Theorem dvafvsca

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	dvafvsca.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvafvsca.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dvafvsca.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dvafvsca.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
	dvafvsca.s	⊢ · = ( ·𝑠 ‘ 𝑈 )
Assertion	dvafvsca	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvafvsca.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvafvsca.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvafvsca.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvafvsca.u	⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
5		dvafvsca.s	⊢ · = ( ·𝑠 ‘ 𝑈 )
6		eqid	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 6 4	dvaset	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
8	7	fveq2d	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
9	3	fvexi	⊢ 𝐸 ∈ V
10	2	fvexi	⊢ 𝑇 ∈ V
11	9 10	mpoex	⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ∈ V
12		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } )
13	12	lmodvsca	⊢ ( ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ∈ V → ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) ) )
14	11 13	ax-mp	⊢ ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝑇 ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑓 ∈ 𝑇 , 𝑔 ∈ 𝑇 ↦ ( 𝑓 ∘ 𝑔 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) ⟩ } ) )
15	8 5 14	3eqtr4g	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → · = ( 𝑠 ∈ 𝐸 , 𝑓 ∈ 𝑇 ↦ ( 𝑠 ‘ 𝑓 ) ) )