Theorems · Theorem · nonassociative algebras

RootPairing.finrank_corootSpan_eq

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Fintype ι] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup N] [inst_3 : CommRing R] [inst_4 : Module R M] [inst_5 : Module R N] (P : RootPairing ι R M N)
  (S : Type u_5) [inst_6 : CommRing S] [IsDomain R] [IsDomain S] [inst_9 : Algebra S R] [FaithfulSMul S R]
  [P.IsValuedIn S] [inst_12 : Module S M] [IsScalarTower S R M] [inst_14 : Module S N] [IsScalarTower S R N]
  [P.IsAnisotropic], Module.finrank S ↥(P.corootSpan S) = Module.finrank S ↥(P.rootSpan S)

Defined in: Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate
Cited by: 1 results in Mathlib
Foundations: Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Fintype AddCommGroup AddCommGroup CommRing Module Module CommRing IsDomain IsDomain Algebra FaithfulSMul RootPairing.IsValuedIn Module IsScalarTower Module IsScalarTower RootPairing.IsAnisotropic

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Cites17

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
Fintypestatement and proof · cited by 7,736
Submodulestatement · cited by 7,192
IsScalarTowerstatement and proof · cited by 3,896
IsDomainstatement and proof · cited by 2,196
le_antisymmproof · cited by 2,068
Module.finrankstatement · cited by 1,770
RootPairingstatement and proof · cited by 710
FaithfulSMulstatement and proof · cited by 340

Cited by1

Results whose statement or proof uses this declaration.

RootPairing.polarizationIn_Injectiveproof · cited by 1