Theorems · Theorem · linear algebra

Module.exists_set_linearIndependent_of_lt_lift_rank

∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {c : Cardinal.{w}},
  Cardinal.lift.{v, w} c < Cardinal.lift.{w, v} (Module.rank R M) →
    ∃ s, Cardinal.lift.{w, v} (Cardinal.mk ↑s) = Cardinal.lift.{v, w} c ∧ LinearIndepOn R id s

Defined in: Mathlib.LinearAlgebra.Dimension.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module

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Cites16

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
LT.lt.leproof · cited by 2,189
Cardinal.mkstatement and proof · cited by 942
Cardinal.liftstatement and proof · cited by 583
Module.rankstatement and proof · cited by 496
LinearIndepOnstatement and proof · cited by 211
Module.rank_defproof · cited by 21

Cited by2

Results whose statement or proof uses this declaration.

Module.exists_set_linearIndependent_of_lt_rankproof · cited by 2
Module.Free.exists_linearMap_injective_of_lift_rank_ltproof · cited by 1