Theorems · Theorem · linear algebra
Module.le_rank_iff_exists_linearMap
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R] {n : ℕ},
↑n ≤ Module.rank R M ↔ ∃ f, Function.Injective ⇑f- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 3 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.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Cardinalstatement · cited by 2,598
- Nontrivialstatement and proof · cited by 2,416
- LinearMap.compproof · cited by 1,642
- LinearEquiv.symmproof · cited by 1,461
- LinearEquiv.toLinearMapproof · cited by 1,171
- LinearIndependentproof · cited by 560
Cited by3
Results whose statement or proof uses this declaration.
- Module.one_le_rank_iffproof · cited by 4
- CommSemiring.rank_selfproof · cited by 2
- strongRankCondition_iff_forall_rank_lt_aleph0proof · cited by 1