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Componentwise mode spaces and diagonal tensors

Lax342547.ComponentSpaces · concepts/Lax342547/ComponentSpaces.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Coordinatewise mode subspaces form a join-stable cover class, and the range of a diagonal tensor map is the product of its component ranges.

    Concept map
    22 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 componentwise_join_closed proven

    2 coordinate_space_bot proven

    3 coordinate_space_sup proven

    4 diagonal_range proven

    Lean source view on GitHub

    1import Lax342547.CoverClasses
    2import Lax342547.DirectRanks
    3
    4/-!
    5---
    6title: Componentwise mode spaces and diagonal tensors
    7type: lemma
    8---
    9Coordinatewise mode subspaces form a join-stable cover class, and the range of a diagonal tensor map is the product of its component ranges.
    10-/
    11
    12namespace Lax342547.ComponentSpaces
    13
    14open Lax342547.CoverClasses
    15open scoped BigOperators
    16
    17noncomputable def coordinateSpace {K e : Type} [Field K] {V : e → Type}
    18 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] (S : ∀ i, Submodule K (V i)) :
    19 Submodule K (∀ i, V i) where
    20 carrier := {v | ∀ i, v i ∈ S i}
    21 zero_mem' := fun i => (S i).zero_mem
    22 add_mem' := fun h g i => (S i).add_mem (h i) (g i)
    23 smul_mem' := fun c _ h i => (S i).smul_mem c (h i)
    24
    25def Componentwise {K e : Type} [Field K] {V : e → Type}
    26 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    27 (T : Submodule K (∀ i, V i)) : Prop := ∃ S, T = coordinateSpace S
    28
    29axiom coordinate_space_bot {K e : Type} [Field K] {V : e → Type}
    30 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] :
    31 coordinateSpace (fun i => (⊥ : Submodule K (V i))) = ⊥
    32
    33axiom coordinate_space_sup {K e : Type} [Field K] {V : e → Type}
    34 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    35 (S T : ∀ i, Submodule K (V i)) :
    36 coordinateSpace S ⊔ coordinateSpace T = coordinateSpace (fun i => S i ⊔ T i)
    37
    38axiom componentwise_join_closed {K e : Type} [Field K] {V : e → Type}
    39 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] : JoinClosed (Componentwise (K := K) (V := V))
    40
    41axiom diagonal_range {K e : Type} [Field K] {V W : e → Type}
    42 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    43 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)] (M : ∀ i, V i →ₗ[K] W i) :
    44 LinearMap.range (LinearMap.piMap M) = coordinateSpace (fun i => LinearMap.range (M i))
    45
    46end Lax342547.ComponentSpaces
    47
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