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import Adam.Metadata
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Data.Real.Basic -- definiert `ℝ
import Mathlib.Data.Fin.VecNotation -- Importiert Matrix/Vektor-Notation
--import Mathlib.LinearAlgebra.FinSupp -- contains `top_le_span_range_iff_forall_exists_fun`
import Mathlib.Tactic.FinCases
import Mathlib.Algebra.BigOperators.Finsupp -- default?
import Mathlib.LinearAlgebra.Span
import Mathlib.Tactic.LibrarySearch
import Mathlib
Game "Adam"
World "Module"
Level 7
Title "Hülle"
Introduction
"
"
-- notation "ℝ²" => Fin 2 → ℝ
open Submodule Set Finsupp
open BigOperators -- Summen Notation
-- TODO: Why is this not in Mathlib?
lemma mem_span_of_mem {V K : Type _} [Field K] [AddCommMonoid V]
[Module K V] (M : Set V) {x : V} (h : x ∈ M) :
x ∈ span K M := by
rw [mem_span]
intro p hp
specialize hp h
assumption
Statement
"Für $x, y \\in M$, zeige dass $x + 2y$ in der $K$-linearen Hülle $\\langle M \\rangle$ liegt."
{V K : Type _} [Field K] [AddCommMonoid V] [Module K V] (M : Set V) {x y : V}
(h₁ : x ∈ M) (h₂ : y ∈ M) :
x + (2 : K) • y ∈ span K M := by
apply add_mem
apply mem_span_of_mem
assumption
apply smul_mem
apply mem_span_of_mem
assumption