Metamath Proof Explorer

Theorem docafvalN

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

Ref Expression
Hypotheses docaval.j ⊢ ∨ = ( join ‘ 𝐾 )
docaval.m ⊢ ∧ = ( meet ‘ 𝐾 )
docaval.o ⊢ ⊥ = ( oc ‘ 𝐾 )
docaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
docaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
docaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
docaval.n ⊢ 𝑁 = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
Assertion docafvalN ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝑁 = ( 𝑥 ∈ 𝒫 𝑇 ↦ ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 docaval.j ⊢ ∨ = ( join ‘ 𝐾 )
2 docaval.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 docaval.o ⊢ ⊥ = ( oc ‘ 𝐾 )
4 docaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 docaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 docaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 docaval.n ⊢ 𝑁 = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 docaffvalN ⊢ ( 𝐾 ∈ 𝑉 → ( ocA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) ) )
9 8 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) ) ‘ 𝑊 ) )
10 7 9 syl5eq ⊢ ( 𝐾 ∈ 𝑉 → 𝑁 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) ) ‘ 𝑊 ) )
11 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
12 11 5 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
13 12 pweqd ⊢ ( 𝑤 = 𝑊 → 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝒫 𝑇 )
14 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) )
15 14 6 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = 𝐼 )
16 15 cnveqd ⊢ ( 𝑤 = 𝑊 → ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = ◡ 𝐼 )
17 15 rneqd ⊢ ( 𝑤 = 𝑊 → ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = ran 𝐼 )
18 17 rabeqdv ⊢ ( 𝑤 = 𝑊 → { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } = { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } )
19 18 inteqd ⊢ ( 𝑤 = 𝑊 → ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } = ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } )
20 16 19 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) = ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) )
21 20 fveq2d ⊢ ( 𝑤 = 𝑊 → ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) = ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) )
22 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ⊥ ‘ 𝑤 ) = ( ⊥ ‘ 𝑊 ) )
23 21 22 oveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) = ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) )
24 id ⊢ ( 𝑤 = 𝑊 → 𝑤 = 𝑊 )
25 23 24 oveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) = ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) )
26 15 25 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) )
27 13 26 mpteq12dv ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) = ( 𝑥 ∈ 𝒫 𝑇 ↦ ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) )
28 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) )
29 5 fvexi ⊢ 𝑇 ∈ V
30 29 pwex ⊢ 𝒫 𝑇 ∈ V
31 30 mptex ⊢ ( 𝑥 ∈ 𝒫 𝑇 ↦ ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) ∈ V
32 27 28 31 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ⊥ ‘ ( ◡ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ ∩ { 𝑧 ∈ ran ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑤 ) ) ∧ 𝑤 ) ) ) ) ‘ 𝑊 ) = ( 𝑥 ∈ 𝒫 𝑇 ↦ ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) )
33 10 32 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝑁 = ( 𝑥 ∈ 𝒫 𝑇 ↦ ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) )