Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( .sf ` W ) = ( .sf ` W ) |
2 |
|
eqid |
|- ( TopOpen ` W ) = ( TopOpen ` W ) |
3 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
4 |
|
eqid |
|- ( TopOpen ` ( Scalar ` W ) ) = ( TopOpen ` ( Scalar ` W ) ) |
5 |
1 2 3 4
|
istlm |
|- ( W e. TopMod <-> ( ( W e. TopMnd /\ W e. LMod /\ ( Scalar ` W ) e. TopRing ) /\ ( .sf ` W ) e. ( ( ( TopOpen ` ( Scalar ` W ) ) tX ( TopOpen ` W ) ) Cn ( TopOpen ` W ) ) ) ) |
6 |
5
|
simplbi |
|- ( W e. TopMod -> ( W e. TopMnd /\ W e. LMod /\ ( Scalar ` W ) e. TopRing ) ) |
7 |
6
|
simp2d |
|- ( W e. TopMod -> W e. LMod ) |