Definition df-padd 40242

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 40241	. 2 class +𝑃
2		vl	. . 3 setvar 𝑙
3		cvv 3429	. . 3 class V
4		vm	. . . 4 setvar 𝑚
5		vn	. . . 4 setvar 𝑛
6	2	cv 1541	. . . . . 6 class 𝑙
7		catm 39709	. . . . . 6 class Atoms
8	6, 7	cfv 6498	. . . . 5 class (Atoms‘𝑙)
9	8	cpw 4541	. . . 4 class 𝒫 (Atoms‘𝑙)
10	4	cv 1541	. . . . . 6 class 𝑚
11	5	cv 1541	. . . . . 6 class 𝑛
12	10, 11	cun 3887	. . . . 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 18277	. . . . . . . . . . 11 class join
20	6, 19	cfv 6498	. . . . . . . . . 10 class (join‘𝑙)
21	16, 18, 20	co 7367	. . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22		cple 17227	. . . . . . . . . 10 class le
23	6, 22	cfv 6498	. . . . . . . . 9 class (le‘𝑙)
24	14, 21, 23	wbr 5085	. . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
25	24, 17, 11	wrex 3061	. . . . . . 7 wff ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
26	25, 15, 10	wrex 3061	. . . . . 6 wff ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
27	26, 13, 8	crab 3389	. . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
28	12, 27	cun 3887	. . . 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 7369	. . 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 5166	. 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  40243