Theorems · Theorem · linear algebra
Submodule.bot_eq_top_of_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], Module.rank R M = 0 → ⊥ = ⊤- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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.topstatement and proof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Submodulestatement · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- Cardinalstatement · 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
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.rank_eq_zeroproof · cited by 2