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Definition df-padd 36814
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 36813 . 2 class +𝑃
2 vl . . 3 setvar 𝑙
3 cvv 3495 . . 3 class V
4 vm . . . 4 setvar 𝑚
5 vn . . . 4 setvar 𝑛
62cv 1527 . . . . . 6 class 𝑙
7 catm 36281 . . . . . 6 class Atoms
86, 7cfv 6349 . . . . 5 class (Atoms‘𝑙)
98cpw 4537 . . . 4 class 𝒫 (Atoms‘𝑙)
104cv 1527 . . . . . 6 class 𝑚
115cv 1527 . . . . . 6 class 𝑛
1210, 11cun 3933 . . . . 5 class (𝑚𝑛)
13 vp . . . . . . . . . 10 setvar 𝑝
1413cv 1527 . . . . . . . . 9 class 𝑝
15 vq . . . . . . . . . . 11 setvar 𝑞
1615cv 1527 . . . . . . . . . 10 class 𝑞
17 vr . . . . . . . . . . 11 setvar 𝑟
1817cv 1527 . . . . . . . . . 10 class 𝑟
19 cjn 17544 . . . . . . . . . . 11 class join
206, 19cfv 6349 . . . . . . . . . 10 class (join‘𝑙)
2116, 18, 20co 7145 . . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22 cple 16562 . . . . . . . . . 10 class le
236, 22cfv 6349 . . . . . . . . 9 class (le‘𝑙)
2414, 21, 23wbr 5058 . . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2524, 17, 11wrex 3139 . . . . . . 7 wff 𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2625, 15, 10wrex 3139 . . . . . 6 wff 𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2726, 13, 8crab 3142 . . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
2812, 27cun 3933 . . . 4 class ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
294, 5, 9, 9, 28cmpo 7147 . . 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 5138 . 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 1528 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  36815
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