Theorems · Theorem · linear algebra
LinearMap.det_eq_one_of_finrank_eq_zero
∀ {𝕜 : Type u_7} [inst : Field 𝕜] {M : Type u_8} [inst_1 : AddCommGroup M] [inst_2 : Module 𝕜 M],
Module.finrank 𝕜 M = 0 → ∀ (f : M →ₗ[𝕜] M), LinearMap.det f = 1- Defined in
- Mathlib.LinearAlgebra.Determinant
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Finsetproof · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- Fieldstatement and proof · cited by 7,404
- MonoidHomstatement · cited by 3,629
- Module.finrankstatement and proof · cited by 1,770
- Module.Basisproof · cited by 1,477
- IsEmptyproof · cited by 759
- LinearMap.detstatement · cited by 127
Cited by3
Results whose statement or proof uses this declaration.
- Orientation.map_eq_iff_det_posproof · cited by 3
- Submodule.det_reflectionproof · cited by 1
- Orientation.map_eq_neg_iff_det_negproof · cited by 0