Theorems · Theorem · linear algebra

Module.finrank_eq_zero_iff_of_free

∀ (R : Type u) (M : Type v) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
  [Module.Free R M] [Module.Finite R M], Module.finrank R M = 0 ↔ Subsingleton M

Defined in: Mathlib.LinearAlgebra.Dimension.StrongRankCondition
Cited by: 4 results in Mathlib
Foundations: Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module StrongRankCondition Module.Free Module.Finite

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Cites11

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
Semiringstatement and proof · cited by 13,802
AddCommMonoidstatement and proof · cited by 12,281
Module.finrankstatement · cited by 1,770
Module.Finitestatement and proof · cited by 1,032
Module.Freestatement and proof · cited by 597
Cardinal.aleph0proof · cited by 521
Module.rankproof · cited by 496
not_leproof · cited by 328
StrongRankConditionstatement and proof · cited by 286
Module.rank_lt_aleph0proof · cited by 23

Cited by4

Results whose statement or proof uses this declaration.

Module.finrank_eq_zero_of_subsingletonproof · cited by 8
LinearMap.IsIdempotentElem.trace_eq_zero_iffproof · cited by 1
LinearMap.transvection.eq_id_of_finrank_le_oneproof · cited by 0
SpecialLinearGroup.centerCongr_toLin_equiv_trans_centerEquivRootsOfUnity_eqproof · cited by 0