`mul`

proved

show as: view math explainer →

module: IndisputableMonolith.RecogSpec.Core
domain: RecogSpec
line: 85 · 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 85.

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

  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
  91
  92lemma div (hx : PhiClosed φ x) (hy : PhiClosed φ y) :
  93    PhiClosed φ (x / y) :=
  94  (phiSubfield φ).div_mem hx hy
  95
  96lemma pow (hx : PhiClosed φ x) (n : ℕ) : PhiClosed φ (x ^ n) := by
  97  simpa using (phiSubfield φ).pow_mem hx n
  98
  99lemma pow_self (φ : ℝ) (n : ℕ) : PhiClosed φ (φ ^ n) :=
 100  pow (self φ) n
 101
 102lemma inv_self (φ : ℝ) : PhiClosed φ (φ⁻¹) :=
 103  inv (self φ)
 104
 105lemma inv_pow_self (φ : ℝ) (n : ℕ) : PhiClosed φ ((φ ^ n)⁻¹) :=
 106  inv (pow_self φ n)
 107
 108lemma of_nat (φ : ℝ) (n : ℕ) : PhiClosed φ (n : ℝ) := by
 109  simpa using of_rat φ n
 110
 111lemma half (φ : ℝ) : PhiClosed φ (1 / (2 : ℝ)) := by
 112  have htwo : PhiClosed φ ((2 : ℚ) : ℝ) := of_rat φ 2
 113  simpa using inv htwo
 114
 115end PhiClosed

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