fermi_den_pos

formal source

  58
  59private theorem sqrt2_pos : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)
  60
  61private theorem fermi_den_pos : 0 < Real.sqrt 2 * vev_canonical ^ 2 := by
  62  have hv : 0 < vev_canonical ^ 2 := sq_pos_of_ne_zero (ne_of_gt vev_canonical_pos)
  63  nlinarith [sqrt2_pos, hv]
  64
  65theorem row_fermi_pred_lower :
  66    (1.16e-5 : ℝ) < row_fermi_pred := by
  67  unfold row_fermi_pred
  68  rw [lt_div_iff₀ fermi_den_pos]
  69  have hs : Real.sqrt 2 < (1.4143 : ℝ) := sqrt2_lt_14143
  70  have hden :
  71      Real.sqrt 2 * vev_canonical ^ 2 < (1.4143 : ℝ) * (246 : ℝ) ^ 2 := by
  72    have hv : vev_canonical = (246 : ℝ) := rfl
  73    have hvpos : 0 < vev_canonical ^ 2 := sq_pos_of_ne_zero (ne_of_gt vev_canonical_pos)
  74    rw [hv]
  75    nlinarith
  76  have hnum : (1.16e-5 : ℝ) * ((1.4143 : ℝ) * (246 : ℝ) ^ 2) < 1 := by
  77    norm_num
  78  nlinarith
  79
  80theorem row_fermi_pred_upper :
  81    row_fermi_pred < (1.17e-5 : ℝ) := by
  82  unfold row_fermi_pred
  83  rw [div_lt_iff₀ fermi_den_pos]
  84  have hs : (1.4142 : ℝ) < Real.sqrt 2 := sqrt2_gt_14142
  85  have hden :
  86      (1.4142 : ℝ) * (246 : ℝ) ^ 2 < Real.sqrt 2 * vev_canonical ^ 2 := by
  87    have hv : vev_canonical = (246 : ℝ) := rfl
  88    rw [hv]
  89    nlinarith
  90  have hnum : 1 < (1.17e-5 : ℝ) * ((1.4142 : ℝ) * (246 : ℝ) ^ 2) := by
  91    norm_num