Theorems · Theorem · linear algebra
Subalgebra.rank_eq_one_iff
∀ {F : Type u_1} {E : Type u_2} [inst : CommRing F] [StrongRankCondition F] [inst_2 : Ring E] [inst_3 : Algebra F E]
{S : Subalgebra F E} [Nontrivial E] [Module.Free F ↥S], Module.rank F ↥S = 1 ↔ S = ⊥- Cited by
- 3 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites37
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Submoduleproof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Cardinalstatement and proof · cited by 2,598
- Nontrivialstatement and proof · cited by 2,416
- le_antisymmproof · cited by 2,068
- Module.Basisproof · cited by 1,477
- Subalgebrastatement and proof · cited by 1,353
- LinearMap.rangeproof · cited by 893
Cited by3
Results whose statement or proof uses this declaration.
- IntermediateField.rank_eq_one_iffproof · cited by 3
- Subalgebra.bot_eq_top_iff_rank_eq_oneproof · cited by 2
- Subalgebra.finrank_eq_one_iffproof · cited by 2