Theorem dvaplusg

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

	Ref	Expression
Hypotheses	dvafplus.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvafplus.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvafplus.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dvafplus.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
	dvafplus.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	dvafplus.p	⊢ + = ( +g ‘ 𝐹 )
Assertion	dvaplusg	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 + 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvafplus.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvafplus.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvafplus.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvafplus.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5		dvafplus.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
6		dvafplus.p	⊢ + = ( +g ‘ 𝐹 )
7	1 2 3 4 5 6	dvafplusg	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) )
8	7	oveqd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 + 𝑆 ) = ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑆 ) )
9		eqid	⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) )
10	9 2	tendopl	⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑔 ) ∘ ( 𝑡 ‘ 𝑔 ) ) ) ) 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )
11	8 10	sylan9eq	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 + 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )