Theorems · Theorem · linear algebra

LinearEquiv.isReflexive_of_equiv_dual_of_isReflexive

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M),
  Module.IsReflexive R N

If N is in perfect pairing with M, then it is reflexive.

Defined in: Mathlib.LinearAlgebra.PerfectPairing.Basic
Cited by: 4 results in Mathlib
Foundations: Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module AddCommGroup Module Module.IsReflexive

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Cites16

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapproof · cited by 10,215
LinearEquivstatement and proof · cited by 3,317
LinearEquiv.symmproof · cited by 1,461
Function.Bijectiveproof · cited by 863
Module.Dualstatement and proof · cited by 583
LinearEquiv.transproof · cited by 298
Module.IsReflexivestatement and proof · cited by 58

Cited by4

Results whose statement or proof uses this declaration.

LinearEquiv.flip_flipstatement · cited by 3
Submodule.map_dualCoannihilator_linearEquiv_flipproof · cited by 1
Submodule.dualAnnihilator_map_linearEquiv_flip_symmproof · cited by 0
Submodule.map_dualAnnihilator_linearEquiv_flip_symmproof · cited by 0