ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subrgintm Unicode version

Theorem subrgintm 13302
Description: The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
Assertion
Ref Expression
subrgintm  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  |^| S  e.  (SubRing `  R ) )
Distinct variable groups:    w, R    w, S

Proof of Theorem subrgintm
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 13286 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  r  e.  (SubGrp `  R ) )
21ssriv 3159 . . . 4  |-  (SubRing `  R
)  C_  (SubGrp `  R
)
3 sstr 3163 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  (SubRing `  R
)  C_  (SubGrp `  R
) )  ->  S  C_  (SubGrp `  R )
)
42, 3mpan2 425 . . 3  |-  ( S 
C_  (SubRing `  R )  ->  S  C_  (SubGrp `  R
) )
5 subgintm 12989 . . 3  |-  ( ( S  C_  (SubGrp `  R
)  /\  E. w  w  e.  S )  ->  |^| S  e.  (SubGrp `  R ) )
64, 5sylan 283 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  |^| S  e.  (SubGrp `  R ) )
7 ssel2 3150 . . . . . 6  |-  ( ( S  C_  (SubRing `  R
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
87adantlr 477 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  r  e.  S )  ->  r  e.  (SubRing `  R )
)
9 eqid 2177 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
109subrg1cl 13288 . . . . 5  |-  ( r  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  r )
118, 10syl 14 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  r  e.  S )  ->  ( 1r `  R )  e.  r )
1211ralrimiva 2550 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  A. r  e.  S  ( 1r `  R )  e.  r )
13 ssel 3149 . . . . . . 7  |-  ( S 
C_  (SubRing `  R )  ->  ( w  e.  S  ->  w  e.  (SubRing `  R
) ) )
14 subrgrcl 13285 . . . . . . 7  |-  ( w  e.  (SubRing `  R
)  ->  R  e.  Ring )
1513, 14syl6 33 . . . . . 6  |-  ( S 
C_  (SubRing `  R )  ->  ( w  e.  S  ->  R  e.  Ring )
)
1615exlimdv 1819 . . . . 5  |-  ( S 
C_  (SubRing `  R )  ->  ( E. w  w  e.  S  ->  R  e.  Ring ) )
1716imp 124 . . . 4  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  R  e.  Ring )
18 ringsrg 13155 . . . 4  |-  ( R  e.  Ring  ->  R  e. SRing
)
19 eqid 2177 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2019, 9srgidcl 13090 . . . . 5  |-  ( R  e. SRing  ->  ( 1r `  R )  e.  (
Base `  R )
)
21 elintg 3852 . . . . 5  |-  ( ( 1r `  R )  e.  ( Base `  R
)  ->  ( ( 1r `  R )  e. 
|^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r ) )
2220, 21syl 14 . . . 4  |-  ( R  e. SRing  ->  ( ( 1r
`  R )  e. 
|^| S  <->  A. r  e.  S  ( 1r `  R )  e.  r ) )
2317, 18, 223syl 17 . . 3  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  ( ( 1r `  R )  e.  |^| S 
<-> 
A. r  e.  S  ( 1r `  R )  e.  r ) )
2412, 23mpbird 167 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  ( 1r `  R
)  e.  |^| S
)
258adantlr 477 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  /\  r  e.  S
)  ->  r  e.  (SubRing `  R ) )
26 simprl 529 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  x  e.  |^| S )
27 elinti 3853 . . . . . . . 8  |-  ( x  e.  |^| S  ->  (
r  e.  S  ->  x  e.  r )
)
2827imp 124 . . . . . . 7  |-  ( ( x  e.  |^| S  /\  r  e.  S
)  ->  x  e.  r )
2926, 28sylan 283 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  /\  r  e.  S
)  ->  x  e.  r )
30 simprr 531 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  y  e.  |^| S )
31 elinti 3853 . . . . . . . 8  |-  ( y  e.  |^| S  ->  (
r  e.  S  -> 
y  e.  r ) )
3231imp 124 . . . . . . 7  |-  ( ( y  e.  |^| S  /\  r  e.  S
)  ->  y  e.  r )
3330, 32sylan 283 . . . . . 6  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  /\  r  e.  S
)  ->  y  e.  r )
34 eqid 2177 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
3534subrgmcl 13292 . . . . . 6  |-  ( ( r  e.  (SubRing `  R
)  /\  x  e.  r  /\  y  e.  r )  ->  ( x
( .r `  R
) y )  e.  r )
3625, 29, 33, 35syl3anc 1238 . . . . 5  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  /\  r  e.  S
)  ->  ( x
( .r `  R
) y )  e.  r )
3736ralrimiva 2550 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  A. r  e.  S  ( x ( .r
`  R ) y )  e.  r )
38 simplr 528 . . . . . 6  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  E. w  w  e.  S )
39 eleq1w 2238 . . . . . . . 8  |-  ( r  =  w  ->  (
r  e.  S  <->  w  e.  S ) )
4039cbvexv 1918 . . . . . . 7  |-  ( E. r  r  e.  S  <->  E. w  w  e.  S
)
4136elexd 2750 . . . . . . . . 9  |-  ( ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  /\  r  e.  S
)  ->  ( x
( .r `  R
) y )  e. 
_V )
4241ex 115 . . . . . . . 8  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( r  e.  S  ->  ( x
( .r `  R
) y )  e. 
_V ) )
4342exlimdv 1819 . . . . . . 7  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( E. r 
r  e.  S  -> 
( x ( .r
`  R ) y )  e.  _V )
)
4440, 43biimtrrid 153 . . . . . 6  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( E. w  w  e.  S  ->  ( x ( .r `  R ) y )  e.  _V ) )
4538, 44mpd 13 . . . . 5  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( x ( .r `  R ) y )  e.  _V )
46 elintg 3852 . . . . 5  |-  ( ( x ( .r `  R ) y )  e.  _V  ->  (
( x ( .r
`  R ) y )  e.  |^| S  <->  A. r  e.  S  ( x ( .r `  R ) y )  e.  r ) )
4745, 46syl 14 . . . 4  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( ( x ( .r `  R
) y )  e. 
|^| S  <->  A. r  e.  S  ( x
( .r `  R
) y )  e.  r ) )
4837, 47mpbird 167 . . 3  |-  ( ( ( S  C_  (SubRing `  R )  /\  E. w  w  e.  S
)  /\  ( x  e.  |^| S  /\  y  e.  |^| S ) )  ->  ( x ( .r `  R ) y )  e.  |^| S )
4948ralrimivva 2559 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  A. x  e.  |^| S A. y  e.  |^| S ( x ( .r `  R ) y )  e.  |^| S )
5019, 9, 34issubrg2 13300 . . 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 ) ) )
5117, 50syl 14 . 2  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  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 ) ) )
526, 24, 49, 51mpbir3and 1180 1  |-  ( ( S  C_  (SubRing `  R
)  /\  E. w  w  e.  S )  ->  |^| S  e.  (SubRing `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978   E.wex 1492    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129   |^|cint 3844   ` cfv 5215  (class class class)co 5872   Basecbs 12454   .rcmulr 12529  SubGrpcsubg 12958   1rcur 13073  SRingcsrg 13077   Ringcrg 13110  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-subrg 13278
This theorem is referenced by:  subrgin  13303
  Copyright terms: Public domain W3C validator