Metamath Proof Explorer


Theorem dvhfvsca

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013) (Revised by Mario Carneiro, 23-Jun-2014)

Ref Expression
Hypotheses dvhfvsca.h 𝐻 = ( LHyp ‘ 𝐾 )
dvhfvsca.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dvhfvsca.s · = ( ·𝑠𝑈 )
Assertion dvhfvsca ( ( 𝐾𝑉𝑊𝐻 ) → · = ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) )

Proof

Step Hyp Ref Expression
1 dvhfvsca.h 𝐻 = ( LHyp ‘ 𝐾 )
2 dvhfvsca.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvhfvsca.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvhfvsca.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dvhfvsca.s · = ( ·𝑠𝑈 )
6 eqid ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 6 4 dvhset ( ( 𝐾𝑉𝑊𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) )
8 7 fveq2d ( ( 𝐾𝑉𝑊𝐻 ) → ( ·𝑠𝑈 ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) ) )
9 3 fvexi 𝐸 ∈ V
10 2 fvexi 𝑇 ∈ V
11 10 9 xpex ( 𝑇 × 𝐸 ) ∈ V
12 9 11 mpoex ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ∈ V
13 eqid ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } )
14 13 lmodvsca ( ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ∈ V → ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) ) )
15 12 14 ax-mp ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1st𝑓 ) ∘ ( 1st𝑔 ) ) , ( 𝑇 ↦ ( ( ( 2nd𝑓 ) ‘ ) ∘ ( ( 2nd𝑔 ) ‘ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) ⟩ } ) )
16 8 5 15 3eqtr4g ( ( 𝐾𝑉𝑊𝐻 ) → · = ( 𝑠𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st𝑓 ) ) , ( 𝑠 ∘ ( 2nd𝑓 ) ) ⟩ ) )