_of_pos_of_le_one_of_nonpos
one_lt_rpow πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.one_lt_rpow
one_lt_rpow_of_pos_of_lt_one_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
instOneNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.one_lt_rpow_of_pos_of_lt_one_of_neg
one_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instOneNNReal
β€”eq
Real.one_rpow
orderIsoRpow_apply πŸ“–mathematicalReal
Real.instLT
Real.instZero
DFunLike.coe
RelIso
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
RelIso.instFunLike
orderIsoRpow
instPowReal
β€”β€”
orderIsoRpow_symm_eq πŸ“–mathematicalReal
Real.instLT
Real.instZero
OrderIso.symm
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
orderIsoRpow
DivInvMonoid.toDiv
Real.instDivInvMonoid
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
DivisionSemiring.toGroupWithZero
Semifield.toDivisionSemiring
Field.toSemifield
Real.instField
Preorder.toLT
Real.partialOrder
MulZeroClass.toZero
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
GroupWithZero.toDivInvMonoid
one_div_pos
PosMulReflectLE.toPosMulReflectLT
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
NonUnitalNormedCommRing.toNonUnitalCommRing
NormedCommRing.toNonUnitalNormedCommRing
Real.normedCommRing
PosMulStrictMono.toPosMulReflectLE
MulZeroClass.toMul
Real.linearOrder
IsStrictOrderedRing.toPosMulStrictMono
Real.semiring
Real.instIsStrictOrderedRing
β€”one_div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
strictMono_rpow_of_pos
one_div_one_div
StrictMono.orderIsoOfRightInverse.congr_simp
pow_rpow_inv_natCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
Real.instInv
Real.instNatCast
β€”coe_rpow
coe_pow
Real.pow_rpow_inv_natCast
rpowMonoidHom_apply πŸ“–mathematicalβ€”DFunLike.coe
MonoidHom
NNReal
MulOneClass.toMulOne
MulZeroOneClass.toMulOneClass
NonAssocSemiring.toMulZeroOneClass
Semiring.toNonAssocSemiring
instSemiringNNReal
MonoidHom.instFunLike
rpowMonoidHom
Real
instPowReal
β€”β€”
rpow_add πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”eq
Real.rpow_add
coe_pos
pos_iff_ne_zero
instCanonicallyOrderedAdd
rpow_add' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”eq
Real.rpow_add'
rpow_add_intCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instIntCast
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
zpow
β€”eq
Real.rpow_add_intCast
Nat.cast_zero
rpow_add_intCast' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instIntCast
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
zpow
β€”eq
Real.rpow_add_intCast'
Nat.cast_zero
rpow_add_natCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instNatCast
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
β€”eq
Real.rpow_add_natCast
Nat.cast_zero
rpow_add_natCast' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instNatCast
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
β€”eq
Real.rpow_add_natCast'
Nat.cast_zero
rpow_add_of_nonneg πŸ“–mathematicalReal
Real.instLE
Real.instZero
NNReal
instPowReal
Real.instAdd
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”eq
Real.rpow_add_of_nonneg
rpow_add_one πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instOne
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”Nat.cast_one
pow_one
rpow_add_natCast
rpow_add_one' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instOne
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”rpow_add'
rpow_one
rpow_eq_pow πŸ“–mathematicalβ€”rpow
NNReal
Real
instPowReal
β€”β€”
rpow_eq_rpow_iff πŸ“–mathematicalβ€”NNReal
Real
instPowReal
β€”rpow_left_injective
rpow_eq_zero πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instZeroNNReal
β€”β€”
rpow_eq_zero_iff πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instZeroNNReal
β€”coe_rpow
coe_eq_zero
Real.rpow_eq_zero_iff_of_nonneg
rpow_intCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instIntCast
zpow
β€”Int.cast_natCast
rpow_natCast
zpow_natCast
Int.cast_negSucc
rpow_neg
zpow_negSucc
rpow_intCast_mul πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instMul
Real.instIntCast
zpow
β€”rpow_mul
rpow_intCast
rpow_inv_eq_iff πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instInv
β€”rpow_eq_rpow_iff
one_div
rpow_self_rpow_inv
rpow_inv_le_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
Real.instInv
β€”rpow_le_rpow_iff
one_div
rpow_self_rpow_inv
LT.lt.ne'
rpow_inv_le_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
Preorder.toLE
instPowReal
Real.instInv
β€”Real.rpow_inv_le_iff_of_neg
rpow_inv_le_iff_of_pos πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowReal
Real.instInv
β€”Real.rpow_inv_le_iff_of_pos
rpow_inv_lt_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
Real.instInv
β€”le_rpow_inv_iff
rpow_inv_lt_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowReal
Real.instInv
β€”Real.rpow_inv_lt_iff_of_neg
rpow_inv_lt_iff_of_pos πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
Preorder.toLT
instPowReal
Real.instInv
β€”Real.rpow_inv_lt_iff_of_pos
rpow_inv_natCast_pow πŸ“–mathematicalβ€”NNReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
Real
instPowReal
Real.instInv
Real.instNatCast
β€”coe_pow
coe_rpow
Real.rpow_inv_natCast_pow
rpow_inv_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instInv
β€”rpow_mul
inv_mul_cancelβ‚€
rpow_one
rpow_inv_rpow_self πŸ“–mathematicalβ€”NNReal
Real
instPowReal
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instOne
β€”rpow_mul
Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq
IsCancelMulZero.toIsLeftCancelMulZero
instIsCancelMulZero
Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul
Mathlib.Tactic.FieldSimp.NF.mul_eq_eval
Mathlib.Tactic.FieldSimp.NF.atom_eq_eval
Mathlib.Tactic.FieldSimp.NF.div_eq_eval
Mathlib.Tactic.FieldSimp.NF.one_eq_eval
Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval
Mathlib.Tactic.FieldSimp.NF.mul_eq_evalβ‚‚
one_mul
Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq
Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero
Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst
div_one
Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons
Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div
Mathlib.Tactic.FieldSimp.NF.eval_cons
Mathlib.Tactic.FieldSimp.zpow'_one
one_ne_zero
NeZero.one
GroupWithZero.toNontrivial
rpow_one
rpow_le_one πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.rpow_le_one
rpow_le_one_of_one_le_of_nonpos πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.rpow_le_one_of_one_le_of_nonpos
rpow_le_rpow πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.rpow_le_rpow
rpow_le_rpow_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
β€”Real.rpow_le_rpow_iff
rpow_le_rpow_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
Preorder.toLE
instPowReal
β€”Real.rpow_le_rpow_iff_of_neg
rpow_le_rpow_of_exponent_ge πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Preorder.toLE
instOneNNReal
Real
Real.instLE
instPowRealβ€”Real.rpow_le_rpow_of_exponent_ge
rpow_le_rpow_of_exponent_le πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLE
instPowRealβ€”Real.rpow_le_rpow_of_exponent_le
rpow_le_rpow_of_nonpos πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Preorder.toLE
Real
Real.instLE
Real.instZero
instPowRealβ€”Real.rpow_le_rpow_of_nonpos
rpow_le_self_of_le_one πŸ“–mathematicalNNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLE
Real.instOne
instPowRealβ€”eq_bot_or_bot_lt
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.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_overlap_zero
Mathlib.Meta.NormNum.IsInt.to_isNat
Mathlib.Meta.NormNum.isInt_add
Mathlib.Tactic.Ring.cast_zero
Nat.cast_zero
Mathlib.Tactic.Ring.add_pf_add_zero
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
sub_eq_zero_of_eq
bot_eq_zero'
instCanonicallyOrderedAdd
zero_rpow
rpow_one
rpow_le_rpow_of_exponent_ge
rpow_left_bijective πŸ“–mathematicalβ€”Function.Bijective
NNReal
Real
instPowReal
β€”rpow_left_injective
rpow_left_surjective
rpow_left_injective πŸ“–mathematicalβ€”NNReal
Real
instPowReal
β€”rpow_inv_rpow_self
rpow_left_surjective πŸ“–mathematicalβ€”NNReal
Real
instPowReal
β€”inv_mul_cancelβ‚€
rpow_one
rpow_lt_one πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.rpow_lt_one
coe_nonneg
rpow_lt_one_of_one_lt_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.rpow_lt_one_of_one_lt_of_neg
rpow_lt_rpow πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.rpow_lt_rpow
rpow_lt_rpow_iff πŸ“–mathematicalReal
Real.instLT
Real.instZero
NNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
β€”Real.rpow_lt_rpow_iff
rpow_lt_rpow_iff_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.rpow_lt_rpow_iff_of_neg
rpow_lt_rpow_of_exponent_gt πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
instOneNNReal
Real
Real.instLT
instPowRealβ€”Real.rpow_lt_rpow_of_exponent_gt
rpow_lt_rpow_of_exponent_lt πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instOneNNReal
Real
Real.instLT
instPowRealβ€”Real.rpow_lt_rpow_of_exponent_lt
rpow_lt_rpow_of_neg πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
Real.instLT
Real.instZero
instPowRealβ€”Real.rpow_lt_rpow_of_neg
rpow_mul πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instMul
β€”eq
Real.rpow_mul
rpow_mul_intCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instMul
Real.instIntCast
zpow
β€”rpow_mul
rpow_intCast
rpow_mul_natCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instMul
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”rpow_mul
rpow_natCast
rpow_natCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”eq
Real.rpow_natCast
rpow_natCast_mul πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instMul
Real.instNatCast
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”rpow_mul
rpow_natCast
rpow_neg πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instNeg
instInv
β€”eq
Real.rpow_neg
rpow_neg_one πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instNeg
Real.instOne
instInv
β€”rpow_neg
rpow_one
rpow_ofNat πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”rpow_natCast
rpow_of_add_eq πŸ“–mathematicalReal
Real.instAdd
NNReal
instPowReal
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”rpow_add'
rpow_one πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instOne
β€”eq
Real.rpow_one
rpow_one_add' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instAdd
Real.instOne
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
instSemiringNNReal
β€”rpow_add'
rpow_one
rpow_one_sub' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instOne
instDiv
β€”rpow_sub'
rpow_one
rpow_pos πŸ“–mathematicalNNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
instPowReal
β€”zero_rpow
LT.lt.ne'
rpow_lt_rpow
lt_trichotomy
rpow_zero
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
instIsStrictOrderedRing
neg_neg
rpow_neg
inv_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
neg_pos
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
rpow_rpow_inv πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instInv
β€”rpow_mul
mul_inv_cancelβ‚€
rpow_one
rpow_self_rpow_inv πŸ“–mathematicalβ€”NNReal
Real
instPowReal
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instOne
β€”rpow_mul
Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq
IsCancelMulZero.toIsLeftCancelMulZero
instIsCancelMulZero
Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul
Mathlib.Tactic.FieldSimp.NF.mul_eq_eval
Mathlib.Tactic.FieldSimp.NF.div_eq_eval
Mathlib.Tactic.FieldSimp.NF.one_eq_eval
Mathlib.Tactic.FieldSimp.NF.atom_eq_eval
Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval
Mathlib.Tactic.FieldSimp.NF.mul_eq_evalβ‚‚
one_mul
Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq
Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero
Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst
div_one
Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons
Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div
Mathlib.Tactic.FieldSimp.NF.eval_cons
Mathlib.Tactic.FieldSimp.zpow'_one
one_ne_zero
NeZero.one
GroupWithZero.toNontrivial
rpow_one
rpow_sub πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
instDiv
β€”eq
Real.rpow_sub
coe_pos
pos_iff_ne_zero
instCanonicallyOrderedAdd
rpow_sub' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
instDiv
β€”eq
Real.rpow_sub'
rpow_sub_intCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instNatCast
instDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”eq
Real.rpow_sub_intCast
Nat.cast_zero
rpow_sub_intCast' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instIntCast
instDiv
zpow
β€”eq
Real.rpow_sub_intCast'
Nat.cast_zero
rpow_sub_natCast πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instNatCast
instDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”eq
Real.rpow_sub_natCast
Nat.cast_zero
rpow_sub_natCast' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instNatCast
instDiv
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”eq
Real.rpow_sub_natCast'
Nat.cast_zero
rpow_sub_one πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instOne
instDiv
β€”Nat.cast_one
pow_one
rpow_sub_natCast
rpow_sub_one' πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instSub
Real.instOne
instDiv
β€”rpow_sub'
rpow_one
rpow_two πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
instSemiringNNReal
β€”rpow_ofNat
Nat.instAtLeastTwoHAddOfNat
rpow_zero πŸ“–mathematicalβ€”NNReal
Real
instPowReal
Real.instZero
instOneNNReal
β€”eq
Real.rpow_zero
rpow_zero_pos πŸ“–mathematicalβ€”NNReal
Preorder.toLT
PartialOrder.toPreorder
instPartialOrderNNReal
instZeroNNReal
Real
instPowReal
Real.instZero
β€”rpow_zero
one_pos
LinearOrderedCommMonoidWithZero.toZeroLeOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
instIsStrictOrderedRing
sqrt_eq_rpow πŸ“–mathematicalβ€”DFunLike.coe
OrderIso
NNReal
Preorder.toLE
PartialOrder.toPreorder
instPartialOrderNNReal
instFunLikeOrderIso
sqrt
Real
instPowReal
DivInvMonoid.toDiv
Real.instDivInvMonoid
Real.instOne
instOfNatAtLeastTwo
Real.instNatCast
Nat.instAtLeastTwoHAddOfNat
β€”eq
Nat.instAtLeastTwoHAddOfNat
Real.coe_sqrt
Real.sqrt_eq_rpow
strictMono_rpow_of_pos πŸ“–mathematicalReal
Real.instLT
Real.instZero
StrictMono
NNReal
PartialOrder.toPreorder
instPartialOrderNNReal
instPowReal
β€”rpow_lt_rpow
zero_rpow πŸ“–mathematicalβ€”NNReal
Real
instPowReal
instZeroNNReal
β€”eq
Real.zero_rpow

Real

Theorems

NameKindAssumesProvesValidatesDepends On
enorm_rpow_of_nonneg πŸ“–mathematicalReal
instLE
instZero
ENorm.enorm
ContinuousENorm.toENorm
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddGroup.toPseudoMetricSpace
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
normedCommRing
SeminormedAddGroup.toContinuousENorm
instPow
ENNReal
ENNReal.instPowReal
β€”nnnorm_rpow_of_nonneg
ENNReal.coe_rpow_of_nonneg
finset_prod_rpow πŸ“–mathematicalReal
instLE
instZero
Finset.prod
instCommMonoid
instPow
β€”multiset_prod_map_rpow
list_prod_map_rpow πŸ“–mathematicalReal
instLE
instZero
instMul
instOne
instPow
β€”CanLift.prf
List.canLift
NNReal.canLift
NNReal.list_prod_map_rpow
NNReal.coe_list_prod
list_prod_map_rpow' πŸ“–mathematicalReal
instLE
instZero
instMul
instOne
instPow
β€”list_prod_map_rpow
multiset_prod_map_rpow πŸ“–mathematicalReal
instLE
instZero
Multiset.prod
instCommMonoid
Multiset.map
instPow
β€”list_prod_map_rpow'
nnnorm_rpow_of_nonneg πŸ“–mathematicalReal
instLE
instZero
NNNorm.nnnorm
SeminormedAddGroup.toNNNorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
normedCommRing
instPow
NNReal
NNReal.instPowReal
β€”NNReal.eq
norm_rpow_of_nonneg
toNNReal_rpow_of_nonneg πŸ“–mathematicalReal
instLE
instZero
toNNReal
instPow
NNReal
NNReal.instPowReal
β€”coe_toNNReal
NNReal.coe_rpow
toNNReal_coe

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