Definition df-doch 40219

Description: Define subspace orthocomplement for DVecH 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, 14-Mar-2014.)

Assertion

Ref	Expression
df-doch	⊢ ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))

Distinct variable group:   𝑤,𝑘,𝑥,𝑦

Detailed syntax breakdown of Definition df-doch

Step	Hyp	Ref	Expression
1		coch 40218	. 2 class ocH
2		vk	. . 3 setvar 𝑘
3		cvv 3475	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1541	. . . . 5 class 𝑘
6		clh 38855	. . . . 5 class LHyp
7	5, 6	cfv 6544	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1541	. . . . . . . 8 class 𝑤
10		cdvh 39949	. . . . . . . . 9 class DVecH
11	5, 10	cfv 6544	. . . . . . . 8 class (DVecH‘𝑘)
12	9, 11	cfv 6544	. . . . . . 7 class ((DVecH‘𝑘)‘𝑤)
13		cbs 17144	. . . . . . 7 class Base
14	12, 13	cfv 6544	. . . . . 6 class (Base‘((DVecH‘𝑘)‘𝑤))
15	14	cpw 4603	. . . . 5 class 𝒫 (Base‘((DVecH‘𝑘)‘𝑤))
16	8	cv 1541	. . . . . . . . . 10 class 𝑥
17		vy	. . . . . . . . . . . 12 setvar 𝑦
18	17	cv 1541	. . . . . . . . . . 11 class 𝑦
19		cdih 40099	. . . . . . . . . . . . 13 class DIsoH
20	5, 19	cfv 6544	. . . . . . . . . . . 12 class (DIsoH‘𝑘)
21	9, 20	cfv 6544	. . . . . . . . . . 11 class ((DIsoH‘𝑘)‘𝑤)
22	18, 21	cfv 6544	. . . . . . . . . 10 class (((DIsoH‘𝑘)‘𝑤)‘𝑦)
23	16, 22	wss 3949	. . . . . . . . 9 wff 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)
24	5, 13	cfv 6544	. . . . . . . . 9 class (Base‘𝑘)
25	23, 17, 24	crab 3433	. . . . . . . 8 class {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}
26		cglb 18263	. . . . . . . . 9 class glb
27	5, 26	cfv 6544	. . . . . . . 8 class (glb‘𝑘)
28	25, 27	cfv 6544	. . . . . . 7 class ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})
29		coc 17205	. . . . . . . 8 class oc
30	5, 29	cfv 6544	. . . . . . 7 class (oc‘𝑘)
31	28, 30	cfv 6544	. . . . . 6 class ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))
32	31, 21	cfv 6544	. . . . 5 class (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))
33	8, 15, 32	cmpt 5232	. . . 4 class (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))
34	4, 7, 33	cmpt 5232	. . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))))
35	2, 3, 34	cmpt 5232	. 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))
36	1, 35	wceq 1542	1 wff ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))

This definition is referenced by:  dochffval  40220