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Theorem sdrgacs 36111
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 2461 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2 eqid 2461 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2issdrg2 36108 . . . . . . 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 995 . . . . . . 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 257 . . . . . 6  |-  ( s  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
65baib 919 . . . . 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 18057 . . . . . . . . 9  |-  ( s  e.  (SubRing `  R
)  ->  s  C_  B )
9 selpw 3969 . . . . . . . . 9  |-  ( s  e.  ~P B  <->  s  C_  B )
108, 9sylibr 217 . . . . . . . 8  |-  ( s  e.  (SubRing `  R
)  ->  s  e.  ~P B )
1110adantl 472 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  s  e.  ~P B )
12 iftrue 3898 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  x )
1312eleq1d 2523 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  x  e.  y ) )
1413biimprd 231 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y ) )
15 eldifsni 4110 . . . . . . . . . . . . . 14  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  ->  x  =/=  ( 0g `  R ) )
1615necon2bi 2665 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  -.  x  e.  ( y  \  { ( 0g `  R ) } ) )
1716pm2.21d 110 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  ( y 
\  { ( 0g
`  R ) } )  ->  ( ( invr `  R ) `  x )  e.  y ) )
1814, 172thd 248 . . . . . . . . . . 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 4109 . . . . . . . . . . . . 13  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  <-> 
( x  e.  y  /\  x  =/=  ( 0g `  R ) ) )
2019rbaibr 921 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  (
x  e.  y  <->  x  e.  ( y  \  {
( 0g `  R
) } ) ) )
21 ifnefalse 3904 . . . . . . . . . . . . 13  |-  ( x  =/=  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  ( ( invr `  R
) `  x )
)
2221eleq1d 2523 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  ( ( invr `  R ) `  x )  e.  y ) )
2320, 22imbi12d 326 . . . . . . . . . . 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 2718 . . . . . . . . . 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 2828 . . . . . . . . 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 3555 . . . . . . . . . 10  |-  ( y  =  s  ->  (
y  \  { ( 0g `  R ) } )  =  ( s 
\  { ( 0g
`  R ) } ) )
27 eleq2 2528 . . . . . . . . . 10  |-  ( y  =  s  ->  (
( ( invr `  R
) `  x )  e.  y  <->  ( ( invr `  R ) `  x
)  e.  s ) )
2826, 27raleqbidv 3012 . . . . . . . . 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 265 . . . . . . . 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 3208 . . . . . . 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 17 . . . . . 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 651 . . . . 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 264 . . . 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 3628 . . . 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 271 . . 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 2459 . 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 5897 . . . . 5  |-  ( Base `  R )  e.  _V
387, 37eqeltri 2535 . . . 4  |-  B  e. 
_V
39 mreacs 15612 . . . 4  |-  ( B  e.  _V  ->  (ACS `  B )  e.  (Moore `  ~P B ) )
4038, 39mp1i 13 . . 3  |-  ( R  e.  DivRing  ->  (ACS `  B
)  e.  (Moore `  ~P B ) )
41 drngring 18030 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  Ring )
427subrgacs 36110 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  e.  (ACS `  B ) )
4341, 42syl 17 . . 3  |-  ( R  e.  DivRing  ->  (SubRing `  R )  e.  (ACS `  B )
)
44 simplr 767 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =  ( 0g `  R ) )  ->  x  e.  B
)
45 df-ne 2634 . . . . . . 7  |-  ( x  =/=  ( 0g `  R )  <->  -.  x  =  ( 0g `  R ) )
467, 2, 1drnginvrcl 18040 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  x  e.  B  /\  x  =/=  ( 0g `  R
) )  ->  (
( invr `  R ) `  x )  e.  B
)
47463expa 1215 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x
)  e.  B )
4845, 47sylan2br 483 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  -.  x  =  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x )  e.  B
)
4944, 48ifclda 3924 . . . . 5  |-  ( ( R  e.  DivRing  /\  x  e.  B )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )
5049ralrimiva 2813 . . . 4  |-  ( R  e.  DivRing  ->  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  B )
51 acsfn1 15615 . . . 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 674 . . 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 15553 . . 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 1276 . 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 2539 1  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   {crab 2752   _Vcvv 3056    \ cdif 3412    i^i cin 3414    C_ wss 3415   ifcif 3892   ~Pcpw 3962   {csn 3979   ` cfv 5600   Basecbs 15169   0gc0g 15386  Moorecmre 15536  ACScacs 15539   Ringcrg 17828   invrcinvr 17947   DivRingcdr 18023  SubRingcsubrg 18052  SubDRingcsdrg 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-subg 16862  df-mgp 17772  df-ur 17784  df-ring 17830  df-oppr 17899  df-dvdsr 17917  df-unit 17918  df-invr 17948  df-drng 18025  df-subrg 18054  df-sdrg 36106
This theorem is referenced by: (None)
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