Theorems · Theorem · nonassociative algebras
RootPairing.linearIndependent_of_add_mem_range_root
∀ {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) [CharZero R]
[IsAddTorsionFree M] [P.IsReduced] {i j : ι},
P.root i + P.root j ∈ Set.range ⇑P.root → LinearIndependent R ![P.root i, P.root j]- Defined in
- Mathlib.LinearAlgebra.RootSystem.Reduced
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
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 and proof · cited by 4,705
- 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
- LinearIndependentstatement · cited by 560
Cited by2
Results whose statement or proof uses this declaration.
- RootPairing.linearIndependent_of_add_mem_range_root'proof · cited by 3
- RootPairing.linearIndependent_of_sub_mem_range_rootproof · cited by 1