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Theorem subrgint 16993
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 16977 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3458 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3462 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  (SubRing `  R
)  C_  (SubGrp `  R
) )  ->  S  C_  (SubGrp `  R )
)
42, 3mpan2 671 . . 3  |-  ( S 
C_  (SubRing `  R )  ->  S  C_  (SubGrp `  R
) )
5 subgint 15807 . . 3  |-  ( ( S  C_  (SubGrp `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
64, 5sylan 471 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
7 ssel2 3449 . . . . . 6  |-  ( ( S  C_  (SubRing `  R
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
87adantlr 714 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
9 eqid 2451 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 16979 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  r )
118, 10syl 16 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  ( 1r `  R )  e.  r )
1211ralrimiva 2822 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 fvex 5799 . . . 4  |-  ( 1r
`  R )  e. 
_V
1413elint2 4233 . . 3  |-  ( ( 1r `  R )  e.  |^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r )
1512, 14sylibr 212 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  ( 1r `  R )  e.  |^| S )
168adantlr 714 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
17 simprl 755 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
18 elinti 4235 . . . . . . . 8  |-  ( x  e.  |^| S  ->  (
r  e.  S  ->  x  e.  r )
)
1918imp 429 . . . . . . 7  |-  ( ( x  e.  |^| S  /\  r  e.  S
)  ->  x  e.  r )
2017, 19sylan 471 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  x  e.  r )
21 simprr 756 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
22 elinti 4235 . . . . . . . 8  |-  ( y  e.  |^| S  ->  (
r  e.  S  -> 
y  e.  r ) )
2322imp 429 . . . . . . 7  |-  ( ( y  e.  |^| S  /\  r  e.  S
)  ->  y  e.  r )
2421, 23sylan 471 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  y  e.  r )
25 eqid 2451 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
2625subrgmcl 16983 . . . . . 6  |-  ( ( r  e.  (SubRing `  R
)  /\  x  e.  r  /\  y  e.  r )  ->  ( x
( .r `  R
) y )  e.  r )
2716, 20, 24, 26syl3anc 1219 . . . . 5  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  (
x ( .r `  R ) y )  e.  r )
2827ralrimiva 2822 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
29 ovex 6215 . . . . 5  |-  ( x ( .r `  R
) y )  e. 
_V
3029elint2 4233 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  |^| S  <->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
3128, 30sylibr 212 . . 3  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( .r `  R ) y )  e.  |^| S )
3231ralrimivva 2904 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R
) y )  e. 
|^| S )
33 ssn0 3768 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  (SubRing `  R
)  =/=  (/) )
34 n0 3744 . . . 4  |-  ( (SubRing `  R )  =/=  (/)  <->  E. r 
r  e.  (SubRing `  R
) )
35 subrgrcl 16976 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  R  e.  Ring )
3635exlimiv 1689 . . . 4  |-  ( E. r  r  e.  (SubRing `  R )  ->  R  e.  Ring )
3734, 36sylbi 195 . . 3  |-  ( (SubRing `  R )  =/=  (/)  ->  R  e.  Ring )
38 eqid 2451 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3938, 9, 25issubrg2 16991 . . 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 20 . 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 1171 1  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   E.wex 1587    e. wcel 1758    =/= wne 2644   A.wral 2795    C_ wss 3426   (/)c0 3735   |^|cint 4226   ` cfv 5516  (class class class)co 6190   Basecbs 14276   .rcmulr 14341  SubGrpcsubg 15777   1rcur 16708   Ringcrg 16751  SubRingcsubrg 16967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-subg 15780  df-mgp 16697  df-ur 16709  df-rng 16753  df-subrg 16969
This theorem is referenced by:  subrgin  16994  subrgmre  16995  aspsubrg  17508  rgspncl  29664
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