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Theorem subrgint 15892
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 15876 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3354 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3358 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  (SubRing `  R
)  C_  (SubGrp `  R
) )  ->  S  C_  (SubGrp `  R )
)
42, 3mpan2 654 . . 3  |-  ( S 
C_  (SubRing `  R )  ->  S  C_  (SubGrp `  R
) )
5 subgint 14966 . . 3  |-  ( ( S  C_  (SubGrp `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
64, 5sylan 459 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  R )
)
7 ssel2 3345 . . . . . 6  |-  ( ( S  C_  (SubRing `  R
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
87adantlr 697 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
9 eqid 2438 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 15878 . . . . 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 2791 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 fvex 5744 . . . 4  |-  ( 1r
`  R )  e. 
_V
1413elint2 4059 . . 3  |-  ( ( 1r `  R )  e.  |^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r )
1512, 14sylibr 205 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  ( 1r `  R )  e.  |^| S )
168adantlr 697 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
17 simprl 734 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  x  e.  |^| S )
18 elinti 4061 . . . . . . . 8  |-  ( x  e.  |^| S  ->  (
r  e.  S  ->  x  e.  r )
)
1918imp 420 . . . . . . 7  |-  ( ( x  e.  |^| S  /\  r  e.  S
)  ->  x  e.  r )
2017, 19sylan 459 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  x  e.  r )
21 simprr 735 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  y  e.  |^| S )
22 elinti 4061 . . . . . . . 8  |-  ( y  e.  |^| S  ->  (
r  e.  S  -> 
y  e.  r ) )
2322imp 420 . . . . . . 7  |-  ( ( y  e.  |^| S  /\  r  e.  S
)  ->  y  e.  r )
2421, 23sylan 459 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  y  e.  r )
25 eqid 2438 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
2625subrgmcl 15882 . . . . . 6  |-  ( ( r  e.  (SubRing `  R
)  /\  x  e.  r  /\  y  e.  r )  ->  ( x
( .r `  R
) y )  e.  r )
2716, 20, 24, 26syl3anc 1185 . . . . 5  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  /\  r  e.  S )  ->  (
x ( .r `  R ) y )  e.  r )
2827ralrimiva 2791 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
29 ovex 6108 . . . . 5  |-  ( x ( .r `  R
) y )  e. 
_V
3029elint2 4059 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  |^| S  <->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
3128, 30sylibr 205 . . 3  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  (
x ( .r `  R ) y )  e.  |^| S )
3231ralrimivva 2800 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R
) y )  e. 
|^| S )
33 ssn0 3662 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  (SubRing `  R
)  =/=  (/) )
34 n0 3639 . . . 4  |-  ( (SubRing `  R )  =/=  (/)  <->  E. r 
r  e.  (SubRing `  R
) )
35 subrgrcl 15875 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  R  e.  Ring )
3635exlimiv 1645 . . . 4  |-  ( E. r  r  e.  (SubRing `  R )  ->  R  e.  Ring )
3734, 36sylbi 189 . . 3  |-  ( (SubRing `  R )  =/=  (/)  ->  R  e.  Ring )
38 eqid 2438 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3938, 9, 25issubrg2 15890 . . 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 19 . 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 1138 1  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   |^|cint 4052   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532  SubGrpcsubg 14940   Ringcrg 15662   1rcur 15664  SubRingcsubrg 15866
This theorem is referenced by:  subrgin  15893  subrgmre  15894  aspsubrg  16392  rgspncl  27353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-mgp 15651  df-rng 15665  df-ur 15667  df-subrg 15868
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