File: Mathlib/Data/Real/ENatENNReal.lean

full_name: ENat.toENNReal
code: /-- Coercion from `ℕ∞` to `ℝ≥0∞`. -/
@[coe] def toENNReal : ℕ∞ → ℝ≥0∞ 
goal: ℕ∞ → ℝ≥0∞

full_name: ENat.hasCoeENNReal
code: instance hasCoeENNReal : CoeTC ℕ∞ ℝ≥0∞ 
goal: CoeTC ℕ∞ ℝ≥0∞

full_name: ENat.map_coe_nnreal
code: @[simp]
theorem map_coe_nnreal : WithTop.map ((↑) : ℕ → ℝ≥0) = ((↑) : ℕ∞ → ℝ≥0∞) 
goal: WithTop.map Nat.cast = ENat.toENNReal

full_name: ENat.toENNRealOrderEmbedding_apply
code: /-- Coercion `ℕ∞ → ℝ≥0∞` as an `OrderEmbedding`. -/
@[simps! (config := .asFn)]
def toENNRealOrderEmbedding : ℕ∞ ↪o ℝ≥0∞ 
goal: ⇑ENat.toENNRealOrderEmbedding = ENat.toENNReal

full_name: ENat.toENNRealOrderEmbedding
code: /-- Coercion `ℕ∞ → ℝ≥0∞` as an `OrderEmbedding`. -/
@[simps! (config := .asFn)]
def toENNRealOrderEmbedding : ℕ∞ ↪o ℝ≥0∞ 
goal: ℕ∞ ↪o ℝ≥0∞

full_name: ENat.toENNRealRingHom
code: /-- Coercion `ℕ∞ → ℝ≥0∞` as a ring homomorphism. -/
@[simps! (config := .asFn)]
def toENNRealRingHom : ℕ∞ →+* ℝ≥0∞ 
goal: ℕ∞ →+* ℝ≥0∞

full_name: ENat.toENNRealRingHom_apply
code: /-- Coercion `ℕ∞ → ℝ≥0∞` as a ring homomorphism. -/
@[simps! (config := .asFn)]
def toENNRealRingHom : ℕ∞ →+* ℝ≥0∞ 
goal: ⇑ENat.toENNRealRingHom = ENat.toENNReal

full_name: ENat.toENNReal_top
code: @[simp, norm_cast]
theorem toENNReal_top : ((⊤ : ℕ∞) : ℝ≥0∞) = ⊤ 
goal: ↑⊤ = ⊤

full_name: ENat.toENNReal_coe
code: @[simp, norm_cast]
theorem toENNReal_coe (n : ℕ) : ((n : ℕ∞) : ℝ≥0∞) = n 
goal: ↑↑n = ↑n

full_name: ENat.toENNReal_ofNat
code: @[simp, norm_cast]
theorem toENNReal_ofNat (n : ℕ) [n.AtLeastTwo] :
    ((no_index (OfNat.ofNat n : ℕ∞)) : ℝ≥0∞) = OfNat.ofNat n 
goal: ↑(OfNat.ofNat n) = OfNat.ofNat n

full_name: ENat.toENNReal_coe_eq_iff
code: @[simp, norm_cast]
theorem toENNReal_coe_eq_iff : (m : ℝ≥0∞) = (n : ℝ≥0∞) ↔ m = n 
goal: ↑m = ↑n ↔ m = n

full_name: ENat.toENNReal_le
code: @[simp, norm_cast]
theorem toENNReal_le : (m : ℝ≥0∞) ≤ n ↔ m ≤ n 
goal: ↑m ≤ ↑n ↔ m ≤ n

full_name: ENat.toENNReal_lt
code: @[simp, norm_cast]
theorem toENNReal_lt : (m : ℝ≥0∞) < n ↔ m < n 
goal: ↑m < ↑n ↔ m < n

full_name: ENat.toENNReal_mono
code: @[mono]
theorem toENNReal_mono : Monotone ((↑) : ℕ∞ → ℝ≥0∞) 
goal: Monotone ENat.toENNReal

full_name: ENat.toENNReal_strictMono
code: @[mono]
theorem toENNReal_strictMono : StrictMono ((↑) : ℕ∞ → ℝ≥0∞) 
goal: StrictMono ENat.toENNReal

full_name: ENat.toENNReal_zero
code: @[simp, norm_cast]
theorem toENNReal_zero : ((0 : ℕ∞) : ℝ≥0∞) = 0 
goal: ↑0 = 0

full_name: ENat.toENNReal_add
code: @[simp, norm_cast]
theorem toENNReal_add (m n : ℕ∞) : ↑(m + n) = (m + n : ℝ≥0∞) 
goal: ↑(m + n) = ↑m + ↑n

full_name: ENat.toENNReal_one
code: @[simp, norm_cast]
theorem toENNReal_one : ((1 : ℕ∞) : ℝ≥0∞) = 1 
goal: ↑1 = 1

full_name: ENat.toENNReal_mul
code: @[simp, norm_cast]
theorem toENNReal_mul (m n : ℕ∞) : ↑(m * n) = (m * n : ℝ≥0∞) 
goal: ↑(m * n) = ↑m * ↑n

full_name: ENat.toENNReal_pow
code: @[simp, norm_cast]
theorem toENNReal_pow (x : ℕ∞) (n : ℕ) : (x ^ n : ℕ∞) = (x : ℝ≥0∞) ^ n 
goal: ↑(x ^ n) = ↑x ^ n

full_name: ENat.toENNReal_min
code: @[simp, norm_cast]
theorem toENNReal_min (m n : ℕ∞) : ↑(min m n) = (min m n : ℝ≥0∞) 
goal: ↑(min m n) = min ↑m ↑n

full_name: ENat.toENNReal_max
code: @[simp, norm_cast]
theorem toENNReal_max (m n : ℕ∞) : ↑(max m n) = (max m n : ℝ≥0∞) 
goal: ↑(max m n) = max ↑m ↑n

full_name: ENat.toENNReal_sub
code: @[simp, norm_cast]
theorem toENNReal_sub (m n : ℕ∞) : ↑(m - n) = (m - n : ℝ≥0∞) 
goal: ↑(m - n) = ↑m - ↑n