File: Mathlib/SetTheory/Cardinal/PartENat.lean

full_name: Cardinal.toPartENat
code: /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
  to `⊤`. -/
noncomputable def toPartENat : Cardinal →+o PartENat 
goal: Cardinal.{u_1} →+o PartENat

full_name: Cardinal.partENatOfENat_toENat
code: @[simp]
theorem partENatOfENat_toENat (c : Cardinal) : (toENat c : PartENat) = toPartENat c 
goal: ↑(Cardinal.toENat c) = Cardinal.toPartENat c

full_name: Cardinal.toPartENat_natCast
code: @[simp]
theorem toPartENat_natCast (n : ℕ) : toPartENat n = n 
goal: Cardinal.toPartENat ↑n = ↑n

full_name: Cardinal.toPartENat_apply_of_lt_aleph0
code: theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c 
goal: Cardinal.toPartENat c = ↑(Cardinal.toNat c)

full_name: Cardinal.toPartENat_eq_top
code: theorem toPartENat_eq_top {c : Cardinal} :
    toPartENat c = ⊤ ↔ ℵ₀ ≤ c 
goal: Cardinal.toPartENat c = ⊤ ↔ Cardinal.aleph0 ≤ c

full_name: Cardinal.toPartENat_apply_of_aleph0_le
code: theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toPartENat c = ⊤ 
goal: Cardinal.toPartENat c = ⊤

full_name: Cardinal.mk_toPartENat_of_infinite
code: @[simp]
theorem mk_toPartENat_of_infinite [h : Infinite α] : toPartENat #α = ⊤ 
goal: Cardinal.toPartENat (Cardinal.mk α) = ⊤

full_name: Cardinal.aleph0_toPartENat
code: @[simp]
theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ 
goal: Cardinal.toPartENat Cardinal.aleph0 = ⊤

full_name: Cardinal.toPartENat_surjective
code: theorem toPartENat_surjective : Surjective toPartENat 
goal: Function.Surjective ⇑Cardinal.toPartENat

full_name: Cardinal.toPartENat_strictMonoOn
code: theorem toPartENat_strictMonoOn : StrictMonoOn toPartENat (Set.Iic ℵ₀) 
goal: StrictMonoOn (⇑Cardinal.toPartENat) (Set.Iic Cardinal.aleph0)

full_name: Cardinal.toPartENat_le_iff_of_le_aleph0
code: lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
    toPartENat c ≤ toPartENat c' ↔ c ≤ c' 
goal: Cardinal.toPartENat c ≤ Cardinal.toPartENat c' ↔ c ≤ c'

full_name: Cardinal.toPartENat_le_iff_of_lt_aleph0
code: lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
    toPartENat c ≤ toPartENat c' ↔ c ≤ c' 
goal: Cardinal.toPartENat c ≤ Cardinal.toPartENat c' ↔ c ≤ c'

full_name: Cardinal.toPartENat_eq_iff_of_le_aleph0
code: lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :
    toPartENat c = toPartENat c' ↔ c = c' 
goal: Cardinal.toPartENat c = Cardinal.toPartENat c' ↔ c = c'

full_name: Cardinal.toPartENat_mono
code: theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') :
    toPartENat c ≤ toPartENat c' 
goal: Cardinal.toPartENat c ≤ Cardinal.toPartENat c'

full_name: Cardinal.toPartENat_lift
code: theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c 
goal: Cardinal.toPartENat (Cardinal.lift.{u, v} c) = Cardinal.toPartENat c

full_name: Cardinal.toPartENat_congr
code: theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat #α = toPartENat #β 
goal: Cardinal.toPartENat (Cardinal.mk α) = Cardinal.toPartENat (Cardinal.mk β)

full_name: Cardinal.mk_toPartENat_eq_coe_card
code: theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat #α = Fintype.card α 
goal: Cardinal.toPartENat (Cardinal.mk α) = ↑(Fintype.card α)