Metamath Proof Explorer


Definition df-lshyp

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)

Ref Expression
Assertion df-lshyp LSHyp = w V s LSubSp w | s Base w v Base w LSpan w s v = Base w

Detailed syntax breakdown

Step Hyp Ref Expression
0 clsh class LSHyp
1 vw setvar w
2 cvv class V
3 vs setvar s
4 clss class LSubSp
5 1 cv setvar w
6 5 4 cfv class LSubSp w
7 3 cv setvar s
8 cbs class Base
9 5 8 cfv class Base w
10 7 9 wne wff s Base w
11 vv setvar v
12 clspn class LSpan
13 5 12 cfv class LSpan w
14 11 cv setvar v
15 14 csn class v
16 7 15 cun class s v
17 16 13 cfv class LSpan w s v
18 17 9 wceq wff LSpan w s v = Base w
19 18 11 9 wrex wff v Base w LSpan w s v = Base w
20 10 19 wa wff s Base w v Base w LSpan w s v = Base w
21 20 3 6 crab class s LSubSp w | s Base w v Base w LSpan w s v = Base w
22 1 2 21 cmpt class w V s LSubSp w | s Base w v Base w LSpan w s v = Base w
23 0 22 wceq wff LSHyp = w V s LSubSp w | s Base w v Base w LSpan w s v = Base w