Definition df-docaN 41144

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 41143	. 2 class ocA
2		vk	. . 3 setvar 𝑘
3		cvv 3464	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1539	. . . . 5 class 𝑘
6		clh 40008	. . . . 5 class LHyp
7	5, 6	cfv 6536	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1539	. . . . . . 7 class 𝑤
10		cltrn 40125	. . . . . . . 8 class LTrn
11	5, 10	cfv 6536	. . . . . . 7 class (LTrn‘𝑘)
12	9, 11	cfv 6536	. . . . . 6 class ((LTrn‘𝑘)‘𝑤)
13	12	cpw 4580	. . . . 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 3931	. . . . . . . . . . . 12 wff 𝑥 ⊆ 𝑧
18		cdia 41052	. . . . . . . . . . . . . . 15 class DIsoA
19	5, 18	cfv 6536	. . . . . . . . . . . . . 14 class (DIsoA‘𝑘)
20	9, 19	cfv 6536	. . . . . . . . . . . . 13 class ((DIsoA‘𝑘)‘𝑤)
21	20	crn 5660	. . . . . . . . . . . 12 class ran ((DIsoA‘𝑘)‘𝑤)
22	17, 15, 21	crab 3420	. . . . . . . . . . 11 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
23	22	cint 4927	. . . . . . . . . 10 class ∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
24	20	ccnv 5658	. . . . . . . . . 10 class ◡((DIsoA‘𝑘)‘𝑤)
25	23, 24	cfv 6536	. . . . . . . . 9 class (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
26		coc 17284	. . . . . . . . . 10 class oc
27	5, 26	cfv 6536	. . . . . . . . 9 class (oc‘𝑘)
28	25, 27	cfv 6536	. . . . . . . 8 class ((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))
29	9, 27	cfv 6536	. . . . . . . 8 class ((oc‘𝑘)‘𝑤)
30		cjn 18328	. . . . . . . . 9 class join
31	5, 30	cfv 6536	. . . . . . . 8 class (join‘𝑘)
32	28, 29, 31	co 7410	. . . . . . 7 class (((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))
33		cmee 18329	. . . . . . . 8 class meet
34	5, 33	cfv 6536	. . . . . . 7 class (meet‘𝑘)
35	32, 9, 34	co 7410	. . . . . 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 6536	. . . . 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 5206	. . . 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 5206	. . 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 5206	. 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  41145