of_rank_le — Mathlib

English

A lattice M in V over R with IsLattice structure over K has the R-rank of M equal to the K-rank of V.

Русский

У решётки M в V над R с структурой IsLattice над K ранг R-части M равен ранг K-V.

LaTeX

$$$\operatorname{rank}_R M = \operatorname{rank}_K V$$$

Lean4

/-- A finitely-generated `R`-submodule of `V` of rank at least the `K`-rank of `V`
is a lattice. -/
theorem of_rank_le [Module.Finite K V] [IsFractionRing R K] {M : Submodule R V} (hfg : M.FG)
    (hr : Module.rank K V ≤ Module.rank R M) : IsLattice K M
    where
  fg := hfg
  span_eq_top := by simpa using Submodule.span_range_eq_top_of_injective_of_rank_le M.injective_subtype hr

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