Theorem dvhvscacl

Description: Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-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	⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion	dvhvscacl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · 𝐹 ) ∈ ( 𝑇 × 𝐸 ) )

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	dvhvsca	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · 𝐹 ) = ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ )
7		simpl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		simprl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → 𝑅 ∈ 𝐸 )
9		xp1st	⊢ ( 𝐹 ∈ ( 𝑇 × 𝐸 ) → ( 1st ‘ 𝐹 ) ∈ 𝑇 )
10	9	ad2antll	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 1st ‘ 𝐹 ) ∈ 𝑇 )
11	1 2 3	tendocl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐸 ∧ ( 1st ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) ∈ 𝑇 )
12	7 8 10 11	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) ∈ 𝑇 )
13		xp2nd	⊢ ( 𝐹 ∈ ( 𝑇 × 𝐸 ) → ( 2nd ‘ 𝐹 ) ∈ 𝐸 )
14	13	ad2antll	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 2nd ‘ 𝐹 ) ∈ 𝐸 )
15	1 3	tendococl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐸 ∧ ( 2nd ‘ 𝐹 ) ∈ 𝐸 ) → ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ∈ 𝐸 )
16	7 8 14 15	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ∈ 𝐸 )
17		opelxpi	⊢ ( ( ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) ∈ 𝑇 ∧ ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ∈ 𝐸 ) → ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ ∈ ( 𝑇 × 𝐸 ) )
18	12 16 17	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ⟨ ( 𝑅 ‘ ( 1st ‘ 𝐹 ) ) , ( 𝑅 ∘ ( 2nd ‘ 𝐹 ) ) ⟩ ∈ ( 𝑇 × 𝐸 ) )
19	6 18	eqeltrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ ( 𝑇 × 𝐸 ) ) ) → ( 𝑅 · 𝐹 ) ∈ ( 𝑇 × 𝐸 ) )