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Metamath Proof Explorer

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 ) ) ) ) ) )