Definition df-doch 42103

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 42102	. 2 class ocH
2		vk	. . 3 setvar 𝑘
3		cvv 3455	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1569	. . . . 5 class 𝑘
6		clh 40739	. . . . 5 class LHyp
7	5, 6	cfv 6538	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1569	. . . . . . . 8 class 𝑤
10		cdvh 41833	. . . . . . . . 9 class DVecH
11	5, 10	cfv 6538	. . . . . . . 8 class (DVecH‘𝑘)
12	9, 11	cfv 6538	. . . . . . 7 class ((DVecH‘𝑘)‘𝑤)
13		cbs 17270	. . . . . . 7 class Base
14	12, 13	cfv 6538	. . . . . 6 class (Base‘((DVecH‘𝑘)‘𝑤))
15	14	cpw 4563	. . . . 5 class 𝒫 (Base‘((DVecH‘𝑘)‘𝑤))
16	8	cv 1569	. . . . . . . . . 10 class 𝑥
17		vy	. . . . . . . . . . . 12 setvar 𝑦
18	17	cv 1569	. . . . . . . . . . 11 class 𝑦
19		cdih 41983	. . . . . . . . . . . . 13 class DIsoH
20	5, 19	cfv 6538	. . . . . . . . . . . 12 class (DIsoH‘𝑘)
21	9, 20	cfv 6538	. . . . . . . . . . 11 class ((DIsoH‘𝑘)‘𝑤)
22	18, 21	cfv 6538	. . . . . . . . . 10 class (((DIsoH‘𝑘)‘𝑤)‘𝑦)
23	16, 22	wss 3906	. . . . . . . . 9 wff 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)
24	5, 13	cfv 6538	. . . . . . . . 9 class (Base‘𝑘)
25	23, 17, 24	crab 3416	. . . . . . . 8 class {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}
26		cglb 18367	. . . . . . . . 9 class glb
27	5, 26	cfv 6538	. . . . . . . 8 class (glb‘𝑘)
28	25, 27	cfv 6538	. . . . . . 7 class ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})
29		coc 17319	. . . . . . . 8 class oc
30	5, 29	cfv 6538	. . . . . . 7 class (oc‘𝑘)
31	28, 30	cfv 6538	. . . . . 6 class ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))
32	31, 21	cfv 6538	. . . . 5 class (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))
33	8, 15, 32	cmpt 5193	. . . 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 5193	. . 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 5193	. 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 1570	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  42104