Theorems · Theorem · nonassociative algebras
RootPairing.root_sub_root_mem_of_pairingIn_pos
∀ {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) [Finite ι]
[CharZero R] [inst_7 : P.IsCrystallographic] {i j : ι} [IsDomain R],
0 < P.pairingIn ℤ i j → i ≠ j → P.root i - P.root j ∈ Set.range ⇑P.root- Cited by
- 7 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites51
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
- mul_oneproof · cited by 3,885
- Finitestatement and proof · cited by 3,029
- one_mulproof · cited by 2,841
- IsDomainstatement and proof · cited by 2,196
- LT.lt.ne'proof · cited by 1,417
- one_smulproof · cited by 1,374
Cited by7
Results whose statement or proof uses this declaration.
- RootPairing.setOfPred_root_add_zsmul_eq_Icc_of_linearIndependentproof · cited by 5
- RootPairing.Base.root_sub_root_mem_of_mem_of_memproof · cited by 3
- RootPairing.Base.IsPos.induction_on_addproof · cited by 2
- RootPairing.Base.pairingIn_le_zero_of_neproof · cited by 2
- RootPairing.root_add_root_mem_of_pairingIn_negproof · cited by 2
- RootPairing.baseOf_pairwise_pairing_le_zeroproof · cited by 1
- RootPairing.pairingIn_eq_zero_of_add_notMem_of_sub_notMemproof · cited by 1