Theorem dvaplusgv

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	dvaplusgv	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑅 + 𝑆 ) ‘ 𝐺 ) = ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) )

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	dvaplusg	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 + 𝑆 ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) )
8	7	fveq1d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( ( 𝑅 + 𝑆 ) ‘ 𝐺 ) = ( ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) ‘ 𝐺 ) )
9	8	3adantr3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑅 + 𝑆 ) ‘ 𝐺 ) = ( ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) ‘ 𝐺 ) )
10		simpr3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐺 ∈ 𝑇 )
11		fveq2	⊢ ( 𝑓 = 𝐺 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐺 ) )
12		fveq2	⊢ ( 𝑓 = 𝐺 → ( 𝑆 ‘ 𝑓 ) = ( 𝑆 ‘ 𝐺 ) )
13	11 12	coeq12d	⊢ ( 𝑓 = 𝐺 → ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) = ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) )
14		eqid	⊢ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) )
15		fvex	⊢ ( 𝑅 ‘ 𝐺 ) ∈ V
16		fvex	⊢ ( 𝑆 ‘ 𝐺 ) ∈ V
17	15 16	coex	⊢ ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ∈ V
18	13 14 17	fvmpt	⊢ ( 𝐺 ∈ 𝑇 → ( ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) ‘ 𝐺 ) = ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) )
19	10 18	syl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑅 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑓 ) ) ) ‘ 𝐺 ) = ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) )
20	9 19	eqtrd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑅 + 𝑆 ) ‘ 𝐺 ) = ( ( 𝑅 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) )