of_le_of_flat_left — Mathlib

English

If M and N are disjoint and flat on one side, the rank bound extends to the infimum intersection with N and M'

Русский

Если M и N линейно раздельны и на одной стороне плоские, граница ранга распространяется на пересечение с M'.

LaTeX

$$Module.rank R (Subtype (fun x => SetLike.instMembership.mem (Submodule.instMin.min M N) x)) ≤ 1$$

Lean4

/-- If `M` and `N` are linearly disjoint, `M'` and `N'` are submodules of `M` and `N`,
respectively, such that `M` and `N'` are flat, then `M'` and `N'` are also linearly disjoint. -/
theorem of_le_of_flat_left (H : M.LinearDisjoint N) {M' N' : Submodule R S} (hm : M' ≤ M) (hn : N' ≤ N)
    [Module.Flat R M] [Module.Flat R N'] : M'.LinearDisjoint N' :=
  (H.of_le_right_of_flat hn).of_le_left_of_flat hm

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