Theorem dvafmulr

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

	Ref	Expression
Hypotheses	dvafmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvafmul.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvafmul.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dvafmul.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
	dvafmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	dvafmul.p	⊢ · = ( .r ‘ 𝐹 )
Assertion	dvafmulr	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvafmul.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvafmul.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvafmul.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvafmul.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5		dvafmul.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
6		dvafmul.p	⊢ · = ( .r ‘ 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 ) → ( .r ‘ 𝐹 ) = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
10	6 9	syl5eq	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
11		eqid	⊢ ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
12	1 2 3 7 11	erngfmul	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) )
13	10 12	eqtrd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑠 ∘ 𝑡 ) ) )