Theorems · Theorem · nonassociative algebras

RootPairing.posRootForm_posForm_apply_apply

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (P : RootPairing ι R M N) (S : Type u_5)
  [inst_5 : CommRing S] [inst_6 : LinearOrder S] [inst_7 : IsStrictOrderedRing S] [inst_8 : Algebra S R]
  [inst_9 : FaithfulSMul S R] [inst_10 : Module S M] [inst_11 : IsScalarTower S R M] [inst_12 : P.IsValuedIn S]
  [inst_13 : Fintype ι] (x y : ↥(P.rootSpan S)),
  ((P.posRootForm S).posForm x) y = ∑ i, (P.coroot'In S i) x * (P.coroot'In S i) y

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Cited by

1 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommRingLinearOrderIsStrictOrderedRingAlgebraFaithfulSMulModuleIsScalarTowerRootPairing.IsValuedInFintype