Theorems · Theorem · linear algebra

Submodule.one_le_finrank_iff

∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsDomain R]
  [Module.IsTorsionFree R M] [StrongRankCondition R] {S : Submodule R M} [Module.Finite R ↥S],
  1 ≤ Module.finrank R ↥S ↔ S ≠ ⊥

Defined in: Mathlib.LinearAlgebra.Dimension.Finite
Cited by: 2 results in Mathlib
Foundations: Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Ring AddCommGroup Module IsDomain Module.IsTorsionFree StrongRankCondition Module.Finite

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Cites11

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Modulestatement and proof · cited by 20,661
AddCommGroupstatement and proof · cited by 12,871
Ringstatement and proof · cited by 7,463
Submodulestatement and proof · cited by 7,192
Bot.botstatement and proof · cited by 4,720
IsDomainstatement and proof · cited by 2,196
Module.finrankstatement · cited by 1,770
Module.Finitestatement and proof · cited by 1,032
Module.IsTorsionFreestatement and proof · cited by 600
StrongRankConditionstatement and proof · cited by 286
Submodule.finrank_eq_zeroproof · cited by 9

Cited by2

Results whose statement or proof uses this declaration.

Module.Dual.isCompl_ker_of_disjoint_of_ne_botproof · cited by 2
LinearMap.ker_ne_bot_of_finrank_ltproof · cited by 1