Definition df-docaN 41121

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 41120	. 2 class ocA
2		vk	. . 3 setvar 𝑘
3		cvv 3450	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1539	. . . . 5 class 𝑘
6		clh 39985	. . . . 5 class LHyp
7	5, 6	cfv 6514	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1539	. . . . . . 7 class 𝑤
10		cltrn 40102	. . . . . . . 8 class LTrn
11	5, 10	cfv 6514	. . . . . . 7 class (LTrn‘𝑘)
12	9, 11	cfv 6514	. . . . . 6 class ((LTrn‘𝑘)‘𝑤)
13	12	cpw 4566	. . . . 5 class 𝒫 ((LTrn‘𝑘)‘𝑤)
14	8	cv 1539	. . . . . . . . . . . . 13 class 𝑥
15		vz	. . . . . . . . . . . . . 14 setvar 𝑧
16	15	cv 1539	. . . . . . . . . . . . 13 class 𝑧
17	14, 16	wss 3917	. . . . . . . . . . . 12 wff 𝑥 ⊆ 𝑧
18		cdia 41029	. . . . . . . . . . . . . . 15 class DIsoA
19	5, 18	cfv 6514	. . . . . . . . . . . . . 14 class (DIsoA‘𝑘)
20	9, 19	cfv 6514	. . . . . . . . . . . . 13 class ((DIsoA‘𝑘)‘𝑤)
21	20	crn 5642	. . . . . . . . . . . 12 class ran ((DIsoA‘𝑘)‘𝑤)
22	17, 15, 21	crab 3408	. . . . . . . . . . 11 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
23	22	cint 4913	. . . . . . . . . 10 class ∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
24	20	ccnv 5640	. . . . . . . . . 10 class ◡((DIsoA‘𝑘)‘𝑤)
25	23, 24	cfv 6514	. . . . . . . . 9 class (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
26		coc 17235	. . . . . . . . . 10 class oc
27	5, 26	cfv 6514	. . . . . . . . 9 class (oc‘𝑘)
28	25, 27	cfv 6514	. . . . . . . 8 class ((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))
29	9, 27	cfv 6514	. . . . . . . 8 class ((oc‘𝑘)‘𝑤)
30		cjn 18279	. . . . . . . . 9 class join
31	5, 30	cfv 6514	. . . . . . . 8 class (join‘𝑘)
32	28, 29, 31	co 7390	. . . . . . 7 class (((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))
33		cmee 18280	. . . . . . . 8 class meet
34	5, 33	cfv 6514	. . . . . . 7 class (meet‘𝑘)
35	32, 9, 34	co 7390	. . . . . 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 6514	. . . . 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 5191	. . . 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 5191	. . 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 5191	. 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 1540	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  41122