Definition df-doch 41330

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 41329	. 2 class ocH
2		vk	. . 3 setvar 𝑘
3		cvv 3477	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1535	. . . . 5 class 𝑘
6		clh 39966	. . . . 5 class LHyp
7	5, 6	cfv 6562	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1535	. . . . . . . 8 class 𝑤
10		cdvh 41060	. . . . . . . . 9 class DVecH
11	5, 10	cfv 6562	. . . . . . . 8 class (DVecH‘𝑘)
12	9, 11	cfv 6562	. . . . . . 7 class ((DVecH‘𝑘)‘𝑤)
13		cbs 17244	. . . . . . 7 class Base
14	12, 13	cfv 6562	. . . . . 6 class (Base‘((DVecH‘𝑘)‘𝑤))
15	14	cpw 4604	. . . . 5 class 𝒫 (Base‘((DVecH‘𝑘)‘𝑤))
16	8	cv 1535	. . . . . . . . . 10 class 𝑥
17		vy	. . . . . . . . . . . 12 setvar 𝑦
18	17	cv 1535	. . . . . . . . . . 11 class 𝑦
19		cdih 41210	. . . . . . . . . . . . 13 class DIsoH
20	5, 19	cfv 6562	. . . . . . . . . . . 12 class (DIsoH‘𝑘)
21	9, 20	cfv 6562	. . . . . . . . . . 11 class ((DIsoH‘𝑘)‘𝑤)
22	18, 21	cfv 6562	. . . . . . . . . 10 class (((DIsoH‘𝑘)‘𝑤)‘𝑦)
23	16, 22	wss 3962	. . . . . . . . 9 wff 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)
24	5, 13	cfv 6562	. . . . . . . . 9 class (Base‘𝑘)
25	23, 17, 24	crab 3432	. . . . . . . 8 class {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}
26		cglb 18367	. . . . . . . . 9 class glb
27	5, 26	cfv 6562	. . . . . . . 8 class (glb‘𝑘)
28	25, 27	cfv 6562	. . . . . . 7 class ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})
29		coc 17305	. . . . . . . 8 class oc
30	5, 29	cfv 6562	. . . . . . 7 class (oc‘𝑘)
31	28, 30	cfv 6562	. . . . . 6 class ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))
32	31, 21	cfv 6562	. . . . 5 class (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))
33	8, 15, 32	cmpt 5230	. . . 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 5230	. . 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 5230	. 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 1536	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  41331