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 ( ( 𝐾𝑋𝑊𝐻 ) → 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 ( ( 𝐾𝑋𝑊𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } ) )
8 7 fveq2d ( ( 𝐾𝑋𝑊𝐻 ) → ( Base ‘ 𝑈 ) = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } ) ) )
9 2 fvexi 𝑇 ∈ V
10 eqid ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } )
11 10 lmodbase ( 𝑇 ∈ V → 𝑇 = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } ) ) )
12 9 11 ax-mp 𝑇 = ( Base ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓𝑇 , 𝑔𝑇 ↦ ( 𝑓𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓𝑇 ↦ ( 𝑠𝑓 ) ) ⟩ } ) )
13 8 4 12 3eqtr4g ( ( 𝐾𝑋𝑊𝐻 ) → 𝑉 = 𝑇 )