Theorem dvavsca

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013)

	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	dvavsca	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ‘ 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	1 2 3 4 5	dvafvsca	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )
7	6	oveqd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) 𝐹 ) )
8		fveq1	⊢ ( 𝑠 = 𝑅 → ( 𝑠 ‘ 𝑓 ) = ( 𝑅 ‘ 𝑓 ) )
9		fveq2	⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) )
10		eqid	⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) )
11		fvex	⊢ ( 𝑅 ‘ 𝐹 ) ∈ V
12	8 9 10 11	ovmpo	⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )
13	7 12	sylan9eq	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑅 · 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )