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Definition df-docaN 36228
Description: Define subspace orthocomplement for DVecA partial 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, 6-Dec-2013.)
Assertion
Ref Expression
df-docaN ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
Distinct variable group:   𝑤,𝑘,𝑥,𝑧

Detailed syntax breakdown of Definition df-docaN
StepHypRef Expression
1 cocaN 36227 . 2 class ocA
2 vk . . 3 setvar 𝑘
3 cvv 3195 . . 3 class V
4 vw . . . 4 setvar 𝑤
52cv 1480 . . . . 5 class 𝑘
6 clh 35089 . . . . 5 class LHyp
75, 6cfv 5876 . . . 4 class (LHyp‘𝑘)
8 vx . . . . 5 setvar 𝑥
94cv 1480 . . . . . . 7 class 𝑤
10 cltrn 35206 . . . . . . . 8 class LTrn
115, 10cfv 5876 . . . . . . 7 class (LTrn‘𝑘)
129, 11cfv 5876 . . . . . 6 class ((LTrn‘𝑘)‘𝑤)
1312cpw 4149 . . . . 5 class 𝒫 ((LTrn‘𝑘)‘𝑤)
148cv 1480 . . . . . . . . . . . . 13 class 𝑥
15 vz . . . . . . . . . . . . . 14 setvar 𝑧
1615cv 1480 . . . . . . . . . . . . 13 class 𝑧
1714, 16wss 3567 . . . . . . . . . . . 12 wff 𝑥𝑧
18 cdia 36136 . . . . . . . . . . . . . . 15 class DIsoA
195, 18cfv 5876 . . . . . . . . . . . . . 14 class (DIsoA‘𝑘)
209, 19cfv 5876 . . . . . . . . . . . . 13 class ((DIsoA‘𝑘)‘𝑤)
2120crn 5105 . . . . . . . . . . . 12 class ran ((DIsoA‘𝑘)‘𝑤)
2217, 15, 21crab 2913 . . . . . . . . . . 11 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}
2322cint 4466 . . . . . . . . . 10 class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}
2420ccnv 5103 . . . . . . . . . 10 class ((DIsoA‘𝑘)‘𝑤)
2523, 24cfv 5876 . . . . . . . . 9 class (((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧})
26 coc 15930 . . . . . . . . . 10 class oc
275, 26cfv 5876 . . . . . . . . 9 class (oc‘𝑘)
2825, 27cfv 5876 . . . . . . . 8 class ((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))
299, 27cfv 5876 . . . . . . . 8 class ((oc‘𝑘)‘𝑤)
30 cjn 16925 . . . . . . . . 9 class join
315, 30cfv 5876 . . . . . . . 8 class (join‘𝑘)
3228, 29, 31co 6635 . . . . . . 7 class (((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))
33 cmee 16926 . . . . . . . 8 class meet
345, 33cfv 5876 . . . . . . 7 class (meet‘𝑘)
3532, 9, 34co 6635 . . . . . 6 class ((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)
3635, 20cfv 5876 . . . . 5 class (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))
378, 13, 36cmpt 4720 . . . 4 class (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))
384, 7, 37cmpt 4720 . . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))
392, 3, 38cmpt 4720 . 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
401, 39wceq 1481 1 wff ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
Colors of variables: wff setvar class
This definition is referenced by:  docaffvalN  36229
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