Theorems · Theorem · nonassociative algebras
RootPairing.Base.cartanMatrix_le_zero_of_ne
∀ {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} (b : P.Base)
[inst_5 : P.IsCrystallographic] [CharZero R] [IsDomain R] [Finite ι] (i j : ↥b.support),
i ≠ j → b.cartanMatrix i j ≤ 0- Cited by
- 3 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- Finsetstatement · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- Finitestatement and proof · cited by 3,029
- IsDomainstatement and proof · cited by 2,196
- CharZerostatement and proof · cited by 932
- RootPairingstatement and proof · cited by 710
- RootPairing.IsCrystallographicstatement and proof · cited by 160
- RootPairing.Basestatement and proof · cited by 148
- RootPairing.Base.supportstatement and proof · cited by 139
- RootPairing.Base.cartanMatrixstatement · cited by 32
Cited by3
Results whose statement or proof uses this declaration.
- RootPairing.Base.abs_cartanMatrix_applyproof · cited by 1
- RootPairing.Base.cartanMatrix_mem_of_neproof · cited by 0
- RootPairing.Base.det_four_sub_cartanMatrix_ne_zeroproof · cited by 0