Metamath Proof Explorer

Theorem djafvalN

Description: Subspace join for DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses djaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
djaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
djaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
djaval.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
djaval.j ⊢ 𝐽 = ( ( vA ‘ 𝐾 ) ‘ 𝑊 )
Assertion djafvalN ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐽 = ( 𝑥 ∈ 𝒫 𝑇 , 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 djaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 djaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 djaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4 djaval.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
5 djaval.j ⊢ 𝐽 = ( ( vA ‘ 𝐾 ) ‘ 𝑊 )
6 1 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 ) ) ) ) ) )
7 6 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( vA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) ‘ 𝑊 ) )
8 5 7 syl5eq ⊢ ( 𝐾 ∈ 𝑉 → 𝐽 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) ‘ 𝑊 ) )
9 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
10 9 2 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
11 10 pweqd ⊢ ( 𝑤 = 𝑊 → 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝒫 𝑇 )
12 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) )
13 12 4 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) = ⊥ )
14 13 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( ⊥ ‘ 𝑥 ) )
15 13 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) = ( ⊥ ‘ 𝑦 ) )
16 14 15 ineq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) = ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) )
17 13 16 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) = ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) )
18 11 11 17 mpoeq123dv ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) = ( 𝑥 ∈ 𝒫 𝑇 , 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) ) )
19 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 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 ) ) ) ) )
20 2 fvexi ⊢ 𝑇 ∈ V
21 20 pwex ⊢ 𝒫 𝑇 ∈ V
22 21 21 mpoex ⊢ ( 𝑥 ∈ 𝒫 𝑇 , 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) ) ∈ V
23 18 19 22 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) , 𝑦 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ∩ ( ( ( ocA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ) ) ) ) ‘ 𝑊 ) = ( 𝑥 ∈ 𝒫 𝑇 , 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) ) )
24 8 23 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐽 = ( 𝑥 ∈ 𝒫 𝑇 , 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘ ( ( ⊥ ‘ 𝑥 ) ∩ ( ⊥ ‘ 𝑦 ) ) ) ) )