Theorems · Theorem · nonassociative algebras
RootPairing.Base.cartanMatrix_apply_eq_zero_iff_pairing
∀ {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] {i j : ↥b.support}, b.cartanMatrix i j = 0 ↔ P.pairing ↑i ↑j = 0- Cited by
- 1 results in Mathlib
- Foundations
- Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- 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
- Algebra.algebraMapproof · cited by 4,706
- Matrixproof · cited by 4,303
- map_zeroproof · cited by 1,614
- CharZerostatement and proof · cited by 932
- RootPairingstatement and proof · cited by 710
- FaithfulSMul.algebraMap_injectiveproof · cited by 198
- RootPairing.IsCrystallographicstatement and proof · cited by 160
Cited by1
Results whose statement or proof uses this declaration.
- RootPairing.Base.induction_on_cartanMatrixproof · cited by 0