Step |
Hyp |
Ref |
Expression |
1 |
|
subrgsubg |
|- ( r e. ( SubRing ` R ) -> r e. ( SubGrp ` R ) ) |
2 |
1
|
ssriv |
|- ( SubRing ` R ) C_ ( SubGrp ` R ) |
3 |
|
sstr |
|- ( ( S C_ ( SubRing ` R ) /\ ( SubRing ` R ) C_ ( SubGrp ` R ) ) -> S C_ ( SubGrp ` R ) ) |
4 |
2 3
|
mpan2 |
|- ( S C_ ( SubRing ` R ) -> S C_ ( SubGrp ` R ) ) |
5 |
|
subgint |
|- ( ( S C_ ( SubGrp ` R ) /\ S =/= (/) ) -> |^| S e. ( SubGrp ` R ) ) |
6 |
4 5
|
sylan |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> |^| S e. ( SubGrp ` R ) ) |
7 |
|
ssel2 |
|- ( ( S C_ ( SubRing ` R ) /\ r e. S ) -> r e. ( SubRing ` R ) ) |
8 |
7
|
adantlr |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ r e. S ) -> r e. ( SubRing ` R ) ) |
9 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
10 |
9
|
subrg1cl |
|- ( r e. ( SubRing ` R ) -> ( 1r ` R ) e. r ) |
11 |
8 10
|
syl |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ r e. S ) -> ( 1r ` R ) e. r ) |
12 |
11
|
ralrimiva |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> A. r e. S ( 1r ` R ) e. r ) |
13 |
|
fvex |
|- ( 1r ` R ) e. _V |
14 |
13
|
elint2 |
|- ( ( 1r ` R ) e. |^| S <-> A. r e. S ( 1r ` R ) e. r ) |
15 |
12 14
|
sylibr |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> ( 1r ` R ) e. |^| S ) |
16 |
8
|
adantlr |
|- ( ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> r e. ( SubRing ` R ) ) |
17 |
|
simprl |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S ) |
18 |
|
elinti |
|- ( x e. |^| S -> ( r e. S -> x e. r ) ) |
19 |
18
|
imp |
|- ( ( x e. |^| S /\ r e. S ) -> x e. r ) |
20 |
17 19
|
sylan |
|- ( ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> x e. r ) |
21 |
|
simprr |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S ) |
22 |
|
elinti |
|- ( y e. |^| S -> ( r e. S -> y e. r ) ) |
23 |
22
|
imp |
|- ( ( y e. |^| S /\ r e. S ) -> y e. r ) |
24 |
21 23
|
sylan |
|- ( ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> y e. r ) |
25 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
26 |
25
|
subrgmcl |
|- ( ( r e. ( SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r ) |
27 |
16 20 24 26
|
syl3anc |
|- ( ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> ( x ( .r ` R ) y ) e. r ) |
28 |
27
|
ralrimiva |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. r e. S ( x ( .r ` R ) y ) e. r ) |
29 |
|
ovex |
|- ( x ( .r ` R ) y ) e. _V |
30 |
29
|
elint2 |
|- ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) |
31 |
28 30
|
sylibr |
|- ( ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( .r ` R ) y ) e. |^| S ) |
32 |
31
|
ralrimivva |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) |
33 |
|
ssn0 |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> ( SubRing ` R ) =/= (/) ) |
34 |
|
n0 |
|- ( ( SubRing ` R ) =/= (/) <-> E. r r e. ( SubRing ` R ) ) |
35 |
|
subrgrcl |
|- ( r e. ( SubRing ` R ) -> R e. Ring ) |
36 |
35
|
exlimiv |
|- ( E. r r e. ( SubRing ` R ) -> R e. Ring ) |
37 |
34 36
|
sylbi |
|- ( ( SubRing ` R ) =/= (/) -> R e. Ring ) |
38 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
39 |
38 9 25
|
issubrg2 |
|- ( R e. Ring -> ( |^| S e. ( SubRing ` R ) <-> ( |^| S e. ( SubGrp ` R ) /\ ( 1r ` R ) e. |^| S /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) ) |
40 |
33 37 39
|
3syl |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> ( |^| S e. ( SubRing ` R ) <-> ( |^| S e. ( SubGrp ` R ) /\ ( 1r ` R ) e. |^| S /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) ) |
41 |
6 15 32 40
|
mpbir3and |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> |^| S e. ( SubRing ` R ) ) |