mem_segment_translate — Mathlib

English

Translation by a fixed vector preserves segment membership: a ∈ [x -[𝕜] y] implies a + translation belongs to [a+y -[𝕜] a+z] iff x ∈ [y -[𝕜] z].

Русский

Сдвиг на фиксированный вектор сохраняет принадлежность к сегменту: a ∈ [x -[𝕜] y] ⇔ ...

LaTeX

$$$$ a ∈ [x -[\mathbb{K}] y] \iff a \in [x -[\mathbb{K}] y]. $$$$

Lean4

@[simp]
theorem mem_segment_translate (a : E) {x b c} : a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c] := by
  simp_rw [← vadd_eq_add, ← vadd_segment, vadd_mem_vadd_set_iff]

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