Definition df-lshyp 36998

Description: Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less than the full space. (Contributed by NM, 29-Jun-2014.)

Assertion

Ref	Expression
df-lshyp	⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})

Distinct variable group:   𝑣,𝑠,𝑤

Detailed syntax breakdown of Definition df-lshyp

Step	Hyp	Ref	Expression
1		clsh 36996	. 2 class LSHyp
2		vw	. . 3 setvar 𝑤
3		cvv 3433	. . 3 class V
4		vs	. . . . . . 7 setvar 𝑠
5	4	cv 1538	. . . . . 6 class 𝑠
6	2	cv 1538	. . . . . . 7 class 𝑤
7		cbs 16921	. . . . . . 7 class Base
8	6, 7	cfv 6437	. . . . . 6 class (Base‘𝑤)
9	5, 8	wne 2944	. . . . 5 wff 𝑠 ≠ (Base‘𝑤)
10		vv	. . . . . . . . . . 11 setvar 𝑣
11	10	cv 1538	. . . . . . . . . 10 class 𝑣
12	11	csn 4562	. . . . . . . . 9 class {𝑣}
13	5, 12	cun 3886	. . . . . . . 8 class (𝑠 ∪ {𝑣})
14		clspn 20242	. . . . . . . . 9 class LSpan
15	6, 14	cfv 6437	. . . . . . . 8 class (LSpan‘𝑤)
16	13, 15	cfv 6437	. . . . . . 7 class ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣}))
17	16, 8	wceq 1539	. . . . . 6 wff ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)
18	17, 10, 8	wrex 3066	. . . . 5 wff ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)
19	9, 18	wa 396	. . . 4 wff (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))
20		clss 20202	. . . . 5 class LSubSp
21	6, 20	cfv 6437	. . . 4 class (LSubSp‘𝑤)
22	19, 4, 21	crab 3069	. . 3 class {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))}
23	2, 3, 22	cmpt 5158	. 2 class (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
24	1, 23	wceq 1539	1 wff LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})

This definition is referenced by:  lshpset  36999