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Theorem subrgint 15567
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 15551 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3184 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3187 . . . 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 14641 . . 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 3175 . . . . . 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 2283 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 15553 . . . . 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 2626 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 fvex 5539 . . . 4  |-  ( 1r
`  R )  e. 
_V
1413elint2 3869 . . 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 3871 . . . . . . . 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 3871 . . . . . . . 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 2283 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
2625subrgmcl 15557 . . . . . 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 2626 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  /\  (
x  e.  |^| S  /\  y  e.  |^| S
) )  ->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r )
29 ovex 5883 . . . . 5  |-  ( x ( .r `  R
) y )  e. 
_V
3029elint2 3869 . . . 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 2635 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R
) y )  e. 
|^| S )
33 ssn0 3487 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  S  =/=  (/) )  ->  (SubRing `  R
)  =/=  (/) )
34 n0 3464 . . . 4  |-  ( (SubRing `  R )  =/=  (/)  <->  E. r 
r  e.  (SubRing `  R
) )
35 subrgrcl 15550 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  R  e.  Ring )
3635exlimiv 1666 . . . 4  |-  ( E. r  r  e.  (SubRing `  R )  ->  R  e.  Ring )
3734, 36sylbi 187 . . 3  |-  ( (SubRing `  R )  =/=  (/)  ->  R  e.  Ring )
38 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3938, 9, 25issubrg2 15565 . . 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 1528    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|cint 3862   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  SubGrpcsubg 14615   Ringcrg 15337   1rcur 15339  SubRingcsubrg 15541
This theorem is referenced by:  subrgin  15568  subrgmre  15569  aspsubrg  16071  rgspncl  26786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543
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