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[Merged by Bors] - chore(LinearAlgebra/ExteriorPower/Basis): avoid use of IsWellFounded.wellOrderExtension #37078

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[Merged by Bors] - chore(LinearAlgebra/ExteriorPower/Basis): avoid use of IsWellFounded.wellOrderExtension #37078

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well well well
vihdzp Mar 24, 2026
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Merge branch 'master' into well
vihdzp Apr 2, 2026
396568b
fixes
vihdzp Apr 2, 2026
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8 changes: 4 additions & 4 deletions Mathlib/LinearAlgebra/ExteriorPower/Basis.lean
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Expand Up @@ -7,7 +7,6 @@ module

public import Mathlib.LinearAlgebra.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.Order.Extension.Well
public import Mathlib.RingTheory.Finiteness.Subalgebra
public import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition

Expand Down Expand Up @@ -148,20 +147,21 @@ lemma basis_repr {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard

/-! ### Freeness and dimension of `⋀[R]^n M`. -/

open Classical in
/-- If `M` is a free module, then so is its `n`th exterior power. -/
instance instFree [Module.Free R M] : Module.Free R (⋀[R]^n M) :=
have ⟨I, b⟩ := Module.Free.exists_basis R M
letI : LinearOrder I := IsWellFounded.wellOrderExtension emptyWf.rel
letI : LinearOrder I := linearOrderOfSTO WellOrderingRel
Module.Free.of_basis (b.exteriorPower n)

variable [Nontrivial R]

open Classical in
/-- If `R` is non-trivial and `M` is finite free of rank `r`, then
the `n`th exterior power of `M` is of finrank `Nat.choose r n`. -/
lemma finrank_eq [Module.Free R M] [Module.Finite R M] :
Module.finrank R (⋀[R]^n M) = Nat.choose (Module.finrank R M) n := by
letI : LinearOrder (Module.Free.ChooseBasisIndex R M) :=
IsWellFounded.wellOrderExtension emptyWf.rel
letI : LinearOrder (Module.Free.ChooseBasisIndex R M) := linearOrderOfSTO WellOrderingRel
let B := (Module.Free.chooseBasis R M).exteriorPower n
rw [Module.finrank_eq_card_basis (Module.Free.chooseBasis R M), Module.finrank_eq_card_basis B,
Fintype.card_eq_nat_card, powersetCard.card, Fintype.card_eq_nat_card]
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