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Metamath Proof Explorer

Theorem dvhsca

Description: The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014)

Ref Expression
Hypotheses dvhsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvhsca.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
dvhsca.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dvhsca.f ⊢ 𝐹 = ( Scalar ‘ 𝑈 )
Assertion dvhsca ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = 𝐷 )

Proof

Step Hyp Ref Expression
1 dvhsca.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvhsca.d ⊢ 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3 dvhsca.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 dvhsca.f ⊢ 𝐹 = ( Scalar ‘ 𝑈 )
5 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7 1 5 6 2 3 dvhset ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
8 7 fveq2d ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
9 2 fvexi ⊢ 𝐷 ∈ V
10 eqid ⊢ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
11 10 lmodsca ⊢ ( 𝐷 ∈ V → 𝐷 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
12 9 11 ax-mp ⊢ 𝐷 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑔 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( ( 1st ‘ 𝑓 ) ∘ ( 1st ‘ 𝑔 ) ) , ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( ( 2nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑓 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
13 8 4 12 3eqtr4g ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = 𝐷 )