posPart_smul_of_nonpos — Mathlib

English

NegPart respects nonnegative scalar action in the sense that NegPart( r a ) = r NegPart(a) for r ∈ NNReal.

Русский

NegPart(r a) = r NegPart(a) для r ∈ NNReal.

LaTeX

$$$NegPart((r) a) = r \; NegPart(a) \text{ for } r \in \mathbb{NNReal}$$$

Lean4

theorem posPart_smul_of_nonpos {r : ℝ} (hr : r ≤ 0) {a : A} : (r • a)⁺ = -r • a⁻ :=
  by
  nth_rw 1 [← neg_neg r]
  rw [neg_smul, ← smul_neg, posPart_smul_of_nonneg (neg_nonneg.mpr hr), posPart_neg]

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