Step |
Hyp |
Ref |
Expression |
1 |
|
sraassa.a |
|- A = ( ( subringAlg ` W ) ` S ) |
2 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
3 |
2
|
subrgss |
|- ( S e. ( SubRing ` W ) -> S C_ ( Base ` W ) ) |
4 |
3
|
adantl |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> S C_ ( Base ` W ) ) |
5 |
|
eqid |
|- ( Cntr ` ( mulGrp ` W ) ) = ( Cntr ` ( mulGrp ` W ) ) |
6 |
2 5
|
crngbascntr |
|- ( W e. CRing -> ( Base ` W ) = ( Cntr ` ( mulGrp ` W ) ) ) |
7 |
6
|
adantr |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> ( Base ` W ) = ( Cntr ` ( mulGrp ` W ) ) ) |
8 |
4 7
|
sseqtrd |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> S C_ ( Cntr ` ( mulGrp ` W ) ) ) |
9 |
|
crngring |
|- ( W e. CRing -> W e. Ring ) |
10 |
9
|
adantr |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> W e. Ring ) |
11 |
|
simpr |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> S e. ( SubRing ` W ) ) |
12 |
1 5 10 11
|
sraassab |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> ( A e. AssAlg <-> S C_ ( Cntr ` ( mulGrp ` W ) ) ) ) |
13 |
8 12
|
mpbird |
|- ( ( W e. CRing /\ S e. ( SubRing ` W ) ) -> A e. AssAlg ) |