Theorems · Theorem · nonassociative algebras
RootPairing.coxeterWeightIn_eq_four_iff_not_linearIndependent
∀ {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)
{i j : ι} [Finite ι] [CharZero R] [IsDomain R] [Module.IsTorsionFree R M] [inst_9 : CommRing S]
[inst_10 : Algebra S R] [FaithfulSMul S R] [inst_12 : P.IsValuedIn S],
P.coxeterWeightIn S i j = 4 ↔ ¬LinearIndependent R ![P.root i, P.root j]- Defined in
- Mathlib.LinearAlgebra.RootSystem.Reduced
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Algebrastatement and proof · cited by 11,388
- Finitestatement and proof · cited by 3,029
- IsDomainstatement and proof · cited by 2,196
- Function.Embeddingstatement · cited by 988
- CharZerostatement and proof · cited by 932
- Matrix.vecConsstatement · cited by 852
- Matrix.vecEmptystatement · cited by 832
- RootPairingstatement and proof · cited by 710
Cited by2
Results whose statement or proof uses this declaration.
- RootPairing.root_sub_root_mem_of_pairingIn_posproof · cited by 7
- RootPairing.coxeterWeightIn_ne_fourproof · cited by 1