English
In a commutative diagram with an exact top row 0 → X1 → X2 → X3 and bottom maps Sum.inl and Sum.inr, if v spans X1 and w spans X3, then u spans X2; more precisely, the span generated by the images of Sum.inl and Sum.inr under u are disjoint in X2.
Русский
В диаграмме 0 → X1 → X2 → X3, если v покрывает X1, w покрывает X3, то u покрывает X2; точнее, подмодули, генерируемые im(u∘Sum.inl) и im(u∘Sum.inr) в X2, являются взаимоисключающими.
LaTeX
$$$\mathrm{Disjoint}(\mathrm{span}_R(\mathrm{range}(u\circ\mathrm{Sum.inl})),\ \mathrm{span}_R(\mathrm{range}(u\circ\mathrm{Sum.inr})))$$$
Lean4
theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl))) (span R (range (u ∘ Sum.inr))) :=
by
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.coe_range, span_eq (LinearMap.range S.f.hom), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
