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Metamath Proof Explorer

Theorem dvafvsca

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvafvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvafvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dvafvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion dvafvsca ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )

Proof

Step Hyp Ref Expression
1 dvafvsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvafvsca.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 dvafvsca.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 dvafvsca.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5 dvafvsca.s ⊢ · = ( ·𝑠 ‘ 𝑈 )
6 eqid ⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 6 4 dvaset ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
8 7 fveq2d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
9 3 fvexi ⊢ 𝐸 ∈ V
10 2 fvexi ⊢ 𝑇 ∈ V
11 9 10 mpoex ⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ∈ V
12 eqid ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } )
13 12 lmodvsca ⊢ ( ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ∈ V → ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
14 11 13 ax-mp ⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
15 8 5 14 3eqtr4g ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )