mabs_mabs_div_mabs_le — Mathlib

English

Absolute value commutes with dependent if-then-else: |if p then a p else b p|_m = if p then |a p|_m else |b p|_m.

Русский

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

LaTeX

$$|if p then a p else b p|_m = if p then |a p|_m else |b p|_m$$

Lean4

@[to_additive]
theorem mabs_mabs_div_mabs_le (a b : α) : |(|a|ₘ / |b|ₘ)|ₘ ≤ |a / b|ₘ :=
  by
  rw [mabs, sup_le_iff]
  constructor
  · apply div_le_iff_le_mul.2
    convert mabs_mul_le (a / b) b
    rw [div_mul_cancel]
  · rw [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_le_iff_le_mul, mabs_div_comm]
    convert mabs_mul_le (b / a) a
    · rw [div_mul_cancel]

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