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Theorem sdrgacs 31394
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 2454 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
2 eqid 2454 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2issdrg2 31391 . . . . . . 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 975 . . . . . . 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 901 . . . . 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 17628 . . . . . . . . 9  |-  ( s  e.  (SubRing `  R
)  ->  s  C_  B )
9 selpw 4006 . . . . . . . . 9  |-  ( s  e.  ~P B  <->  s  C_  B )
108, 9sylibr 212 . . . . . . . 8  |-  ( s  e.  (SubRing `  R
)  ->  s  e.  ~P B )
1110adantl 464 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  s  e.  (SubRing `  R )
)  ->  s  e.  ~P B )
12 iftrue 3935 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . 12  |-  ( x  =  ( 0g `  R )  ->  (
x  e.  y  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  y ) )
15 eldifsni 4142 . . . . . . . . . . . . . 14  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  ->  x  =/=  ( 0g `  R ) )
1615necon2bi 2691 . . . . . . . . . . . . 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 4141 . . . . . . . . . . . . 13  |-  ( x  e.  ( y  \  { ( 0g `  R ) } )  <-> 
( x  e.  y  /\  x  =/=  ( 0g `  R ) ) )
2019rbaibr 903 . . . . . . . . . . . 12  |-  ( x  =/=  ( 0g `  R )  ->  (
x  e.  y  <->  x  e.  ( y  \  {
( 0g `  R
) } ) ) )
21 ifnefalse 3941 . . . . . . . . . . . . 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 318 . . . . . . . . . . 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 2767 . . . . . . . . . 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 2883 . . . . . . . . 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 3601 . . . . . . . . . 10  |-  ( y  =  s  ->  (
y  \  { ( 0g `  R ) } )  =  ( s 
\  { ( 0g
`  R ) } ) )
27 eleq2 2527 . . . . . . . . . 10  |-  ( y  =  s  ->  (
( ( invr `  R
) `  x )  e.  y  <->  ( ( invr `  R ) `  x
)  e.  s ) )
2826, 27raleqbidv 3065 . . . . . . . . 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 3255 . . . . . . 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 639 . . . . 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 3673 . . . 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 2451 . 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 5858 . . . . 5  |-  ( Base `  R )  e.  _V
387, 37eqeltri 2538 . . . 4  |-  B  e. 
_V
39 mreacs 15150 . . . 4  |-  ( B  e.  _V  ->  (ACS `  B )  e.  (Moore `  ~P B ) )
4038, 39mp1i 12 . . 3  |-  ( R  e.  DivRing  ->  (ACS `  B
)  e.  (Moore `  ~P B ) )
41 drngring 17601 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  Ring )
427subrgacs 31393 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  e.  (ACS `  B ) )
4341, 42syl 16 . . 3  |-  ( R  e.  DivRing  ->  (SubRing `  R )  e.  (ACS `  B )
)
44 simplr 753 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =  ( 0g `  R ) )  ->  x  e.  B
)
45 df-ne 2651 . . . . . . 7  |-  ( x  =/=  ( 0g `  R )  <->  -.  x  =  ( 0g `  R ) )
467, 2, 1drnginvrcl 17611 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  x  e.  B  /\  x  =/=  ( 0g `  R
) )  ->  (
( invr `  R ) `  x )  e.  B
)
47463expa 1194 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  x  =/=  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x
)  e.  B )
4845, 47sylan2br 474 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  x  e.  B )  /\  -.  x  =  ( 0g `  R ) )  ->  ( ( invr `  R ) `  x )  e.  B
)
4944, 48ifclda 3961 . . . . 5  |-  ( ( R  e.  DivRing  /\  x  e.  B )  ->  if ( x  =  ( 0g `  R ) ,  x ,  ( (
invr `  R ) `  x ) )  e.  B )
5049ralrimiva 2868 . . . 4  |-  ( R  e.  DivRing  ->  A. x  e.  B  if ( x  =  ( 0g `  R ) ,  x ,  ( ( invr `  R
) `  x )
)  e.  B )
51 acsfn1 15153 . . . 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 661 . . 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 15091 . . 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 1226 . 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 2542 1  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  e.  (ACS `  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   ifcif 3929   ~Pcpw 3999   {csn 4016   ` cfv 5570   Basecbs 14719   0gc0g 14932  Moorecmre 15074  ACScacs 15077   Ringcrg 17396   invrcinvr 17518   DivRingcdr 17594  SubRingcsubrg 17623  SubDRingcsdrg 31388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-subg 16400  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-drng 17596  df-subrg 17625  df-sdrg 31389
This theorem is referenced by: (None)
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