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Theorem isdivrngo 22019
Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Assertion
Ref Expression
isdivrngo  |-  ( H  e.  A  ->  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) ) )

Proof of Theorem isdivrngo
Dummy variables  g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4213 . . . . 5  |-  ( G DivRingOps H 
<-> 
<. G ,  H >.  e.  DivRingOps
)
2 df-drngo 21994 . . . . . . 7  |-  DivRingOps  =  { <. x ,  y >.  |  ( <. x ,  y >.  e.  RingOps  /\  ( y  |`  (
( ran  x  \  {
(GId `  x ) } )  X.  ( ran  x  \  { (GId
`  x ) } ) ) )  e. 
GrpOp ) }
32relopabi 5000 . . . . . 6  |-  Rel  DivRingOps
43brrelexi 4918 . . . . 5  |-  ( G DivRingOps H  ->  G  e.  _V )
51, 4sylbir 205 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  ->  G  e.  _V )
65anim1i 552 . . 3  |-  ( (
<. G ,  H >.  e.  DivRingOps  /\  H  e.  A
)  ->  ( G  e.  _V  /\  H  e.  A ) )
76ancoms 440 . 2  |-  ( ( H  e.  A  /\  <. G ,  H >.  e.  DivRingOps
)  ->  ( G  e.  _V  /\  H  e.  A ) )
8 rngoablo2 22010 . . . . 5  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
9 elex 2964 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
108, 9syl 16 . . . 4  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  _V )
1110ad2antrl 709 . . 3  |-  ( ( H  e.  A  /\  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) )  ->  G  e.  _V )
12 simpl 444 . . 3  |-  ( ( H  e.  A  /\  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) )  ->  H  e.  A )
1311, 12jca 519 . 2  |-  ( ( H  e.  A  /\  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) )  -> 
( G  e.  _V  /\  H  e.  A ) )
14 df-drngo 21994 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
1514eleq2i 2500 . . 3  |-  ( <. G ,  H >.  e.  DivRingOps  <->  <. G ,  H >.  e. 
{ <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  e. 
GrpOp ) } )
16 opeq1 3984 . . . . . 6  |-  ( g  =  G  ->  <. g ,  h >.  =  <. G ,  h >. )
1716eleq1d 2502 . . . . 5  |-  ( g  =  G  ->  ( <. g ,  h >.  e.  RingOps  <->  <. G ,  h >.  e.  RingOps ) )
18 rneq 5095 . . . . . . . . 9  |-  ( g  =  G  ->  ran  g  =  ran  G )
19 fveq2 5728 . . . . . . . . . 10  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
2019sneqd 3827 . . . . . . . . 9  |-  ( g  =  G  ->  { (GId
`  g ) }  =  { (GId `  G ) } )
2118, 20difeq12d 3466 . . . . . . . 8  |-  ( g  =  G  ->  ( ran  g  \  { (GId
`  g ) } )  =  ( ran 
G  \  { (GId `  G ) } ) )
2221, 21xpeq12d 4903 . . . . . . 7  |-  ( g  =  G  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )
2322reseq2d 5146 . . . . . 6  |-  ( g  =  G  ->  (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  =  ( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) ) )
2423eleq1d 2502 . . . . 5  |-  ( g  =  G  ->  (
( h  |`  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  e. 
GrpOp 
<->  ( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) )
2517, 24anbi12d 692 . . . 4  |-  ( g  =  G  ->  (
( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp )  <->  ( <. G ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) ) )
26 opeq2 3985 . . . . . 6  |-  ( h  =  H  ->  <. G ,  h >.  =  <. G ,  H >. )
2726eleq1d 2502 . . . . 5  |-  ( h  =  H  ->  ( <. G ,  h >.  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
28 reseq1 5140 . . . . . 6  |-  ( h  =  H  ->  (
h  |`  ( ( ran 
G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId `  G
) } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) ) )
2928eleq1d 2502 . . . . 5  |-  ( h  =  H  ->  (
( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp 
<->  ( H  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) )
3027, 29anbi12d 692 . . . 4  |-  ( h  =  H  ->  (
( <. G ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp )  <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) ) )
3125, 30opelopabg 4473 . . 3  |-  ( ( G  e.  _V  /\  H  e.  A )  ->  ( <. G ,  H >.  e.  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps 
/\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }  <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) ) )
3215, 31syl5bb 249 . 2  |-  ( ( G  e.  _V  /\  H  e.  A )  ->  ( <. G ,  H >.  e.  DivRingOps 
<->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) ) )
337, 13, 32pm5.21nd 869 1  |-  ( H  e.  A  ->  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317   {csn 3814   <.cop 3817   class class class wbr 4212   {copab 4265    X. cxp 4876   ran crn 4879    |` cres 4880   ` cfv 5454   GrpOpcgr 21774  GIdcgi 21775   AbelOpcablo 21869   RingOpscrngo 21963   DivRingOpscdrng 21993
This theorem is referenced by:  zrdivrng  22020  isdrngo1  26572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-rngo 21964  df-drngo 21994
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