Theorems · Theorem · nonassociative algebras
RootPairing.chainBotCoeff_sub_chainTopCoeff
∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Finite ι] [inst_1 : CommRing R]
[inst_2 : CharZero R] [inst_3 : IsDomain 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.IsCrystallographic] {i j : ι},
LinearIndependent R ![P.root i, P.root j] →
↑(RootPairing.chainBotCoeff i j) - ↑(RootPairing.chainTopCoeff i j) = P.pairingIn ℤ j i- Defined in
- Mathlib.LinearAlgebra.RootSystem.Chain
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites47
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · 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
- Algebra.algebraMapproof · cited by 4,706
- Set.rangeproof · cited by 4,705
- Finitestatement and proof · cited by 3,029
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- IsDomainstatement and proof · cited by 2,196
- le_antisymmproof · cited by 2,068
- Function.Embeddingstatement · cited by 988
Cited by4
Results whose statement or proof uses this declaration.
- RootPairing.chainBotCoeff_if_one_zeroproof · cited by 2
- RootPairing.chainBotCoeff_add_chainTopCoeff_eq_pairingIn_chainTopIdxproof · cited by 2
- RootPairing.chainTopCoeff_sub_chainBotCoeffproof · cited by 1
- RootPairing.pairingIn_le_zero_of_root_add_memproof · cited by 1