Theorems · Theorem · linear algebra
Module.finrank_pos
∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [StrongRankCondition R]
[Module.Finite R M] [IsDomain R] [Module.IsTorsionFree R M] [h : Nontrivial M], 0 < Module.finrank R MA nontrivial finite-dimensional space has positive finrank.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 62 results in Mathlib
- Foundations
- Depth 102 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.
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Nontrivialstatement and proof · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Module.finrankstatement · cited by 1,770
- Module.Finitestatement and proof · cited by 1,032
- Module.IsTorsionFreestatement and proof · cited by 600
- StrongRankConditionstatement and proof · cited by 286
- Module.finrank_pos_iffproof · cited by 8
Cited by62
Results whose statement or proof uses this declaration.
- Ideal.inertiaDeg_posproof · cited by 8
- FiniteField.nonempty_algHom_of_finrank_dvdproof · cited by 3
- FiniteField.orderOf_frobeniusAlgHomproof · cited by 3
- Submodule.finrank_ltproof · cited by 3
- IntermediateField.exists_lt_finrank_of_infinite_dimensionalproof · cited by 3
- InnerProductSpace.volume_ballproof · cited by 3
- IntermediateField.LinearDisjoint.finrank_left_eq_finrankproof · cited by 2
- FiniteField.finrank_zmod_extensionproof · cited by 2
- NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_of_norm_leproof · cited by 2
- integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrableproof · cited by 2
- MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProdproof · cited by 2