File: Mathlib/Analysis/SpecialFunctions/Log/ENNRealLogExp.lean

full_name: EReal.log_exp
code: @[simp] lemma EReal.log_exp (x : EReal) : log (exp x) = x 
goal: x.exp.log = x

full_name: ENNReal.exp_log
code: @[simp] lemma ENNReal.exp_log (x : ℝ≥0∞) : exp (log x) = x 
goal: x.log.exp = x

full_name: EReal.exp_nmul
code: lemma exp_nmul (x : EReal) (n : ℕ) : exp (n * x) = (exp x) ^ n 
goal: (↑n * x).exp = x.exp ^ n

full_name: EReal.exp_mul
code: lemma exp_mul (x : EReal) (y : ℝ) : exp (x * y) = (exp x) ^ y 
goal: (x * ↑y).exp = x.exp ^ y

full_name: ENNReal.logOrderIso
code: /-- `ENNReal.log` and its inverse `EReal.exp` are an order isomorphism between `ℝ≥0∞` and
`EReal`. -/
noncomputable
def logOrderIso : ℝ≥0∞ ≃o EReal 
goal: ℝ≥0∞ ≃o EReal

full_name: ENNReal.logOrderIso_apply
code: @[simp] lemma logOrderIso_apply (x : ℝ≥0∞) : logOrderIso x = log x 
goal: ENNReal.logOrderIso x = x.log

full_name: EReal.expOrderIso
code: /-- `EReal.exp` and its inverse `ENNReal.log` are an order isomorphism between `EReal` and
`ℝ≥0∞`. -/
noncomputable
def _root_.EReal.expOrderIso 
goal: EReal ≃o ℝ≥0∞

full_name: EReal.expOrderIso_apply
code: @[simp] lemma _root_.EReal.expOrderIso_apply (x : EReal) : expOrderIso x = exp x 
goal: EReal.expOrderIso x = x.exp

full_name: ENNReal.logOrderIso_symm
code: @[simp] lemma logOrderIso_symm : logOrderIso.symm = expOrderIso 
goal: ENNReal.logOrderIso.symm = EReal.expOrderIso

full_name: EReal.expOrderIso_symm
code: @[simp] lemma _root_.EReal.expOrderIso_symm : expOrderIso.symm = logOrderIso 
goal: EReal.expOrderIso.symm = ENNReal.logOrderIso

full_name: ENNReal.logHomeomorph
code: /-- `log` as a homeomorphism. -/
noncomputable def logHomeomorph : ℝ≥0∞ ≃ₜ EReal 
goal: ℝ≥0∞ ≃ₜ EReal

full_name: ENNReal.logHomeomorph_apply
code: @[simp] lemma logHomeomorph_apply (x : ℝ≥0∞) : logHomeomorph x = log x 
goal: ENNReal.logHomeomorph x = x.log

full_name: EReal.expHomeomorph
code: /-- `exp` as a homeomorphism. -/
noncomputable def _root_.EReal.expHomeomorph : EReal ≃ₜ ℝ≥0∞ 
goal: EReal ≃ₜ ℝ≥0∞

full_name: EReal.expHomeomorph_apply
code: @[simp] lemma _root_.EReal.expHomeomorph_apply (x : EReal) : expHomeomorph x = exp x 
goal: EReal.expHomeomorph x = x.exp

full_name: ENNReal.logHomeomorph_symm
code: @[simp] lemma logHomeomorph_symm : logHomeomorph.symm = expHomeomorph 
goal: ENNReal.logHomeomorph.symm = EReal.expHomeomorph

full_name: EReal.expHomeomorph_symm
code: @[simp] lemma _root_.EReal.expHomeomorph_symm : expHomeomorph.symm = logHomeomorph 
goal: EReal.expHomeomorph.symm = ENNReal.logHomeomorph

full_name: ENNReal.continuous_log
code: @[continuity, fun_prop]
lemma continuous_log : Continuous log 
goal: Continuous ENNReal.log

full_name: ENNReal.continuous_exp
code: @[continuity, fun_prop]
lemma continuous_exp : Continuous exp 
goal: Continuous EReal.exp

full_name: ENNReal.measurable_log
code: @[measurability, fun_prop]
lemma measurable_log : Measurable log 
goal: Measurable ENNReal.log

full_name: EReal.measurable_exp
code: @[measurability, fun_prop]
lemma _root_.EReal.measurable_exp : Measurable exp 
goal: Measurable EReal.exp

full_name: Measurable.ennreal_log
code: @[measurability, fun_prop]
lemma _root_.Measurable.ennreal_log {α : Type*} {_ : MeasurableSpace α}
    {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x ↦ log (f x) 
goal: ∀ {x : MeasurableSpace α} {f : α → ℝ≥0∞}, Measurable f → Measurable fun x => (f x).log

full_name: Measurable.ereal_exp
code: @[measurability, fun_prop]
lemma _root_.Measurable.ereal_exp {α : Type*} {_ : MeasurableSpace α}
    {f : α → EReal} (hf : Measurable f) :
    Measurable fun x ↦ exp (f x) 
goal: ∀ {x : MeasurableSpace α} {f : α → EReal}, Measurable f → Measurable fun x => (f x).exp

full_name: instPolishSpaceEReal
code: instance : PolishSpace EReal 
goal: PolishSpace EReal