Definition df-docaN 41584

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 41583	. 2 class ocA
2		vk	. . 3 setvar 𝑘
3		cvv 3430	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1541	. . . . 5 class 𝑘
6		clh 40448	. . . . 5 class LHyp
7	5, 6	cfv 6494	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1541	. . . . . . 7 class 𝑤
10		cltrn 40565	. . . . . . . 8 class LTrn
11	5, 10	cfv 6494	. . . . . . 7 class (LTrn‘𝑘)
12	9, 11	cfv 6494	. . . . . 6 class ((LTrn‘𝑘)‘𝑤)
13	12	cpw 4542	. . . . 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 3890	. . . . . . . . . . . 12 wff 𝑥 ⊆ 𝑧
18		cdia 41492	. . . . . . . . . . . . . . 15 class DIsoA
19	5, 18	cfv 6494	. . . . . . . . . . . . . 14 class (DIsoA‘𝑘)
20	9, 19	cfv 6494	. . . . . . . . . . . . 13 class ((DIsoA‘𝑘)‘𝑤)
21	20	crn 5627	. . . . . . . . . . . 12 class ran ((DIsoA‘𝑘)‘𝑤)
22	17, 15, 21	crab 3390	. . . . . . . . . . 11 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
23	22	cint 4890	. . . . . . . . . 10 class ∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}
24	20	ccnv 5625	. . . . . . . . . 10 class ◡((DIsoA‘𝑘)‘𝑤)
25	23, 24	cfv 6494	. . . . . . . . 9 class (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
26		coc 17223	. . . . . . . . . 10 class oc
27	5, 26	cfv 6494	. . . . . . . . 9 class (oc‘𝑘)
28	25, 27	cfv 6494	. . . . . . . 8 class ((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))
29	9, 27	cfv 6494	. . . . . . . 8 class ((oc‘𝑘)‘𝑤)
30		cjn 18272	. . . . . . . . 9 class join
31	5, 30	cfv 6494	. . . . . . . 8 class (join‘𝑘)
32	28, 29, 31	co 7362	. . . . . . 7 class (((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))
33		cmee 18273	. . . . . . . 8 class meet
34	5, 33	cfv 6494	. . . . . . 7 class (meet‘𝑘)
35	32, 9, 34	co 7362	. . . . . 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 6494	. . . . 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 5167	. . . 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 5167	. . 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 5167	. 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  41585