prodProdProdComm — Mathlib

English

Reiteration of the top-product criterion in the lattice of submodules.

Русский

Повторение условия верхнего произведения в решётке подмодулей.

LaTeX

$$$p_1.\mathrm{prod} p_2 = \top \iff p_1 = \top \land p_2 = \top$$$

Lean4

/-- Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. -/
@[simps apply]
def prodProdProdComm : ((M × M₂) × M₃ × M₄) ≃ₗ[R] (M × M₃) × M₂ × M₄ :=
  {
    AddEquiv.prodProdProdComm M M₂ M₃
      M₄ with
    toFun := fun mnmn => ((mnmn.1.1, mnmn.2.1), (mnmn.1.2, mnmn.2.2))
    invFun := fun mmnn => ((mmnn.1.1, mmnn.2.1), (mmnn.1.2, mmnn.2.2))
    map_smul' := fun _c _mnmn => rfl }

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