Description: A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | isnvc2.1 | |- F = ( Scalar ` W ) |
|
Assertion | isnvc2 | |- ( W e. NrmVec <-> ( W e. NrmMod /\ F e. DivRing ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvc2.1 | |- F = ( Scalar ` W ) |
|
2 | isnvc | |- ( W e. NrmVec <-> ( W e. NrmMod /\ W e. LVec ) ) |
|
3 | nlmlmod | |- ( W e. NrmMod -> W e. LMod ) |
|
4 | 1 | islvec | |- ( W e. LVec <-> ( W e. LMod /\ F e. DivRing ) ) |
5 | 4 | baib | |- ( W e. LMod -> ( W e. LVec <-> F e. DivRing ) ) |
6 | 3 5 | syl | |- ( W e. NrmMod -> ( W e. LVec <-> F e. DivRing ) ) |
7 | 6 | pm5.32i | |- ( ( W e. NrmMod /\ W e. LVec ) <-> ( W e. NrmMod /\ F e. DivRing ) ) |
8 | 2 7 | bitri | |- ( W e. NrmVec <-> ( W e. NrmMod /\ F e. DivRing ) ) |