Definition df-padd 39790

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 39789	. 2 class +𝑃
2		vl	. . 3 setvar 𝑙
3		cvv 3447	. . 3 class V
4		vm	. . . 4 setvar 𝑚
5		vn	. . . 4 setvar 𝑛
6	2	cv 1539	. . . . . 6 class 𝑙
7		catm 39256	. . . . . 6 class Atoms
8	6, 7	cfv 6511	. . . . 5 class (Atoms‘𝑙)
9	8	cpw 4563	. . . 4 class 𝒫 (Atoms‘𝑙)
10	4	cv 1539	. . . . . 6 class 𝑚
11	5	cv 1539	. . . . . 6 class 𝑛
12	10, 11	cun 3912	. . . . 5 class (𝑚 ∪ 𝑛)
13		vp	. . . . . . . . . 10 setvar 𝑝
14	13	cv 1539	. . . . . . . . 9 class 𝑝
15		vq	. . . . . . . . . . 11 setvar 𝑞
16	15	cv 1539	. . . . . . . . . 10 class 𝑞
17		vr	. . . . . . . . . . 11 setvar 𝑟
18	17	cv 1539	. . . . . . . . . 10 class 𝑟
19		cjn 18272	. . . . . . . . . . 11 class join
20	6, 19	cfv 6511	. . . . . . . . . 10 class (join‘𝑙)
21	16, 18, 20	co 7387	. . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22		cple 17227	. . . . . . . . . 10 class le
23	6, 22	cfv 6511	. . . . . . . . 9 class (le‘𝑙)
24	14, 21, 23	wbr 5107	. . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
25	24, 17, 11	wrex 3053	. . . . . . 7 wff ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
26	25, 15, 10	wrex 3053	. . . . . 6 wff ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
27	26, 13, 8	crab 3405	. . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
28	12, 27	cun 3912	. . . 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 7389	. . 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 5188	. 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 1540	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  39791