weight_eq_zero_iff_eq_zero — Mathlib

English

If M is canonically ordered and non-torsion, then weight w f = 0 iff f = 0.

Русский

Если M канонически упорядочено и безtorsion, то weight w f = 0 тогда и только тогда, когда f = 0.

LaTeX

$$weight w f = 0 ↔ f = 0$$

Lean4

/-- If `M` is a `CanonicallyOrderedAddCommMonoid`, then `weight f` is zero iff `f = 0`. -/
theorem weight_eq_zero_iff_eq_zero (w : σ → M) [NonTorsionWeight ℕ w] {f : σ →₀ ℕ} : weight w f = 0 ↔ f = 0 := by
  classical
  constructor
  · intro h
    ext s
    simp only [Finsupp.coe_zero, Pi.zero_apply]
    by_contra hs
    apply NonTorsionWeight.ne_zero ℕ w s
    rw [← nonpos_iff_eq_zero, ← h]
    exact le_weight_of_ne_zero' w hs
  · intro h
    rw [h, map_zero]

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