Definition df-padd 39995

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 39994	. 2 class +𝑃
2		vl	. . 3 setvar 𝑙
3		cvv 3438	. . 3 class V
4		vm	. . . 4 setvar 𝑚
5		vn	. . . 4 setvar 𝑛
6	2	cv 1540	. . . . . 6 class 𝑙
7		catm 39462	. . . . . 6 class Atoms
8	6, 7	cfv 6490	. . . . 5 class (Atoms‘𝑙)
9	8	cpw 4552	. . . 4 class 𝒫 (Atoms‘𝑙)
10	4	cv 1540	. . . . . 6 class 𝑚
11	5	cv 1540	. . . . . 6 class 𝑛
12	10, 11	cun 3897	. . . . 5 class (𝑚 ∪ 𝑛)
13		vp	. . . . . . . . . 10 setvar 𝑝
14	13	cv 1540	. . . . . . . . 9 class 𝑝
15		vq	. . . . . . . . . . 11 setvar 𝑞
16	15	cv 1540	. . . . . . . . . 10 class 𝑞
17		vr	. . . . . . . . . . 11 setvar 𝑟
18	17	cv 1540	. . . . . . . . . 10 class 𝑟
19		cjn 18232	. . . . . . . . . . 11 class join
20	6, 19	cfv 6490	. . . . . . . . . 10 class (join‘𝑙)
21	16, 18, 20	co 7356	. . . . . . . . 9 class (𝑞(join‘𝑙)𝑟)
22		cple 17182	. . . . . . . . . 10 class le
23	6, 22	cfv 6490	. . . . . . . . 9 class (le‘𝑙)
24	14, 21, 23	wbr 5096	. . . . . . . 8 wff 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
25	24, 17, 11	wrex 3058	. . . . . . 7 wff ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
26	25, 15, 10	wrex 3058	. . . . . 6 wff ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)
27	26, 13, 8	crab 3397	. . . . 5 class {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}
28	12, 27	cun 3897	. . . 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 7358	. . 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 5177	. 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 1541	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  39996