Theorem djaffvalN

Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypothesis	djaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	djaffvalN	⊢ ( 𝐾 ∈ 𝑉 → ( vA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		djaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		elex	⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
3		fveq2	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
4	3 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
5		fveq2	⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
6	5	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
7	6	pweqd	⊢ ( 𝑘 = 𝐾 → 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
8		fveq2	⊢ ( 𝑘 = 𝐾 → ( ocA ‘ 𝑘 ) = ( ocA ‘ 𝐾 ) )
9	8	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) = ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) )
10	9	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) )
11	9	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) )
12	10 11	ineq12d	⊢ ( 𝑘 = 𝐾 → ( ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) ) = ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) )
13	9 12	fveq12d	⊢ ( 𝑘 = 𝐾 → ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) = ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) )
14	7 7 13	mpoeq123dv	⊢ ( 𝑘 = 𝐾 → ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) = ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) )
15	4 14	mpteq12dv	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) )
16		df-djaN	⊢ vA = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) )
17	15 16 1	mptfvmpt	⊢ ( 𝐾 ∈ V → ( vA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) )
18	2 17	syl	⊢ ( 𝐾 ∈ 𝑉 → ( vA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) )