Theorems · Theorem · linear algebra
Submodule.rank_eq_zero
∀ {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] {S : Submodule R M}, Module.rank R ↥S = 0 ↔ S = ⊥See rank_subsingleton for the reason that Nontrivial R is needed.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Top.topproof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Cardinalstatement and proof · cited by 2,598
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- Module.rankstatement and proof · cited by 496
- Submodule.mem_topproof · cited by 58
- Submodule.eq_bot_iffproof · cited by 31
Cited by2
Results whose statement or proof uses this declaration.
- Submodule.finrank_eq_zeroproof · cited by 9
- Submodule.disjoint_ker_of_finrank_leproof · cited by 2