Theorems · Theorem · nonassociative algebras
RootPairing.neg_root_mem
∀ {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) (i : ι),
-P.root i ∈ Set.range ⇑P.root- Defined in
- Mathlib.LinearAlgebra.RootSystem.Defs
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Setstatement · cited by 53,352
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Set.rangestatement · cited by 4,705
- Function.Embeddingstatement · cited by 988
- RootPairingstatement and proof · cited by 710
- RootPairing.rootstatement and proof · cited by 326
- RootPairing.reflectionPermproof · cited by 87
- RootPairing.root_reflectionPermproof · cited by 57
- RootPairing.reflection_apply_selfproof · cited by 34
Cited by2
Results whose statement or proof uses this declaration.
- RootPairing.root_sub_root_mem_of_pairingIn_posproof · cited by 7
- RootPairing.neg_coroot_memproof · cited by 0