Theorems · Theorem · linear algebra
LinearMap.finite_of_det_ne_one
∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {f : M →ₗ[R] M},
LinearMap.det f ≠ 1 → Module.Finite R MIf a linear map has determinant different from 1, then the space is finite-dimensional.
- Defined in
- Mathlib.LinearAlgebra.Determinant
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- MonoidHomstatement · cited by 3,629
- Module.Basisproof · cited by 1,477
- Module.Finitestatement · cited by 1,032
- LinearMap.detstatement and proof · cited by 127
- Module.Finite.of_basisproof · cited by 22
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.bot_lt_ker_of_det_eq_zeroproof · cited by 3
- LinearEquiv.transvection.det_eq_oneproof · cited by 0