Definition df-padd 40176

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) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))

Distinct variable group:   𝑚,𝑙,𝑛,𝑝,𝑞,𝑟

Detailed syntax breakdown of Definition df-padd

Step	Hyp	Ref	Expression
1		cpadd 40175	. 2 class +𝑃
2		vl	. . 3 setvar 𝑙
3		cvv 3442	. . 3 class V
4		vm	. . . 4 setvar 𝑚
5		vn	. . . 4 setvar 𝑛
6	2	cv 1541	. . . . . 6 class 𝑙
7		catm 39643	. . . . . 6 class Atoms
8	6, 7	cfv 6500	. . . . 5 class (Atoms‘𝑙)
9	8	cpw 4556	. . . 4 class 𝒫 (Atoms‘𝑙)
10	4	cv 1541	. . . . . 6 class 𝑚
11	5	cv 1541	. . . . . 6 class 𝑛
12	10, 11	cun 3901	. . . . 5 class (𝑚 ∪ 𝑛)
13		vp	. . . . . . . . . 10 setvar 𝑝
14	13	cv 1541	. . . . . . . . 9 class 𝑝
15		vq	. . . . . . . . . . 11 setvar 𝑞
16	15	cv 1541	. . . . . . . . . 10 class 𝑞
17		vr	. . . . . . . . . . 11 setvar 𝑟
18	17	cv 1541	. . . . . . . . . 10 class 𝑟
19		cjn 18246	. . . . . . . . . . 11 class join
20	6, 19	cfv 6500	. . . . . . . . . 10 class (join‘𝑙)
21	16, 18, 20	co 7368	. . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22		cple 17196	. . . . . . . . . 10 class le
23	6, 22	cfv 6500	. . . . . . . . 9 class (le‘𝑙)
24	14, 21, 23	wbr 5100	. . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
25	24, 17, 11	wrex 3062	. . . . . . 7 wff ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
26	25, 15, 10	wrex 3062	. . . . . 6 wff ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
27	26, 13, 8	crab 3401	. . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
28	12, 27	cun 3901	. . . 4 class ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})
29	4, 5, 9, 9, 28	cmpo 7370	. . 3 class (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}))
30	2, 3, 29	cmpt 5181	. 2 class (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
31	1, 30	wceq 1542	1 wff +𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))

This definition is referenced by:  paddfval  40177