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Definition df-padd 33894
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 33893 . 2 class +𝑃
2 vl . . 3 setvar 𝑙
3 cvv 3173 . . 3 class V
4 vm . . . 4 setvar 𝑚
5 vn . . . 4 setvar 𝑛
62cv 1474 . . . . . 6 class 𝑙
7 catm 33362 . . . . . 6 class Atoms
86, 7cfv 5790 . . . . 5 class (Atoms‘𝑙)
98cpw 4108 . . . 4 class 𝒫 (Atoms‘𝑙)
104cv 1474 . . . . . 6 class 𝑚
115cv 1474 . . . . . 6 class 𝑛
1210, 11cun 3538 . . . . 5 class (𝑚𝑛)
13 vp . . . . . . . . . 10 setvar 𝑝
1413cv 1474 . . . . . . . . 9 class 𝑝
15 vq . . . . . . . . . . 11 setvar 𝑞
1615cv 1474 . . . . . . . . . 10 class 𝑞
17 vr . . . . . . . . . . 11 setvar 𝑟
1817cv 1474 . . . . . . . . . 10 class 𝑟
19 cjn 16716 . . . . . . . . . . 11 class join
206, 19cfv 5790 . . . . . . . . . 10 class (join‘𝑙)
2116, 18, 20co 6527 . . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22 cple 15724 . . . . . . . . . 10 class le
236, 22cfv 5790 . . . . . . . . 9 class (le‘𝑙)
2414, 21, 23wbr 4578 . . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2524, 17, 11wrex 2897 . . . . . . 7 wff 𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2625, 15, 10wrex 2897 . . . . . 6 wff 𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2726, 13, 8crab 2900 . . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
2812, 27cun 3538 . . . 4 class ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
294, 5, 9, 9, 28cmpt2 6529 . . 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 4638 . 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 1475 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  33895
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