Step |
Hyp |
Ref |
Expression |
1 |
|
submss.b |
|- B = ( Base ` M ) |
2 |
|
submrcl |
|- ( S e. ( SubMnd ` M ) -> M e. Mnd ) |
3 |
|
eqid |
|- ( 0g ` M ) = ( 0g ` M ) |
4 |
|
eqid |
|- ( M |`s S ) = ( M |`s S ) |
5 |
1 3 4
|
issubm2 |
|- ( M e. Mnd -> ( S e. ( SubMnd ` M ) <-> ( S C_ B /\ ( 0g ` M ) e. S /\ ( M |`s S ) e. Mnd ) ) ) |
6 |
2 5
|
syl |
|- ( S e. ( SubMnd ` M ) -> ( S e. ( SubMnd ` M ) <-> ( S C_ B /\ ( 0g ` M ) e. S /\ ( M |`s S ) e. Mnd ) ) ) |
7 |
6
|
ibi |
|- ( S e. ( SubMnd ` M ) -> ( S C_ B /\ ( 0g ` M ) e. S /\ ( M |`s S ) e. Mnd ) ) |
8 |
7
|
simp1d |
|- ( S e. ( SubMnd ` M ) -> S C_ B ) |