mabs_mul_le — Mathlib

English

Absolute value commutes with a piecewise choice: |if p then a else b|_m = if p then |a|_m else |b|_m.

Русский

Модуль сохраняется через ветвление: |if p then a else b|_м = if p then |a|_м else |b|_м.

LaTeX

$$|\mathrm{abs} (\mathrm{ite}~p~a~b) = \mathrm{ite}~p~|a|_m~|b|_m$$

Lean4

/-- The absolute value satisfies the triangle inequality. -/
@[to_additive /-- The absolute value satisfies the triangle inequality. -/
    ]
theorem mabs_mul_le (a b : α) : |a * b|ₘ ≤ |a|ₘ * |b|ₘ :=
  by
  apply sup_le
  · exact mul_le_mul' (le_mabs_self a) (le_mabs_self b)
  · rw [mul_inv]
    exact mul_le_mul' (inv_le_mabs _) (inv_le_mabs _)

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