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	eqtrid	⊢ ( 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 ) ) ) )