Theorems · Theorem · linear algebra

Orientation.eq_or_eq_neg_of_isEmpty

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] {ι : Type u_3} [inst_5 : IsEmpty ι] (o : Orientation R M ι),
  o = positiveOrientation ∨ o = -positiveOrientation

A module M over a linearly ordered commutative ring has precisely two "orientations" with respect to an empty index type. (Note that these are only orientations of M of in the conventional mathematical sense if M is zero-dimensional.)

Defined in

Mathlib.LinearAlgebra.Orientation

Cited by

6 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRingAddCommGroupModuleIsEmpty