Theorems · Theorem · linear algebra
Submodule.finrank_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] [StrongRankCondition R] {S : Submodule R M} [Module.Finite R ↥S],
Module.finrank R ↥S = 0 ↔ S = ⊥- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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 · cited by 4,720
- Cardinalproof · cited by 2,598
- IsDomainstatement and proof · cited by 2,196
- Nat.cast_zeroproof · cited by 1,870
- Module.finrankstatement and proof · 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
Cited by9
Results whose statement or proof uses this declaration.
- Submodule.finrank_sup_span_singletonproof · cited by 2
- LinearMap.continuous_of_isClosed_kerproof · cited by 2
- Submodule.one_le_finrank_iffproof · cited by 2
- Submodule.isAtom_iff_finrank_eq_oneproof · cited by 1
- LinearMap.injective_iff_surjective_of_finrank_eq_finrankproof · cited by 1
- Module.exists_basis_of_basis_baseChangeproof · cited by 1
- Module.End.iSup_maxGenEigenspace_eq_topproof · cited by 1
- LinearIsometryEquiv.reflections_generate_dim_auxproof · cited by 1
- LinearMap.singularValues_eq_zero_iffproof · cited by 0