Theorems · Theorem · nonassociative algebras

RootPairing.Base.apply_mem_range_root_of_cartanMatrixEq

∀ {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] [CharZero R] [IsDomain R] [Finite ι]
  {ι₂ : Type u_6} {M₂ : Type u_7} {N₂ : Type u_8} [inst_8 : AddCommGroup M₂] [inst_9 : Module R M₂]
  [inst_10 : AddCommGroup N₂] [inst_11 : Module R N₂] {P : RootPairing ι R M N} [P.IsRootSystem]
  [inst_13 : P.IsCrystallographic] [P.IsReduced] (b : P.Base) {P₂ : RootPairing ι₂ R M₂ N₂}
  [inst_15 : P₂.IsCrystallographic] (b₂ : P₂.Base) (e : ↥b.support ≃ ↥b₂.support) (f : M ≃ₗ[R] M₂),
  (∀ (i : ↥b.support), f (P.root ↑i) = P₂.root ↑(e i)) →
    ∀ m ∈ Set.range ⇑P.root,
      (∀ (i j : ↥b.support), b₂.cartanMatrix (e i) (e j) = b.cartanMatrix i j) → f m ∈ Set.range ⇑P₂.root

Defined in

Mathlib.LinearAlgebra.RootSystem.CartanMatrix

Cited by

0 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCharZeroIsDomainFiniteAddCommGroupModuleAddCommGroupModuleRootPairing.IsRootSystemRootPairing.IsCrystallographicRootPairing.IsReducedRootPairing.IsCrystallographic