Metamath Proof Explorer


Theorem subrgint

Description: The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014) (Revised by Mario Carneiro, 7-Dec-2014)

Ref Expression
Assertion subrgint
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> |^| S e. ( SubRing ` R ) )

Proof

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 ) )