Theorem dvhmulr

Description: Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

	Ref	Expression
Hypotheses	dvhfmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvhfmul.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvhfmul.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dvhfmul.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dvhfmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	dvhfmul.m	⊢ · = ( .r ‘ 𝐹 )
Assertion	dvhmulr	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		dvhfmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvhfmul.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvhfmul.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvhfmul.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dvhfmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
6		dvhfmul.m	⊢ · = ( .r ‘ 𝐹 )
7	1 2 3 4 5 6	dvhfmulr	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) )
8	7	oveqd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) )
9		coexg	⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ∘ 𝑆 ) ∈ V )
10		coeq1	⊢ ( 𝑟 = 𝑅 → ( 𝑟 ∘ 𝑠 ) = ( 𝑅 ∘ 𝑠 ) )
11		coeq2	⊢ ( 𝑠 = 𝑆 → ( 𝑅 ∘ 𝑠 ) = ( 𝑅 ∘ 𝑆 ) )
12		eqid	⊢ ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) = ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) )
13	10 11 12	ovmpog	⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ ( 𝑅 ∘ 𝑆 ) ∈ V ) → ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )
14	9 13	mpd3an3	⊢ ( ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) → ( 𝑅 ( 𝑟 ∈ 𝐸 , 𝑠 ∈ 𝐸 ↦ ( 𝑟 ∘ 𝑠 ) ) 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )
15	8 14	sylan9eq	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ) ) → ( 𝑅 · 𝑆 ) = ( 𝑅 ∘ 𝑆 ) )