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Definition df-lines 33605
Description: Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.)
Assertion
Ref Expression
df-lines Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
Distinct variable group:   𝑘,𝑝,𝑞,𝑟,𝑠

Detailed syntax breakdown of Definition df-lines
StepHypRef Expression
1 clines 33598 . 2 class Lines
2 vk . . 3 setvar 𝑘
3 cvv 3169 . . 3 class V
4 vq . . . . . . . . 9 setvar 𝑞
54cv 1473 . . . . . . . 8 class 𝑞
6 vr . . . . . . . . 9 setvar 𝑟
76cv 1473 . . . . . . . 8 class 𝑟
85, 7wne 2776 . . . . . . 7 wff 𝑞𝑟
9 vs . . . . . . . . 9 setvar 𝑠
109cv 1473 . . . . . . . 8 class 𝑠
11 vp . . . . . . . . . . 11 setvar 𝑝
1211cv 1473 . . . . . . . . . 10 class 𝑝
132cv 1473 . . . . . . . . . . . 12 class 𝑘
14 cjn 16710 . . . . . . . . . . . 12 class join
1513, 14cfv 5787 . . . . . . . . . . 11 class (join‘𝑘)
165, 7, 15co 6524 . . . . . . . . . 10 class (𝑞(join‘𝑘)𝑟)
17 cple 15718 . . . . . . . . . . 11 class le
1813, 17cfv 5787 . . . . . . . . . 10 class (le‘𝑘)
1912, 16, 18wbr 4574 . . . . . . . . 9 wff 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)
20 catm 33368 . . . . . . . . . 10 class Atoms
2113, 20cfv 5787 . . . . . . . . 9 class (Atoms‘𝑘)
2219, 11, 21crab 2896 . . . . . . . 8 class {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}
2310, 22wceq 1474 . . . . . . 7 wff 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}
248, 23wa 382 . . . . . 6 wff (𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})
2524, 6, 21wrex 2893 . . . . 5 wff 𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})
2625, 4, 21wrex 2893 . . . 4 wff 𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})
2726, 9cab 2592 . . 3 class {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})}
282, 3, 27cmpt 4634 . 2 class (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
291, 28wceq 1474 1 wff Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞𝑟𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
Colors of variables: wff setvar class
This definition is referenced by:  lineset  33842
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