Theorem dvafplusg

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	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	dvafplusg	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 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		eqid	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
8	1 7 4 5	dvasca	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
9	8	fveq2d	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐹 ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
10	6 9	syl5eq	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
11		eqid	⊢ ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
12	1 2 3 7 11	erngfplus	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )
13	10 12	eqtrd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → + = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )