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Definition df-padd 35400
Description: Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
Assertion
Ref Expression
df-padd +𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
Distinct variable group:   𝑚,𝑙,𝑛,𝑝,𝑞,𝑟

Detailed syntax breakdown of Definition df-padd
StepHypRef Expression
1 cpadd 35399 . 2 class +𝑃
2 vl . . 3 setvar 𝑙
3 cvv 3231 . . 3 class V
4 vm . . . 4 setvar 𝑚
5 vn . . . 4 setvar 𝑛
62cv 1522 . . . . . 6 class 𝑙
7 catm 34868 . . . . . 6 class Atoms
86, 7cfv 5926 . . . . 5 class (Atoms‘𝑙)
98cpw 4191 . . . 4 class 𝒫 (Atoms‘𝑙)
104cv 1522 . . . . . 6 class 𝑚
115cv 1522 . . . . . 6 class 𝑛
1210, 11cun 3605 . . . . 5 class (𝑚𝑛)
13 vp . . . . . . . . . 10 setvar 𝑝
1413cv 1522 . . . . . . . . 9 class 𝑝
15 vq . . . . . . . . . . 11 setvar 𝑞
1615cv 1522 . . . . . . . . . 10 class 𝑞
17 vr . . . . . . . . . . 11 setvar 𝑟
1817cv 1522 . . . . . . . . . 10 class 𝑟
19 cjn 16991 . . . . . . . . . . 11 class join
206, 19cfv 5926 . . . . . . . . . 10 class (join‘𝑙)
2116, 18, 20co 6690 . . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22 cple 15995 . . . . . . . . . 10 class le
236, 22cfv 5926 . . . . . . . . 9 class (le‘𝑙)
2414, 21, 23wbr 4685 . . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2524, 17, 11wrex 2942 . . . . . . 7 wff 𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2625, 15, 10wrex 2942 . . . . . 6 wff 𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2726, 13, 8crab 2945 . . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
2812, 27cun 3605 . . . 4 class ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
294, 5, 9, 9, 28cmpt2 6692 . . 3 class (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}))
302, 3, 29cmpt 4762 . 2 class (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
311, 30wceq 1523 1 wff +𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
Colors of variables: wff setvar class
This definition is referenced by:  paddfval  35401
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