Definition df-doch 41470

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 41469	. 2 class ocH
2		vk	. . 3 setvar 𝑘
3		cvv 3437	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1540	. . . . 5 class 𝑘
6		clh 40106	. . . . 5 class LHyp
7	5, 6	cfv 6488	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1540	. . . . . . . 8 class 𝑤
10		cdvh 41200	. . . . . . . . 9 class DVecH
11	5, 10	cfv 6488	. . . . . . . 8 class (DVecH‘𝑘)
12	9, 11	cfv 6488	. . . . . . 7 class ((DVecH‘𝑘)‘𝑤)
13		cbs 17124	. . . . . . 7 class Base
14	12, 13	cfv 6488	. . . . . 6 class (Base‘((DVecH‘𝑘)‘𝑤))
15	14	cpw 4551	. . . . 5 class 𝒫 (Base‘((DVecH‘𝑘)‘𝑤))
16	8	cv 1540	. . . . . . . . . 10 class 𝑥
17		vy	. . . . . . . . . . . 12 setvar 𝑦
18	17	cv 1540	. . . . . . . . . . 11 class 𝑦
19		cdih 41350	. . . . . . . . . . . . 13 class DIsoH
20	5, 19	cfv 6488	. . . . . . . . . . . 12 class (DIsoH‘𝑘)
21	9, 20	cfv 6488	. . . . . . . . . . 11 class ((DIsoH‘𝑘)‘𝑤)
22	18, 21	cfv 6488	. . . . . . . . . 10 class (((DIsoH‘𝑘)‘𝑤)‘𝑦)
23	16, 22	wss 3898	. . . . . . . . 9 wff 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)
24	5, 13	cfv 6488	. . . . . . . . 9 class (Base‘𝑘)
25	23, 17, 24	crab 3396	. . . . . . . 8 class {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}
26		cglb 18220	. . . . . . . . 9 class glb
27	5, 26	cfv 6488	. . . . . . . 8 class (glb‘𝑘)
28	25, 27	cfv 6488	. . . . . . 7 class ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})
29		coc 17173	. . . . . . . 8 class oc
30	5, 29	cfv 6488	. . . . . . 7 class (oc‘𝑘)
31	28, 30	cfv 6488	. . . . . 6 class ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))
32	31, 21	cfv 6488	. . . . 5 class (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))
33	8, 15, 32	cmpt 5176	. . . 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 5176	. . 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 5176	. 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 1541	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  41471