Documentation

Mathlib.SetTheory.Cardinal.Continuum

Cardinality of continuum #

In this file we define Cardinal.continuum (notation: 𝔠, localized in Cardinal) to be 2 ^ ℵ₀. 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
Instances For
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    @[simp]
    theorem Cardinal.two_power_aleph0 :
    2 ^ Cardinal.aleph0 = Cardinal.continuum
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    @[simp]
    theorem Cardinal.lift_continuum :
    Cardinal.lift.{v, u_1} Cardinal.continuum = Cardinal.continuum
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    @[simp]
    theorem Cardinal.continuum_le_lift {c : Cardinal.{u}} :
    Cardinal.continuum ≤ Cardinal.lift.{v, u} c ↔ Cardinal.continuum ≤ c
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    @[simp]
    theorem Cardinal.lift_le_continuum {c : Cardinal.{u}} :
    Cardinal.lift.{v, u} c ≤ Cardinal.continuum ↔ c ≤ Cardinal.continuum
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    @[simp]
    theorem Cardinal.continuum_lt_lift {c : Cardinal.{u}} :
    Cardinal.continuum < Cardinal.lift.{v, u} c ↔ Cardinal.continuum < c
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    @[simp]
    theorem Cardinal.lift_lt_continuum {c : Cardinal.{u}} :
    Cardinal.lift.{v, u} c < Cardinal.continuum ↔ c < Cardinal.continuum

    Inequalities #

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    theorem Cardinal.aleph0_lt_continuum :
    Cardinal.aleph0 < Cardinal.continuum
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    theorem Cardinal.aleph0_le_continuum :
    Cardinal.aleph0 ≤ Cardinal.continuum
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    @[simp]
    theorem Cardinal.beth_one :
    Cardinal.beth 1 = Cardinal.continuum
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    theorem Cardinal.nat_lt_continuum (n : ℕ) :
    ↑n < Cardinal.continuum
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    theorem Cardinal.mk_set_nat :
    Cardinal.mk (Set ℕ) = Cardinal.continuum
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    theorem Cardinal.continuum_pos :
    0 < Cardinal.continuum
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    theorem Cardinal.continuum_ne_zero :
    Cardinal.continuum ≠ 0
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    theorem Cardinal.aleph_one_le_continuum :
    Cardinal.aleph 1 ≤ Cardinal.continuum
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    @[simp]
    theorem Cardinal.continuum_toNat :
    Cardinal.toNat Cardinal.continuum = 0
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    @[simp]
    theorem Cardinal.continuum_toPartENat :
    Cardinal.toPartENat Cardinal.continuum = ⊤

    Addition #

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

    Multiplication #

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

    Power #

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    @[simp]
    theorem Cardinal.aleph0_power_aleph0 :
    Cardinal.aleph0 ^ Cardinal.aleph0 = Cardinal.continuum
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    @[simp]
    theorem Cardinal.nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) :
    ↑n ^ Cardinal.aleph0 = Cardinal.continuum
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    @[simp]
    theorem Cardinal.continuum_power_aleph0 :
    Cardinal.continuum ^ Cardinal.aleph0 = Cardinal.continuum
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    theorem Cardinal.power_aleph0_of_le_continuum {x : Cardinal.{u_1}} (h₁ : 2 ≤ x) (h₂ : x ≤ Cardinal.continuum) :
    x ^ Cardinal.aleph0 = Cardinal.continuum