| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rgspnval.r |
|- ( ph -> R e. Ring ) |
| 2 |
|
rgspnval.b |
|- ( ph -> B = ( Base ` R ) ) |
| 3 |
|
rgspnval.ss |
|- ( ph -> A C_ B ) |
| 4 |
|
rgspnval.n |
|- ( ph -> N = ( RingSpan ` R ) ) |
| 5 |
|
rgspnval.sp |
|- ( ph -> U = ( N ` A ) ) |
| 6 |
|
rgspnmin.sr |
|- ( ph -> S e. ( SubRing ` R ) ) |
| 7 |
|
rgspnmin.ss |
|- ( ph -> A C_ S ) |
| 8 |
1 2 3 4 5
|
rgspnval |
|- ( ph -> U = |^| { t e. ( SubRing ` R ) | A C_ t } ) |
| 9 |
|
sseq2 |
|- ( t = S -> ( A C_ t <-> A C_ S ) ) |
| 10 |
9
|
elrab |
|- ( S e. { t e. ( SubRing ` R ) | A C_ t } <-> ( S e. ( SubRing ` R ) /\ A C_ S ) ) |
| 11 |
6 7 10
|
sylanbrc |
|- ( ph -> S e. { t e. ( SubRing ` R ) | A C_ t } ) |
| 12 |
|
intss1 |
|- ( S e. { t e. ( SubRing ` R ) | A C_ t } -> |^| { t e. ( SubRing ` R ) | A C_ t } C_ S ) |
| 13 |
11 12
|
syl |
|- ( ph -> |^| { t e. ( SubRing ` R ) | A C_ t } C_ S ) |
| 14 |
8 13
|
eqsstrd |
|- ( ph -> U C_ S ) |