Theorems · Theorem · nonassociative algebras
RootPairing.pairingIn_eq_zero_iff
∀ {ι : Type u_1} {R : Type u_2} {M : Type u_4} {N : Type u_5} [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) {S : Type u_7}
[inst_5 : CommRing S] [inst_6 : Algebra S R] [FaithfulSMul S R] [inst_8 : P.IsValuedIn S] [IsDomain R]
[Module.IsTorsionFree R M] [NeZero 2] {i j : ι}, P.pairingIn S i j = 0 ↔ P.pairingIn S j i = 0- Cited by
- 4 results in Mathlib
- Foundations
- Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Algebrastatement and proof · cited by 11,388
- IsDomainstatement and proof · cited by 2,196
- RootPairingstatement and proof · cited by 710
- Module.IsTorsionFreestatement and proof · cited by 600
- FaithfulSMulstatement and proof · cited by 340
- RootPairing.IsValuedInstatement and proof · cited by 108
- RootPairing.pairingInstatement and proof · cited by 100
- RootPairing.algebraMap_pairingInproof · cited by 41
- FaithfulSMul.algebraMap_eq_zero_iffproof · cited by 8
Cited by4
Results whose statement or proof uses this declaration.
- RootPairing.chainBotCoeff_if_one_zeroproof · cited by 2
- RootPairing.pairingIn_le_zero_iffproof · cited by 0
- RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_neproof · cited by 0
- RootPairing.GeckConstruction.coe_genWeightSpace_zero_eq_span_range_uproof · cited by 0