Description: The span of a singleton is an additive subgroup (frequently used special case of lspcl ). (Contributed by Mario Carneiro, 21-Apr-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lspsnsubg.v | |- V = ( Base ` W ) |
|
lspsnsubg.n | |- N = ( LSpan ` W ) |
||
Assertion | lspsnsubg | |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( SubGrp ` W ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnsubg.v | |- V = ( Base ` W ) |
|
2 | lspsnsubg.n | |- N = ( LSpan ` W ) |
|
3 | eqid | |- ( LSubSp ` W ) = ( LSubSp ` W ) |
|
4 | 1 3 2 | lspsncl | |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` W ) ) |
5 | 3 | lsssubg | |- ( ( W e. LMod /\ ( N ` { X } ) e. ( LSubSp ` W ) ) -> ( N ` { X } ) e. ( SubGrp ` W ) ) |
6 | 4 5 | syldan | |- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( SubGrp ` W ) ) |