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Metamath Proof Explorer

Theorem dvhvsca

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013)

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 ⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion dvhvsca ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · 𝐹 ) = ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ )

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 ⊢ · = ( ·𝑠 ‘ 𝑈 )
6 1 2 3 4 5 dvhfvsca ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) )
7 6 oveqd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) 𝐹 ) )
8 eqid ⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ )
9 8 dvhvscaval ⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) → ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2nd ‘ 𝑓 ) ) ⟩ ) 𝐹 ) = ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ )
10 7 9 sylan9eq ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · 𝐹 ) = ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ )