English
The direction of the infimum over a family is bounded above by the infimum of the individual directions.
Русский
Направление iInf семейства подпро пространств ограничено сверху инфимом отдельных направлений.
LaTeX
$$direction (iInf s) ≤ ⨅ i, (s i).direction$$
Lean4
theorem direction_sInf_of_mem (t : Set (AffineSubspace k P)) (p : P) (h : ∀ s ∈ t, p ∈ s) :
direction (sInf t) = ⨅ s ∈ t, s.direction :=
by
apply (direction_sInf t).antisymm
intro v hv
rw [← vadd_mem_iff_mem_direction v ((mem_sInf_iff p t).mpr h), mem_sInf_iff]
intro s hs
rw [vadd_mem_iff_mem_direction v (h s hs)]
simp only [Submodule.mem_iInf] at hv
exact hv s hs