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Definition df-padd 35576
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 35575 . 2 class +𝑃
2 vl . . 3 setvar 𝑙
3 cvv 3391 . . 3 class V
4 vm . . . 4 setvar 𝑚
5 vn . . . 4 setvar 𝑛
62cv 1636 . . . . . 6 class 𝑙
7 catm 35043 . . . . . 6 class Atoms
86, 7cfv 6101 . . . . 5 class (Atoms‘𝑙)
98cpw 4351 . . . 4 class 𝒫 (Atoms‘𝑙)
104cv 1636 . . . . . 6 class 𝑚
115cv 1636 . . . . . 6 class 𝑛
1210, 11cun 3767 . . . . 5 class (𝑚𝑛)
13 vp . . . . . . . . . 10 setvar 𝑝
1413cv 1636 . . . . . . . . 9 class 𝑝
15 vq . . . . . . . . . . 11 setvar 𝑞
1615cv 1636 . . . . . . . . . 10 class 𝑞
17 vr . . . . . . . . . . 11 setvar 𝑟
1817cv 1636 . . . . . . . . . 10 class 𝑟
19 cjn 17149 . . . . . . . . . . 11 class join
206, 19cfv 6101 . . . . . . . . . 10 class (join‘𝑙)
2116, 18, 20co 6874 . . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22 cple 16160 . . . . . . . . . 10 class le
236, 22cfv 6101 . . . . . . . . 9 class (le‘𝑙)
2414, 21, 23wbr 4844 . . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2524, 17, 11wrex 3097 . . . . . . 7 wff 𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2625, 15, 10wrex 3097 . . . . . 6 wff 𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
2726, 13, 8crab 3100 . . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
2812, 27cun 3767 . . . 4 class ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
294, 5, 9, 9, 28cmpt2 6876 . . 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 4923 . 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 1637 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  35577
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