Theorems · Theorem · linear algebra
LinearEquiv.det_mul_det_symm
∀ {M : Type u_2} [inst : AddCommGroup M] {A : Type u_5} [inst_1 : CommRing A] [inst_2 : Module A M] (f : M ≃ₗ[A] M),
LinearMap.det ↑f * LinearMap.det ↑f.symm = 1The determinants of a LinearEquiv and its inverse multiply to 1.
- Defined in
- Mathlib.LinearAlgebra.Determinant
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommGroupCommRingModule
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
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- MonoidHomstatement · cited by 3,629
- LinearEquivstatement and proof · cited by 3,317
- LinearEquiv.symmstatement and proof · cited by 1,461
- LinearEquiv.toLinearMapstatement and proof · cited by 1,171
- LinearMap.detstatement and proof · cited by 127
- LinearEquiv.symm_trans_selfproof · cited by 18
Cited by2
Results whose statement or proof uses this declaration.
- LinearEquiv.isUnit_det'proof · cited by 3
- Module.Basis.det_invproof · cited by 2