Theorems · Theorem · linear algebra
Module.exists_set_linearIndependent_of_lt_rank
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {c : Cardinal.{v}},
c < Module.rank R M → ∃ s, Cardinal.mk ↑s = c ∧ LinearIndepOn R id s- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Set.Elemstatement and proof · cited by 7,166
- Cardinalstatement and proof · cited by 2,598
- Cardinal.mkstatement and proof · cited by 942
- Module.rankstatement and proof · cited by 496
- LinearIndepOnstatement and proof · cited by 211
- Cardinal.lift_idproof · cited by 163
- Cardinal.lift_ltproof · cited by 39
- Module.exists_set_linearIndependent_of_lt_lift_rankproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- exists_finset_linearIndependent_of_le_rankproof · cited by 3
- exists_set_linearIndependent_of_lt_rankproof · cited by 0