| import Mathlib.LinearAlgebra.Pi | |
| namespace OAI | |
| noncomputable section | |
| namespace LittleFinitistic.FinitePiSupport | |
| universe u v w | |
| variable {R : Type u} [Ring R] {ι : Type v} [DecidableEq ι] | |
| variable (V : ι → Type w) [∀ i, AddCommGroup (V i)] [∀ i, Module R (V i)] | |
| variable (i₀ : ι) (hV : ∀ i, i ≠ i₀ → Subsingleton (V i)) | |
| /-- A product with all but one coordinate zero is exactly its remaining coordinate. -/ | |
| def equivalence : (∀ i, V i) ≃ₗ[R] V i₀ := | |
| LinearEquiv.ofBijective (LinearMap.proj i₀) ⟨by | |
| intro x y h | |
| funext i | |
| by_cases hi : i = i₀ | |
| · subst i | |
| exact h | |
| · exact (hV i hi).elim _ _, by | |
| intro x | |
| refine ⟨fun i => if h : i = i₀ then h.symm ▸ x else 0, ?_⟩ | |
| simp⟩ | |
| @[simp] theorem equivalence_apply (x : ∀ i, V i) : equivalence (R := R) V i₀ hV x = x i₀ := rfl | |
| end LittleFinitistic.FinitePiSupport | |
| end | |
| end OAI | |