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import TestGame.Metadata
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import Mathlib.Algebra.Module.Submodule.Lattice
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import Mathlib.Data.Real.Basic
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import Mathlib.LinearAlgebra.Span
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Game "TestGame"
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World "Module2"
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Level 9
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Title "Lineare Abbildung"
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Introduction
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"
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"
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universe u
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variable {K : Type u} [Field K]
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variable {ι : Type u} {V : ι → Type u}
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variable [∀i, AddCommMonoid (V i)] [∀i, Module K (V i)]
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/-
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Ein externes Summe von Vektorräumen schreibt man mit `\Pi\0`, also `Π₀ i, V i`.
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Das Suffix `_₀` wird in Mathlib häufig dafür verwendet "endlichen Support" zu bezeichnen.
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-/
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example : Module K (Π₀ i, V i) := inferInstance
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variable {U : Type u} [AddCommMonoid U] [Module K U]
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Statement
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"" : True := by
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sorry
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-- -- :(
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-- variable [decidable_eq ι]
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-- variable [Π (i : ι) (x : V i), decidable (x ≠ 0)]
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-- def my_sum_map (f : Π i, V i →ₗ[K] U) : (Π₀ i, V i) →ₗ[K] U :=
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-- { to_fun := λ x, x.sum (λ i, (f i)),
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-- map_add' :=
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-- begin
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-- intros,
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-- funext,
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-- sorry,
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-- end,
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-- map_smul' :=
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-- begin
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-- intros,
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-- funext,
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-- simp,
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-- sorry
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-- end }
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-- Statement
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-- "Sei `U` ein `K`-Vektorraum und `fᵢ : Vᵢ → U` eine Familie von `K`-lineare Abbildungen
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-- in `K`-Vektorräume. Dann gibt es genau eine Abbildung `f : (Π₀ i, V i) → U`, die mit
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-- allen kommutiert."
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-- (f : ∀ i, V i →ₗ[K] U) :
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-- ∃! (g : (Π₀ i, V i) →ₗ[K] U), (∀ i v, f i v = g (dfinsupp.single i v)) :=
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-- by
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-- let g := my_sum_map f,
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-- use g,
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-- constructor,
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-- { simp,
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-- intros,
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-- sorry },
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-- { intros g' h,
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-- apply linear_map.ext,
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-- intro x,
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-- sorry
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-- -- change (λ i, g' x i) = λ i, f i x, -- Wieso?
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-- -- funext,
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-- -- symmetry,
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-- -- apply h,
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-- }
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