Definition df-lshyp 34264

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 that the full space. (Contributed by NM, 29-Jun-2014.)

Assertion

Ref	Expression
df-lshyp	|- LSHyp = ( w e. _V |-> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } )

Distinct variable group:   v, s, w

Detailed syntax breakdown of Definition df-lshyp

Step	Hyp	Ref	Expression
1		clsh 34262	. 2 class LSHyp
2		vw	. . 3 setvar w
3		cvv 3200	. . 3 class _V
4		vs	. . . . . . 7 setvar s
5	4	cv 1482	. . . . . 6 class s
6	2	cv 1482	. . . . . . 7 class w
7		cbs 15857	. . . . . . 7 class Base
8	6, 7	cfv 5888	. . . . . 6 class ( Base ` w )
9	5, 8	wne 2794	. . . . 5 wff s =/= ( Base ` w )
10		vv	. . . . . . . . . . 11 setvar v
11	10	cv 1482	. . . . . . . . . 10 class v
12	11	csn 4177	. . . . . . . . 9 class { v }
13	5, 12	cun 3572	. . . . . . . 8 class ( s u. { v } )
14		clspn 18971	. . . . . . . . 9 class LSpan
15	6, 14	cfv 5888	. . . . . . . 8 class ( LSpan ` w )
16	13, 15	cfv 5888	. . . . . . 7 class ( ( LSpan ` w ) ` ( s u. { v } ) )
17	16, 8	wceq 1483	. . . . . 6 wff ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w )
18	17, 10, 8	wrex 2913	. . . . 5 wff E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w )
19	9, 18	wa 384	. . . 4 wff ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) )
20		clss 18932	. . . . 5 class LSubSp
21	6, 20	cfv 5888	. . . 4 class ( LSubSp ` w )
22	19, 4, 21	crab 2916	. . 3 class { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) }
23	2, 3, 22	cmpt 4729	. 2 class ( w e. _V |-> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } )
24	1, 23	wceq 1483	1 wff LSHyp = ( w e. _V |-> { s e. ( LSubSp ` w ) | ( s =/= ( Base ` w ) /\ E. v e. ( Base ` w ) ( ( LSpan ` w ) ` ( s u. { v } ) ) = ( Base ` w ) ) } )

This definition is referenced by:  lshpset  34265