`self`

proved

show as: view math explainer →

module: IndisputableMonolith.RecogSpec.Core
domain: RecogSpec
line: 60 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.RecogSpec.Core on GitHub at line 60.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

used by

formal source

  57
  58@[simp] lemma mem : PhiClosed φ x ↔ x ∈ phiSubfield φ := Iff.rfl
  59
  60lemma self (φ : ℝ) : PhiClosed φ φ := by
  61  change φ ∈ phiSubfield φ
  62  exact Subfield.subset_closure (by simp)
  63
  64lemma of_rat (φ : ℝ) (q : ℚ) : PhiClosed φ (q : ℝ) := by
  65  change ((algebraMap ℚ ℝ) q) ∈ phiSubfield φ
  66  simpa using (phiSubfield φ).algebraMap_mem q
  67
  68lemma zero (φ : ℝ) : PhiClosed φ (0 : ℝ) :=
  69  (phiSubfield φ).zero_mem
  70
  71lemma one (φ : ℝ) : PhiClosed φ (1 : ℝ) :=
  72  (phiSubfield φ).one_mem
  73
  74lemma add (hx : PhiClosed φ x) (hy : PhiClosed φ y) :
  75    PhiClosed φ (x + y) :=
  76  (phiSubfield φ).add_mem hx hy
  77
  78lemma sub (hx : PhiClosed φ x) (hy : PhiClosed φ y) :
  79    PhiClosed φ (x - y) :=
  80  (phiSubfield φ).sub_mem hx hy
  81
  82lemma neg (hx : PhiClosed φ x) : PhiClosed φ (-x) :=
  83  (phiSubfield φ).neg_mem hx
  84
  85lemma mul (hx : PhiClosed φ x) (hy : PhiClosed φ y) :
  86    PhiClosed φ (x * y) :=
  87  (phiSubfield φ).mul_mem hx hy
  88
  89lemma inv (hx : PhiClosed φ x) : PhiClosed φ x⁻¹ :=
  90  (phiSubfield φ).inv_mem hx

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