Theorems · Theorem · linear algebra

Module.Basis.adjustToOrientation_apply_eq_or_eq_neg

∀ {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 : Fintype ι] [inst_6 : DecidableEq ι]
  [inst_7 : Nonempty ι] (e : Module.Basis ι R M) (x : Orientation R M ι) (i : ι),
  (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i

Every basis vector from adjust_to_orientation is either that from the original basis or its negation.

Defined in

Mathlib.LinearAlgebra.Orientation

Cited by

2 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRingAddCommGroupModuleFintypeDecidableEqNonempty