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Theorem sdrgacs 27377
Description: Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypothesis
Ref Expression
subrgacs.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
sdrgacs  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)

Proof of Theorem sdrgacs
Dummy variables  x  s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2 eqid 2404 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2issdrg2 27374 . . . . . . 7  |-  ( s  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( ( invr `  R
) `  x )  e.  s ) )
4 3anass 940 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s )  <->  ( R  e.  DivRing 
/\  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
53, 4bitri 241 . . . . . 6  |-  ( s  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
65baib 872 . . . . 5  |-  ( R  e.  DivRing  ->  ( s  e.  (SubDRing `  R )  <->  ( s  e.  (SubRing `  R
)  /\  A. x  e.  ( s  \  {
( 0g `  R
) } ) ( ( invr `  R
) `  x )  e.  s ) ) )
7 subrgacs.b . . . . . . . . . 10  |-  B  =  ( Base `  R
)
87subrgss 15824 . . . . . . . . 9  |-  ( s  e.  (SubRing `  R
)  ->  s  C_  B )
9 vex 2919 . . . . . . . . . 10  |-  s  e. 
_V
109elpw 3765 . . . . . . . . 9  |-  ( s  e.  ~P B  <->  s  C_  B )
118, 10sylibr 204 . . . . . . . 8  |-  ( s  e.  (SubRing `  R
)  ->  s  e.  ~P B )
1211adantl 453 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  s  e.  ~P B )
13 iftrue 3705 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  x )
1413eleq1d 2470 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  x  e.  y ) )
1514biimprd 215 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y ) )
16 eldifsni 3888 . . . . . . . . . . . . . 14  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  ->  x  =/=  ( 0g `  R ) )
1716necon2bi 2613 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  -.  x  e.  ( y  \  { ( 0g `  R ) } ) )
1817pm2.21d 100 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  ( y 
\  { ( 0g
`  R ) } )  ->  ( ( invr `  R ) `  x )  e.  y ) )
1915, 182thd 232 . . . . . . . . . . 11  |-  ( x  =  ( 0g `  R )  ->  (
( x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R ) `  x
) )  e.  y )  <->  ( x  e.  ( y  \  {
( 0g `  R
) } )  -> 
( ( invr `  R
) `  x )  e.  y ) ) )
20 eldifsn 3887 . . . . . . . . . . . . 13  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  <-> 
( x  e.  y  /\  x  =/=  ( 0g `  R ) ) )
2120rbaibr 875 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  (
x  e.  y  <->  x  e.  ( y  \  {
( 0g `  R
) } ) ) )
22 ifnefalse 3707 . . . . . . . . . . . . 13  |-  ( x  =/=  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  ( ( invr `  R
) `  x )
)
2322eleq1d 2470 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  ( ( invr `  R ) `  x )  e.  y ) )
2421, 23imbi12d 312 . . . . . . . . . . 11  |-  ( x  =/=  ( 0g `  R )  ->  (
( x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R ) `  x
) )  e.  y )  <->  ( x  e.  ( y  \  {
( 0g `  R
) } )  -> 
( ( invr `  R
) `  x )  e.  y ) ) )
2519, 24pm2.61ine 2643 . . . . . . . . . 10  |-  ( ( x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y )  <-> 
( x  e.  ( y  \  { ( 0g `  R ) } )  ->  (
( invr `  R ) `  x )  e.  y ) )
2625ralbii2 2694 . . . . . . . . 9  |-  ( A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y  <->  A. x  e.  ( y  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  y )
27 difeq1 3418 . . . . . . . . . 10  |-  ( y  =  s  ->  (
y  \  { ( 0g `  R ) } )  =  ( s 
\  { ( 0g
`  R ) } ) )
28 eleq2 2465 . . . . . . . . . 10  |-  ( y  =  s  ->  (
( ( invr `  R
) `  x )  e.  y  <->  ( ( invr `  R ) `  x
)  e.  s ) )
2927, 28raleqbidv 2876 . . . . . . . . 9  |-  ( y  =  s  ->  ( A. x  e.  (
y  \  { ( 0g `  R ) } ) ( ( invr `  R ) `  x
)  e.  y  <->  A. x  e.  ( s  \  {
( 0g `  R
) } ) ( ( invr `  R
) `  x )  e.  s ) )
3026, 29syl5bb 249 . . . . . . . 8  |-  ( y  =  s  ->  ( A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  A. x  e.  ( s  \  {
( 0g `  R
) } ) ( ( invr `  R
) `  x )  e.  s ) )
3130elrab3 3053 . . . . . . 7  |-  ( s  e.  ~P B  -> 
( s  e.  {
y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R
) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y }  <->  A. x  e.  (
s  \  { ( 0g `  R ) } ) ( ( invr `  R ) `  x
)  e.  s ) )
3212, 31syl 16 . . . . . 6  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  ( s  e.  { y  e.  ~P B  |  A. x  e.  y  if (
x  =  ( 0g
`  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y }  <->  A. x  e.  ( s  \  {
( 0g `  R
) } ) ( ( invr `  R
) `  x )  e.  s ) )
3332pm5.32da 623 . . . . 5  |-  ( R  e.  DivRing  ->  ( ( s  e.  (SubRing `  R
)  /\  s  e.  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R
) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y } )  <->  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
346, 33bitr4d 248 . . . 4  |-  ( R  e.  DivRing  ->  ( s  e.  (SubDRing `  R )  <->  ( s  e.  (SubRing `  R
)  /\  s  e.  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R
) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y } ) ) )
35 elin 3490 . . . 4  |-  ( s  e.  ( (SubRing `  R
)  i^i  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y } )  <->  ( s  e.  (SubRing `  R )  /\  s  e.  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y } ) )
3634, 35syl6bbr 255 . . 3  |-  ( R  e.  DivRing  ->  ( s  e.  (SubDRing `  R )  <->  s  e.  ( (SubRing `  R
)  i^i  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y } ) ) )
3736eqrdv 2402 . 2  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  =  ( (SubRing `  R
)  i^i  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y } ) )
38 fvex 5701 . . . . 5  |-  ( Base `  R )  e.  _V
397, 38eqeltri 2474 . . . 4  |-  B  e. 
_V
40 mreacs 13838 . . . 4  |-  ( B  e.  _V  ->  (ACS `  B )  e.  (Moore `  ~P B ) )
4139, 40mp1i 12 . . 3  |-  ( R  e.  DivRing  ->  (ACS `  B
)  e.  (Moore `  ~P B ) )
42 drngrng 15797 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  Ring )
437subrgacs 27376 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  e.  (ACS `  B ) )
4442, 43syl 16 . . 3  |-  ( R  e.  DivRing  ->  (SubRing `  R )  e.  (ACS `  B )
)
45 simplr 732 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =  ( 0g `  R ) )  ->  x  e.  B
)
46 df-ne 2569 . . . . . . 7  |-  ( x  =/=  ( 0g `  R )  <->  -.  x  =  ( 0g `  R ) )
477, 2, 1drnginvrcl 15807 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  x  e.  B  /\  x  =/=  ( 0g `  R
) )  ->  (
( invr `  R ) `  x )  e.  B
)
48473expa 1153 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x
)  e.  B )
4946, 48sylan2br 463 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  -.  x  =  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x )  e.  B
)
5045, 49ifclda 3726 . . . . 5  |-  ( ( R  e.  DivRing  /\  x  e.  B )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )
5150ralrimiva 2749 . . . 4  |-  ( R  e.  DivRing  ->  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  B )
52 acsfn1 13841 . . . 4  |-  ( ( B  e.  _V  /\  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )  ->  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y }  e.  (ACS `  B
) )
5339, 51, 52sylancr 645 . . 3  |-  ( R  e.  DivRing  ->  { y  e. 
~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y }  e.  (ACS
`  B ) )
54 mreincl 13779 . . 3  |-  ( ( (ACS `  B )  e.  (Moore `  ~P B )  /\  (SubRing `  R
)  e.  (ACS `  B )  /\  {
y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R
) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y }  e.  (ACS `  B
) )  ->  (
(SubRing `  R )  i^i 
{ y  e.  ~P B  |  A. x  e.  y  if (
x  =  ( 0g
`  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y } )  e.  (ACS `  B )
)
5541, 44, 53, 54syl3anc 1184 . 2  |-  ( R  e.  DivRing  ->  ( (SubRing `  R
)  i^i  { y  e.  ~P B  |  A. x  e.  y  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  y } )  e.  (ACS `  B )
)
5637, 55eqeltrd 2478 1  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   ifcif 3699   ~Pcpw 3759   {csn 3774   ` cfv 5413   Basecbs 13424   0gc0g 13678  Moorecmre 13762  ACScacs 13765   Ringcrg 15615   invrcinvr 15731   DivRingcdr 15790  SubRingcsubrg 15819  SubDRingcsdrg 27371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-subrg 15821  df-sdrg 27372
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