Definition df-doch 41393

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 41392	. 2 class ocH
2		vk	. . 3 setvar 𝑘
3		cvv 3436	. . 3 class V
4		vw	. . . 4 setvar 𝑤
5	2	cv 1540	. . . . 5 class 𝑘
6		clh 40029	. . . . 5 class LHyp
7	5, 6	cfv 6481	. . . 4 class (LHyp‘𝑘)
8		vx	. . . . 5 setvar 𝑥
9	4	cv 1540	. . . . . . . 8 class 𝑤
10		cdvh 41123	. . . . . . . . 9 class DVecH
11	5, 10	cfv 6481	. . . . . . . 8 class (DVecH‘𝑘)
12	9, 11	cfv 6481	. . . . . . 7 class ((DVecH‘𝑘)‘𝑤)
13		cbs 17120	. . . . . . 7 class Base
14	12, 13	cfv 6481	. . . . . 6 class (Base‘((DVecH‘𝑘)‘𝑤))
15	14	cpw 4550	. . . . 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 41273	. . . . . . . . . . . . 13 class DIsoH
20	5, 19	cfv 6481	. . . . . . . . . . . 12 class (DIsoH‘𝑘)
21	9, 20	cfv 6481	. . . . . . . . . . 11 class ((DIsoH‘𝑘)‘𝑤)
22	18, 21	cfv 6481	. . . . . . . . . 10 class (((DIsoH‘𝑘)‘𝑤)‘𝑦)
23	16, 22	wss 3902	. . . . . . . . 9 wff 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)
24	5, 13	cfv 6481	. . . . . . . . 9 class (Base‘𝑘)
25	23, 17, 24	crab 3395	. . . . . . . 8 class {𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}
26		cglb 18216	. . . . . . . . 9 class glb
27	5, 26	cfv 6481	. . . . . . . 8 class (glb‘𝑘)
28	25, 27	cfv 6481	. . . . . . 7 class ((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})
29		coc 17169	. . . . . . . 8 class oc
30	5, 29	cfv 6481	. . . . . . 7 class (oc‘𝑘)
31	28, 30	cfv 6481	. . . . . 6 class ((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))
32	31, 21	cfv 6481	. . . . 5 class (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))
33	8, 15, 32	cmpt 5172	. . . 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 5172	. . 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 5172	. 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  41394