Theorems · Theorem · linear algebra
rank_zero_iff_forall_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 ↔ ∀ (x : M), x = 0- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Cardinalstatement · cited by 2,598
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- Module.rankstatement · cited by 496
- exists_neproof · cited by 101
Cited by5
Results whose statement or proof uses this declaration.
- rank_zero_iffproof · cited by 2
- rank_pos_iff_exists_ne_zeroproof · cited by 2
- ModularForm.sturm_bound_levelOne_natproof · cited by 1
- ModularForm.sturm_bound_levelOneproof · cited by 0
- ModularForm.levelOne_odd_weight_rank_zeroproof · cited by 0