Theorems · Theorem · nonassociative algebras

RootPairing.EmbeddedG2.pairingIn_shortAddLong_left

∀ {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)
  [inst_5 : P.EmbeddedG2] [Finite ι] [CharZero R] [IsDomain R] (i : ι),
  P.pairingIn ℤ (RootPairing.EmbeddedG2.shortAddLong P) i =
    P.pairingIn ℤ (RootPairing.EmbeddedG2.short P) i + P.pairingIn ℤ (RootPairing.EmbeddedG2.long P) i

Defined in: Mathlib.LinearAlgebra.RootSystem.Finite.G2
Cited by: 1 results in Mathlib
Foundations: Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module AddCommGroup Module RootPairing.EmbeddedG2 Finite CharZero IsDomain

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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
Finitestatement and proof · cited by 3,029
IsDomainstatement and proof · cited by 2,196
CharZerostatement and proof · cited by 932
RootPairingstatement and proof · cited by 710
RootPairing.pairingInstatement and proof · cited by 100
RootPairing.EmbeddedG2statement and proof · cited by 35
RootPairing.EmbeddedG2.longstatement and proof · cited by 28
RootPairing.EmbeddedG2.shortstatement and proof · cited by 28
RootPairing.EmbeddedG2.shortAddLongstatement · cited by 5

Cited by1

Results whose statement or proof uses this declaration.

RootPairing.EmbeddedG2.isOrthogonal_short_and_longproof · cited by 2