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 ) ) ⟩ )