subsingleton_of_rank_zero — Mathlib

English

If M and M' are finite-dimensional and finrank_R M = finrank_R M', then there exists a linear equivalence M ≃ₗ[R] M'.

Русский

Если M и M' конечномерны и finrank_R M = finrank_R M', существует линейное эквивале между ними.

LaTeX

$$[Module.Finite R M] [Module.Finite R M'] (cond : finrank R M = finrank R M') : Nonempty (M ≃ₗ[R] M')$$

Lean4

/-- A free module of rank zero is trivial. -/
theorem subsingleton_of_rank_zero (h : Module.rank R M = 0) : Subsingleton M :=
  by
  rw [← Basis.mk_eq_rank'' (Module.Free.chooseBasis R M), Cardinal.mk_eq_zero_iff] at h
  exact (Module.Free.chooseBasis R M).repr.subsingleton

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