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Theorem subrgint 15583
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.)
Assertion
Ref Expression
subrgint  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)

Proof of Theorem subrgint
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 15567 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3197 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3200 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  (SubRing `  R
)  C_  (SubGrp `  R
) )  ->  S  C_  (SubGrp `  R )
)
42, 3mpan2 652 . . 3  |-  ( S 
C_  (SubRing `  R )  ->  S  C_  (SubGrp `  R
) )
5 subgint 14657 . . 3  |-  ( ( S  C_  (SubGrp `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
64, 5sylan 457 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
7 ssel2 3188 . . . . . 6  |-  ( ( S  C_  (SubRing `  R
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
87adantlr 695 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
9 eqid 2296 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 15569 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  r )
118, 10syl 15 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  ( 1r `  R )  e.  r )
1211ralrimiva 2639 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 fvex 5555 . . . 4  |-  ( 1r
`  R )  e. 
_V
1413elint2 3885 . . 3  |-  ( ( 1r `  R )  e.  |^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r )
1512, 14sylibr 203 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  ( 1r `  R )  e.  |^| S )
168adantlr 695 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
17 simprl 732 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
18 elinti 3887 . . . . . . . 8  |-  ( x  e.  |^| S  ->  (
r  e.  S  ->  x  e.  r )
)
1918imp 418 . . . . . . 7  |-  ( ( x  e.  |^| S  /\  r  e.  S
)  ->  x  e.  r )
2017, 19sylan 457 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  x  e.  r )
21 simprr 733 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
22 elinti 3887 . . . . . . . 8  |-  ( y  e.  |^| S  ->  (
r  e.  S  -> 
y  e.  r ) )
2322imp 418 . . . . . . 7  |-  ( ( y  e.  |^| S  /\  r  e.  S
)  ->  y  e.  r )
2421, 23sylan 457 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  y  e.  r )
25 eqid 2296 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
2625subrgmcl 15573 . . . . . 6  |-  ( ( r  e.  (SubRing `  R
)  /\  x  e.  r  /\  y  e.  r )  ->  ( x
( .r `  R
) y )  e.  r )
2716, 20, 24, 26syl3anc 1182 . . . . 5  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  (
x ( .r `  R ) y )  e.  r )
2827ralrimiva 2639 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
29 ovex 5899 . . . . 5  |-  ( x ( .r `  R
) y )  e. 
_V
3029elint2 3885 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  |^| S  <->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
3128, 30sylibr 203 . . 3  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( .r `  R ) y )  e.  |^| S )
3231ralrimivva 2648 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R
) y )  e. 
|^| S )
33 ssn0 3500 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  (SubRing `  R
)  =/=  (/) )
34 n0 3477 . . . 4  |-  ( (SubRing `  R )  =/=  (/)  <->  E. r 
r  e.  (SubRing `  R
) )
35 subrgrcl 15566 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  R  e.  Ring )
3635exlimiv 1624 . . . 4  |-  ( E. r  r  e.  (SubRing `  R )  ->  R  e.  Ring )
3734, 36sylbi 187 . . 3  |-  ( (SubRing `  R )  =/=  (/)  ->  R  e.  Ring )
38 eqid 2296 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3938, 9, 25issubrg2 15581 . . 3  |-  ( 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 ) ) )
4033, 37, 393syl 18 . 2  |-  ( ( 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 ) ) )
416, 15, 32, 40mpbir3and 1135 1  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   |^|cint 3878   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  SubGrpcsubg 14631   Ringcrg 15353   1rcur 15355  SubRingcsubrg 15557
This theorem is referenced by:  subrgin  15584  subrgmre  15585  aspsubrg  16087  rgspncl  27477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559
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