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Theorem sdrgacs 27500
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 27497 . . . . . . 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 941 . . . . . . 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 242 . . . . . 6  |-  ( s  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( s  e.  (SubRing `  R )  /\  A. x  e.  ( s  \  { ( 0g `  R ) } ) ( (
invr `  R ) `  x )  e.  s ) ) )
65baib 873 . . . . 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 15874 . . . . . . . . 9  |-  ( s  e.  (SubRing `  R
)  ->  s  C_  B )
9 vex 2961 . . . . . . . . . 10  |-  s  e. 
_V
109elpw 3807 . . . . . . . . 9  |-  ( s  e.  ~P B  <->  s  C_  B )
118, 10sylibr 205 . . . . . . . 8  |-  ( s  e.  (SubRing `  R
)  ->  s  e.  ~P B )
1211adantl 454 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  s  e.  ~P B )
13 iftrue 3747 . . . . . . . . . . . . . 14  |-  ( x  =  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  x )
1413eleq1d 2504 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  x  e.  y ) )
1514biimprd 216 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y ) )
16 eldifsni 3930 . . . . . . . . . . . . . 14  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  ->  x  =/=  ( 0g `  R ) )
1716necon2bi 2652 . . . . . . . . . . . . 13  |-  ( x  =  ( 0g `  R )  ->  -.  x  e.  ( y  \  { ( 0g `  R ) } ) )
1817pm2.21d 101 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  ( y 
\  { ( 0g
`  R ) } )  ->  ( ( invr `  R ) `  x )  e.  y ) )
1915, 182thd 233 . . . . . . . . . . 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 3929 . . . . . . . . . . . . 13  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  <-> 
( x  e.  y  /\  x  =/=  ( 0g `  R ) ) )
2120rbaibr 876 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  (
x  e.  y  <->  x  e.  ( y  \  {
( 0g `  R
) } ) ) )
22 ifnefalse 3749 . . . . . . . . . . . . 13  |-  ( x  =/=  ( 0g `  R )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  =  ( ( invr `  R
) `  x )
)
2322eleq1d 2504 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  ( if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y  <->  ( ( invr `  R ) `  x )  e.  y ) )
2421, 23imbi12d 313 . . . . . . . . . . 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 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 ) )
2625ralbii2 2735 . . . . . . . . 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 3460 . . . . . . . . . 10  |-  ( y  =  s  ->  (
y  \  { ( 0g `  R ) } )  =  ( s 
\  { ( 0g
`  R ) } ) )
28 eleq2 2499 . . . . . . . . . 10  |-  ( y  =  s  ->  (
( ( invr `  R
) `  x )  e.  y  <->  ( ( invr `  R ) `  x
)  e.  s ) )
2927, 28raleqbidv 2918 . . . . . . . . 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 250 . . . . . . . 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 3095 . . . . . . 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 624 . . . . 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 249 . . . 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 3532 . . . 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 256 . . 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 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 } ) )
38 fvex 5745 . . . . 5  |-  ( Base `  R )  e.  _V
397, 38eqeltri 2508 . . . 4  |-  B  e. 
_V
40 mreacs 13888 . . . 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 15847 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  Ring )
437subrgacs 27499 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  e.  (ACS `  B ) )
4442, 43syl 16 . . 3  |-  ( R  e.  DivRing  ->  (SubRing `  R )  e.  (ACS `  B )
)
45 simplr 733 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =  ( 0g `  R ) )  ->  x  e.  B
)
46 df-ne 2603 . . . . . . 7  |-  ( x  =/=  ( 0g `  R )  <->  -.  x  =  ( 0g `  R ) )
477, 2, 1drnginvrcl 15857 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  x  e.  B  /\  x  =/=  ( 0g `  R
) )  ->  (
( invr `  R ) `  x )  e.  B
)
48473expa 1154 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x
)  e.  B )
4946, 48sylan2br 464 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  -.  x  =  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x )  e.  B
)
5045, 49ifclda 3768 . . . . 5  |-  ( ( R  e.  DivRing  /\  x  e.  B )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )
5150ralrimiva 2791 . . . 4  |-  ( R  e.  DivRing  ->  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  B )
52 acsfn1 13891 . . . 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 646 . . 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 13829 . . 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 1185 . 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 2512 1  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   ifcif 3741   ~Pcpw 3801   {csn 3816   ` cfv 5457   Basecbs 13474   0gc0g 13728  Moorecmre 13812  ACScacs 13815   Ringcrg 15665   invrcinvr 15781   DivRingcdr 15840  SubRingcsubrg 15869  SubDRingcsdrg 27494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-0g 13732  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-subg 14946  df-mgp 15654  df-rng 15668  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-drng 15842  df-subrg 15871  df-sdrg 27495
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