Definition df-docaN 41525

Description: Define subspace orthocomplement for DVecA partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)

Assertion

Ref	Expression
df-docaN	⊢ ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))

Distinct variable group:   𝑤,𝑘,𝑥,𝑧

Detailed syntax breakdown of Definition df-docaN

Step	Hyp	Ref	Expression
1		cocaN 41524	. 2 class ocA
2		vk	. . 3 setvar 𝑘
3		cvv 3442	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1541	. . . . 5 class 𝑘
6		clh 40389	. . . . 5 class LHyp
7	5, 6	cfv 6502	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1541	. . . . . . 7 class 𝑤
10		cltrn 40506	. . . . . . . 8 class LTrn
11	5, 10	cfv 6502	. . . . . . 7 class (LTrn‘𝑘)
12	9, 11	cfv 6502	. . . . . 6 class ((LTrn‘𝑘)‘𝑤)
13	12	cpw 4556	. . . . 5 class 𝒫 ((LTrn‘𝑘)‘𝑤)
14	8	cv 1541	. . . . . . . . . . . . 13 class 𝑥
15		vz	. . . . . . . . . . . . . 14 setvar 𝑧
16	15	cv 1541	. . . . . . . . . . . . 13 class 𝑧
17	14, 16	wss 3903	. . . . . . . . . . . 12 wff 𝑥 ⊆ 𝑧
18		cdia 41433	. . . . . . . . . . . . . . 15 class DIsoA
19	5, 18	cfv 6502	. . . . . . . . . . . . . 14 class (DIsoA‘𝑘)
20	9, 19	cfv 6502	. . . . . . . . . . . . 13 class ((DIsoA‘𝑘)‘𝑤)
21	20	crn 5635	. . . . . . . . . . . 12 class ran ((DIsoA‘𝑘)‘𝑤)
22	17, 15, 21	crab 3401	. . . . . . . . . . 11 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
23	22	cint 4904	. . . . . . . . . 10 class ∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
24	20	ccnv 5633	. . . . . . . . . 10 class ◡((DIsoA‘𝑘)‘𝑤)
25	23, 24	cfv 6502	. . . . . . . . 9 class (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
26		coc 17199	. . . . . . . . . 10 class oc
27	5, 26	cfv 6502	. . . . . . . . 9 class (oc‘𝑘)
28	25, 27	cfv 6502	. . . . . . . 8 class ((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))
29	9, 27	cfv 6502	. . . . . . . 8 class ((oc‘𝑘)‘𝑤)
30		cjn 18248	. . . . . . . . 9 class join
31	5, 30	cfv 6502	. . . . . . . 8 class (join‘𝑘)
32	28, 29, 31	co 7370	. . . . . . 7 class (((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))
33		cmee 18249	. . . . . . . 8 class meet
34	5, 33	cfv 6502	. . . . . . 7 class (meet‘𝑘)
35	32, 9, 34	co 7370	. . . . . 6 class ((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)
36	35, 20	cfv 6502	. . . . 5 class (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))
37	8, 13, 36	cmpt 5181	. . . 4 class (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))
38	4, 7, 37	cmpt 5181	. . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))
39	2, 3, 38	cmpt 5181	. 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
40	1, 39	wceq 1542	1 wff ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))

This definition is referenced by:  docaffvalN  41526