Theorems · Theorem · linear algebra
Module.nontrivial_of_finrank_pos
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R],
0 < Module.finrank R M → Nontrivial MA finite-dimensional space is nontrivial if it has positive finrank.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Nontrivialstatement and proof · cited by 2,416
- Module.finrankstatement and proof · cited by 1,770
- Eq.leproof · cited by 605
- Module.finrank_zero_of_subsingletonproof · cited by 6
Cited by7
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.addHaar_smulproof · cited by 13
- Module.nontrivial_of_finrank_eq_succproof · cited by 4
- InnerProductSpace.volume_ball_of_dim_oddproof · cited by 2
- MeasureTheory.integrableOn_ball_of_norm_le_rpowproof · cited by 1
- Subalgebra.isSimpleOrder_of_finrankproof · cited by 1
- LinearIsometryEquiv.reflections_generate_dim_auxproof · cited by 1
- InnerProductSpace.volume_closedBall_of_dim_oddproof · cited by 0