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Definition df-padd 37064
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 37063 . 2 class +𝑃
2 vl . . 3 setvar 𝑙
3 cvv 3480 . . 3 class V
4 vm . . . 4 setvar 𝑚
5 vn . . . 4 setvar 𝑛
62cv 1537 . . . . . 6 class 𝑙
7 catm 36531 . . . . . 6 class Atoms
86, 7cfv 6345 . . . . 5 class (Atoms‘𝑙)
98cpw 4522 . . . 4 class 𝒫 (Atoms‘𝑙)
104cv 1537 . . . . . 6 class 𝑚
115cv 1537 . . . . . 6 class 𝑛
1210, 11cun 3917 . . . . 5 class (𝑚𝑛)
13 vp . . . . . . . . . 10 setvar 𝑝
1413cv 1537 . . . . . . . . 9 class 𝑝
15 vq . . . . . . . . . . 11 setvar 𝑞
1615cv 1537 . . . . . . . . . 10 class 𝑞
17 vr . . . . . . . . . . 11 setvar 𝑟
1817cv 1537 . . . . . . . . . 10 class 𝑟
19 cjn 17556 . . . . . . . . . . 11 class join
206, 19cfv 6345 . . . . . . . . . 10 class (join‘𝑙)
2116, 18, 20co 7151 . . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22 cple 16574 . . . . . . . . . 10 class le
236, 22cfv 6345 . . . . . . . . 9 class (le‘𝑙)
2414, 21, 23wbr 5053 . . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2524, 17, 11wrex 3134 . . . . . . 7 wff 𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2625, 15, 10wrex 3134 . . . . . 6 wff 𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2726, 13, 8crab 3137 . . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
2812, 27cun 3917 . . . 4 class ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
294, 5, 9, 9, 28cmpo 7153 . . 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 5133 . 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 1538 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  37065
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