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Mathlib.SetTheory.Cardinal.Continuum

Cardinality of continuum #

In this file we define Cardinal.continuum (notation: ๐” , localized in Cardinal) to be 2 ^ โ„ตโ‚€. We also prove some simp lemmas about cardinal arithmetic involving ๐” .

Notation #

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def Cardinal.continuum :
Cardinal.{u}

Cardinality of the continuum.

Equations
Instances For
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    def Cardinal.term๐”  :
    Lean.ParserDescr

    Cardinality of the continuum.

    Equations
    Instances For
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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 โ‰ค lift.{v, u} c โ†” continuum โ‰ค c
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      @[simp]
      theorem Cardinal.lift_le_continuum {c : Cardinal.{u}} :
      lift.{v, u} c โ‰ค continuum โ†” c โ‰ค continuum
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      @[simp]
      theorem Cardinal.continuum_lt_lift {c : Cardinal.{u}} :
      continuum < lift.{v, u} c โ†” continuum < c
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      @[simp]
      theorem Cardinal.lift_lt_continuum {c : Cardinal.{u}} :
      lift.{v, u} c < continuum โ†” c < continuum

      Inequalities #

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

      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 : โ„•) :
      โ†‘n + continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_add_nat (n : โ„•) :
      continuum + โ†‘n = continuum
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      @[simp]
      theorem Cardinal.ofNat_add_continuum {n : โ„•} [n.AtLeastTwo] :
      OfNat.ofNat n + continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_add_ofNat {n : โ„•} [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 : โ„•} (hn : n โ‰  0) :
      โ†‘n * continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_mul_nat {n : โ„•} (hn : n โ‰  0) :
      continuum * โ†‘n = continuum
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      @[simp]
      theorem Cardinal.ofNat_mul_continuum {n : โ„•} [n.AtLeastTwo] :
      OfNat.ofNat n * continuum = continuum
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      @[simp]
      theorem Cardinal.continuum_mul_ofNat {n : โ„•} [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 : โ„•} (hn : 2 โ‰ค n) :
      โ†‘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}} (hโ‚ : 2 โ‰ค x) (hโ‚‚ : x โ‰ค continuum) :
      x ^ aleph0 = continuum