English
The span of the orbit of x under G lies in the invariant submodule corresponding to the action by g.
Русский
Образ порожденный орбитой x по действию G лежит в инвариантной подмодуле, соответствующей действию g.
LaTeX
$$x : M, g : G \;\Rightarrow\; \text{span}_R (MulAction.orbit G x) \in invtSubmodule (DistribMulAction.toLinearMap R M g)$$
Lean4
theorem span_orbit_mem_invtSubmodule {G : Type*} [Monoid G] [DistribMulAction G M] [SMulCommClass G R M] (x : M)
(g : G) : span R (MulAction.orbit G x) ∈ invtSubmodule (DistribMulAction.toLinearMap R M g) :=
by
rw [mem_invtSubmodule, Submodule.span_le, Submodule.comap_coe]
intro y hy
simp only [Set.mem_preimage, DistribMulAction.toLinearMap_apply, SetLike.mem_coe]
exact Submodule.subset_span <| MulAction.mem_orbit_of_mem_orbit g hy