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Theorem sdrgacs 29511
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 2438 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2 eqid 2438 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2issdrg2 29508 . . . . . . 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 969 . . . . . . 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 249 . . . . . 6  |-  ( s  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
65baib 896 . . . . 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 16844 . . . . . . . . 9  |-  ( s  e.  (SubRing `  R
)  ->  s  C_  B )
9 selpw 3862 . . . . . . . . 9  |-  ( s  e.  ~P B  <->  s  C_  B )
108, 9sylibr 212 . . . . . . . 8  |-  ( s  e.  (SubRing `  R
)  ->  s  e.  ~P B )
1110adantl 466 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  s  e.  ~P B )
12 iftrue 3792 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  x )
1312eleq1d 2504 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  x  e.  y ) )
1413biimprd 223 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y ) )
15 eldifsni 3996 . . . . . . . . . . . . . 14  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  ->  x  =/=  ( 0g `  R ) )
1615necon2bi 2652 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  -.  x  e.  ( y  \  { ( 0g `  R ) } ) )
1716pm2.21d 106 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  ( y 
\  { ( 0g
`  R ) } )  ->  ( ( invr `  R ) `  x )  e.  y ) )
1814, 172thd 240 . . . . . . . . . . 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 ) ) )
19 eldifsn 3995 . . . . . . . . . . . . 13  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  <-> 
( x  e.  y  /\  x  =/=  ( 0g `  R ) ) )
2019rbaibr 899 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  (
x  e.  y  <->  x  e.  ( y  \  {
( 0g `  R
) } ) ) )
21 ifnefalse 3796 . . . . . . . . . . . . 13  |-  ( x  =/=  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  ( ( invr `  R
) `  x )
)
2221eleq1d 2504 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  ( ( invr `  R ) `  x )  e.  y ) )
2320, 22imbi12d 320 . . . . . . . . . . 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 ) ) )
2418, 23pm2.61ine 2682 . . . . . . . . . 10  |-  ( ( x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y )  <-> 
( x  e.  ( y  \  { ( 0g `  R ) } )  ->  (
( invr `  R ) `  x )  e.  y ) )
2524ralbii2 2738 . . . . . . . . 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 )
26 difeq1 3462 . . . . . . . . . 10  |-  ( y  =  s  ->  (
y  \  { ( 0g `  R ) } )  =  ( s 
\  { ( 0g
`  R ) } ) )
27 eleq2 2499 . . . . . . . . . 10  |-  ( y  =  s  ->  (
( ( invr `  R
) `  x )  e.  y  <->  ( ( invr `  R ) `  x
)  e.  s ) )
2826, 27raleqbidv 2926 . . . . . . . . 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 ) )
2925, 28syl5bb 257 . . . . . . . 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 ) )
3029elrab3 3113 . . . . . . 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 ) )
3111, 30syl 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 ) )
3231pm5.32da 641 . . . . 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 ) ) )
336, 32bitr4d 256 . . . 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 } ) ) )
34 elin 3534 . . . 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 } ) )
3533, 34syl6bbr 263 . . 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 } ) ) )
3635eqrdv 2436 . 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 } ) )
37 fvex 5696 . . . . 5  |-  ( Base `  R )  e.  _V
387, 37eqeltri 2508 . . . 4  |-  B  e. 
_V
39 mreacs 14588 . . . 4  |-  ( B  e.  _V  ->  (ACS `  B )  e.  (Moore `  ~P B ) )
4038, 39mp1i 12 . . 3  |-  ( R  e.  DivRing  ->  (ACS `  B
)  e.  (Moore `  ~P B ) )
41 drngrng 16817 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  Ring )
427subrgacs 29510 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  e.  (ACS `  B ) )
4341, 42syl 16 . . 3  |-  ( R  e.  DivRing  ->  (SubRing `  R )  e.  (ACS `  B )
)
44 simplr 754 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =  ( 0g `  R ) )  ->  x  e.  B
)
45 df-ne 2603 . . . . . . 7  |-  ( x  =/=  ( 0g `  R )  <->  -.  x  =  ( 0g `  R ) )
467, 2, 1drnginvrcl 16827 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  x  e.  B  /\  x  =/=  ( 0g `  R
) )  ->  (
( invr `  R ) `  x )  e.  B
)
47463expa 1187 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x
)  e.  B )
4845, 47sylan2br 476 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  -.  x  =  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x )  e.  B
)
4944, 48ifclda 3816 . . . . 5  |-  ( ( R  e.  DivRing  /\  x  e.  B )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )
5049ralrimiva 2794 . . . 4  |-  ( R  e.  DivRing  ->  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  B )
51 acsfn1 14591 . . . 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
) )
5238, 50, 51sylancr 663 . . 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 ) )
53 mreincl 14529 . . 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 )
)
5440, 43, 52, 53syl3anc 1218 . 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 )
)
5536, 54eqeltrd 2512 1  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   {crab 2714   _Vcvv 2967    \ cdif 3320    i^i cin 3322    C_ wss 3323   ifcif 3786   ~Pcpw 3855   {csn 3872   ` cfv 5413   Basecbs 14166   0gc0g 14370  Moorecmre 14512  ACScacs 14515   Ringcrg 16633   invrcinvr 16751   DivRingcdr 16810  SubRingcsubrg 16839  SubDRingcsdrg 29505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-subg 15669  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-subrg 16841  df-sdrg 29506
This theorem is referenced by: (None)
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