English
Abbreviated definition: given a linearly ordered index type I, a function v: I → M, and a Finset s ⊆ I with card s = n, ιMulti_family(R,n)(v)(s) equals the exterior product of v(i) over i in s, arranged by the order on s.
Русский
Упрощение определения: given линейно упорядоченный множество индексов I, функция v: I → M и конечный подмножество s ⊆ I размера n, ιMulti_family(R,n)(v)(s) равно внешнему произведению v(i) по i ∈ s в порядке, заданном s.
LaTeX
$$$ \\iotaMulti\\_family(R)(n)(v)(s) = \\iotaMulti(R,n)\\big(i \\mapsto v(\\mathrm{orderIsoOfFin} _ s.prop i)\\big). $$$
Lean4
theorem toTrivSqZeroExt_comp_map [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N] [IsCentralScalar R N]
(f : M →ₗ[R] N) : toTrivSqZeroExt.comp (map f) = (TrivSqZeroExt.map f).comp toTrivSqZeroExt :=
by
apply hom_ext
apply LinearMap.ext
simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, Function.comp_apply, AlgHom.toLinearMap_apply, map_apply_ι,
toTrivSqZeroExt_ι, TrivSqZeroExt.map_inr, forall_const]
