Documentation

Mathlib.SetTheory.Cardinal.Continuum

Cardinality of continuum #

In this file we define Cardinal.continuum (notation: 𝔠, localized in Cardinal) to be 2 ^ β„΅β‚€. We also prove some simp lemmas about cardinal arithmetic involving 𝔠.

Notation #

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def Cardinal.continuum :
Cardinal.{u}

Cardinality of the continuum.

Equations
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def Cardinal.term𝔠 :
Lean.ParserDescr

Cardinality of the continuum.

Equations
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@[simp]
theorem Cardinal.two_power_aleph0 :
2 ^ aleph0 = continuum
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@[simp]
theorem Cardinal.lift_continuum :
lift.{v, u_1} continuum = continuum
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@[simp]
theorem Cardinal.continuum_le_lift {c : Cardinal.{u}} :
continuum ≀ lift.{v, u} c ↔ continuum ≀ c
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@[simp]
theorem Cardinal.lift_le_continuum {c : Cardinal.{u}} :
lift.{v, u} c ≀ continuum ↔ c ≀ continuum
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@[simp]
theorem Cardinal.continuum_lt_lift {c : Cardinal.{u}} :
continuum < lift.{v, u} c ↔ continuum < c
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@[simp]
theorem Cardinal.lift_lt_continuum {c : Cardinal.{u}} :
lift.{v, u} c < continuum ↔ c < continuum

Inequalities #

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theorem Cardinal.aleph0_lt_continuum :
aleph0 < continuum
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theorem Cardinal.aleph0_le_continuum :
aleph0 ≀ continuum
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@[simp]
theorem Cardinal.beth_one :
beth 1 = continuum
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theorem Cardinal.nat_lt_continuum (n : β„•) :
↑n < continuum
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theorem Cardinal.mk_set_nat :
mk (Set β„•) = continuum
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theorem Cardinal.continuum_pos :
0 < continuum
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theorem Cardinal.continuum_ne_zero :
continuum β‰  0
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theorem Cardinal.aleph_one_le_continuum :
aleph 1 ≀ continuum
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@[simp]
theorem Cardinal.continuum_toNat :
toNat continuum = 0
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@[simp]
theorem Cardinal.continuum_toENat :
toENat continuum = ⊀

Addition #

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@[simp]
theorem Cardinal.aleph0_add_continuum :
aleph0 + continuum = continuum
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@[simp]
theorem Cardinal.continuum_add_aleph0 :
continuum + aleph0 = continuum
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@[simp]
theorem Cardinal.continuum_add_self :
continuum + continuum = continuum
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@[simp]
theorem Cardinal.nat_add_continuum (n : β„•) :
↑n + continuum = continuum
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@[simp]
theorem Cardinal.continuum_add_nat (n : β„•) :
continuum + ↑n = continuum
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@[simp]
theorem Cardinal.ofNat_add_continuum {n : β„•} [n.AtLeastTwo] :
OfNat.ofNat n + continuum = continuum
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@[simp]
theorem Cardinal.continuum_add_ofNat {n : β„•} [n.AtLeastTwo] :
continuum + OfNat.ofNat n = continuum

Multiplication #

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@[simp]
theorem Cardinal.continuum_mul_self :
continuum * continuum = continuum
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@[simp]
theorem Cardinal.continuum_mul_aleph0 :
continuum * aleph0 = continuum
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@[simp]
theorem Cardinal.aleph0_mul_continuum :
aleph0 * continuum = continuum
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@[simp]
theorem Cardinal.nat_mul_continuum {n : β„•} (hn : n β‰  0) :
↑n * continuum = continuum
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@[simp]
theorem Cardinal.continuum_mul_nat {n : β„•} (hn : n β‰  0) :
continuum * ↑n = continuum
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@[simp]
theorem Cardinal.ofNat_mul_continuum {n : β„•} [n.AtLeastTwo] :
OfNat.ofNat n * continuum = continuum
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@[simp]
theorem Cardinal.continuum_mul_ofNat {n : β„•} [n.AtLeastTwo] :
continuum * OfNat.ofNat n = continuum

Power #

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@[simp]
theorem Cardinal.aleph0_power_aleph0 :
aleph0 ^ aleph0 = continuum
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@[simp]
theorem Cardinal.nat_power_aleph0 {n : β„•} (hn : 2 ≀ n) :
↑n ^ aleph0 = continuum
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@[simp]
theorem Cardinal.continuum_power_aleph0 :
continuum ^ aleph0 = continuum
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theorem Cardinal.power_aleph0_of_le_continuum {x : Cardinal.{u_1}} (h₁ : 2 ≀ x) (hβ‚‚ : x ≀ continuum) :
x ^ aleph0 = continuum