Theorems · Definition · nonassociative algebras

RootPairing.RootPositiveForm.recOn

{ι : Type u_1} →
  {R : Type u_2} →
    {S : Type u_3} →
      {M : Type u_4} →
        {N : Type u_5} →
          [inst : CommRing S] →
            [inst_1 : LinearOrder S] →
              [inst_2 : CommRing R] →
                [inst_3 : Algebra S R] →
                  [inst_4 : AddCommGroup M] →
                    [inst_5 : Module R M] →
                      [inst_6 : AddCommGroup N] →
                        [inst_7 : Module R N] →
                          {P : RootPairing ι R M N} →
                            [inst_8 : P.IsValuedIn S] →
                              {motive : RootPairing.RootPositiveForm S P → Sort u} →
                                (t : RootPairing.RootPositiveForm S P) →
                                  ((form : LinearMap.BilinForm R M) →
                                      (symm : LinearMap.IsSymm form) →
                                        (isOrthogonal_reflection :
                                            ∀ (i : ι), LinearMap.IsOrthogonal form ⇑(P.reflection i)) →
                                          (exists_eq :
                                              ∀ (i j : ι), ∃ s, (algebraMap S R) s = (form (P.root i)) (P.root j)) →
                                            (exists_pos_eq :
                                                ∀ (i : ι), ∃ s > 0, (algebraMap S R) s = (form (P.root i)) (P.root i)) →
                                              motive
                                                { form := form, symm := symm,
                                                  isOrthogonal_reflection := isOrthogonal_reflection,
                                                  exists_eq := exists_eq, exists_pos_eq := exists_pos_eq }) →
                                    motive t

Defined in

Mathlib.LinearAlgebra.RootSystem.RootPositive

Cited by

0 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderCommRingAlgebraAddCommGroupModuleAddCommGroupModuleRootPairing.IsValuedIn