Theorems · Theorem · linear algebra
Module.finrank_zero_of_subsingleton
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R]
[Subsingleton M], Module.finrank R M = 0A (finite-dimensional) space that is a subsingleton has zero finrank.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Cardinalproof · cited by 2,598
- Nontrivialstatement and proof · cited by 2,416
- Module.finrankstatement · cited by 1,770
- map_zeroproof · cited by 1,614
- Cardinal.toNatproof · cited by 153
- rank_subsingleton'proof · cited by 13
Cited by6
Results whose statement or proof uses this declaration.
- Module.nontrivial_of_finrank_posproof · cited by 7
- Module.rankAtStalk_eq_zero_iff_notMem_supportproof · cited by 2
- Module.rankAtStalk_eq_zero_of_subsingletonproof · cited by 2
- CategoryTheory.finrank_hom_simple_simple_le_oneproof · cited by 2
- MeasureTheory.measure_unitBall_eq_integral_div_gammaproof · cited by 1
- CategoryTheory.finrank_hom_simple_simple_eq_zero_of_not_isoproof · cited by 0