Theorems · Theorem · linear algebra
finrank_eq_one_iff_of_nonzero
∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] (v : V),
v ≠ 0 → (Module.finrank K V = 1 ↔ K ∙ v = ⊤)A vector space with a nonzero vector v has dimension 1 iff v spans.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Top.topstatement and proof · cited by 9,680
- Submodulestatement · cited by 7,192
- Module.finrankstatement and proof · cited by 1,770
- Submodule.spanstatement and proof · cited by 1,504
- DivisionRingstatement and proof · cited by 1,062
- Matrix.vecEmptyproof · cited by 832
- Set.union_emptyproof · cited by 78
- Module.Basis.span_eqproof · cited by 44
- Module.finrank_eq_card_basisproof · cited by 40
Cited by3
Results whose statement or proof uses this declaration.
- finrank_eq_one_iff_of_nonzero'proof · cited by 11
- IsSimpleModule.finrank_eq_one_of_isMulCommutativeproof · cited by 1
- WittVector.isocrystal_classificationproof · cited by 0