Theorems · Theorem · linear algebra

LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMap

∀ {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]
  [inst_3 : AddCommGroup M] [inst_4 : AddCommGroup N] [inst_5 : Module L M] [inst_6 : Module L N] [inst_7 : Module K M]
  [inst_8 : Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) [p.IsPerfPair] (M' : Submodule K M)
  (N' : Submodule K N),
  Submodule.span L ↑M' = ⊤ →
    Submodule.span L ↑N' = ⊤ →
      (∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range) → ∃ n b b', ∀ (i : Fin n), b i = ↑(b' i)

If a perfect pairing over a field L takes values in a subfield K along two K-subspaces whose L span is full, then these subspaces induce a K-structure in the sense of [Algebra I, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989].

Defined in

Mathlib.LinearAlgebra.PerfectPairing.Restrict

Cited by

1 results in Mathlib

Foundations

Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraAddCommGroupAddCommGroupModuleModuleModuleModuleIsScalarTowerLinearMap.IsPerfPair