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Metamath Proof Explorer

Theorem dvavbase

Description: The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom W ). (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvavbase.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvavbase.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvavbase.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dvavbase.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
Assertion dvavbase ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑉 = 𝑇 )

Proof

Step Hyp Ref Expression
1 dvavbase.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvavbase.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvavbase.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
4 dvavbase.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 5 6 3 dvaset ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
8 7 fveq2d ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
9 2 fvexi ⊢ 𝑇 ∈ V
10 eqid ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } )
11 10 lmodbase ⊢ ( 𝑇 ∈ V → 𝑇 = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
12 9 11 ax-mp ⊢ 𝑇 = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
13 8 4 12 3eqtr4g ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑉 = 𝑇 )