English
Let f be a compactly supported real-valued function and a ≤ 0. Then the nnrealPart of a smoothed negated function equals the NNReal multiple of the nnrealPart of the negation: (a · f).nnrealPart = (-a).toNNReal · (-f).nnrealPart.
Русский
Пусть f — компактно поддерживаемая веще-valuящая функция, и a ≤ 0. Тогда nnrealPart от произведения на a равен произведению (-a) по NNReal к nnrealPart от (-f): (a · f).nnrealPart = (-a).toNNReal · (-f).nnrealPart.
LaTeX
$$$ (a \\cdot f)^{\\mathrm{nnrealPart}} = (-a)^{\\mathrm{toNNReal}} \\cdot (-f)^{\\mathrm{nnrealPart}}, \\quad a \\le 0,$$$
Lean4
theorem nnrealPart_smul_neg (f : C_c(α, ℝ)) {a : ℝ} (ha : a ≤ 0) :
(a • f).nnrealPart = (-a).toNNReal • (-f).nnrealPart :=
by
ext x
simp only [nnrealPart_apply, coe_smul, Pi.smul_apply, smul_eq_mul, Real.coe_toNNReal', coe_neg, Pi.neg_apply,
NNReal.coe_mul]
rcases le_total 0 (f x) with hfx | hfx
· simp [mul_nonpos_iff, ha, hfx]
· simp [ha, hfx, mul_nonneg_of_nonpos_of_nonpos]
