Theorem dvavadd

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

	Ref	Expression
Hypotheses	dvafvadd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvafvadd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvafvadd.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
	dvafvadd.v	⊢ + = ( +g ‘ 𝑈 )
Assertion	dvavadd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐹 + 𝐺 ) = ( 𝐹 ∘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		dvafvadd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvafvadd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvafvadd.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
4		dvafvadd.v	⊢ + = ( +g ‘ 𝑈 )
5	1 2 3 4	dvafvadd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) )
6	5	oveqd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 + 𝐺 ) = ( 𝐹 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝐺 ) )
7		coexg	⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ V )
8		coeq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ∘ 𝑔 ) = ( 𝐹 ∘ 𝑔 ) )
9		coeq2	⊢ ( 𝑔 = 𝐺 → ( 𝐹 ∘ 𝑔 ) = ( 𝐹 ∘ 𝐺 ) )
10		eqid	⊢ ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) = ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) )
11	8 9 10	ovmpog	⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ∘ 𝐺 ) ∈ V ) → ( 𝐹 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝐺 ) = ( 𝐹 ∘ 𝐺 ) )
12	7 11	mpd3an3	⊢ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) 𝐺 ) = ( 𝐹 ∘ 𝐺 ) )
13	6 12	sylan9eq	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐹 + 𝐺 ) = ( 𝐹 ∘ 𝐺 ) )