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Theorem isdivrngo 21114
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 4040 . . . . 5  |-  ( G DivRingOps H 
<-> 
<. G ,  H >.  e.  DivRingOps
)
2 df-drngo 21089 . . . . . . 7  |-  DivRingOps  =  { <. x ,  y >.  |  ( <. x ,  y >.  e.  RingOps  /\  ( y  |`  (
( ran  x  \  {
(GId `  x ) } )  X.  ( ran  x  \  { (GId
`  x ) } ) ) )  e. 
GrpOp ) }
32relopabi 4827 . . . . . 6  |-  Rel  DivRingOps
43brrelexi 4745 . . . . 5  |-  ( G DivRingOps H  ->  G  e.  _V )
51, 4sylbir 204 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  ->  G  e.  _V )
65anim1i 551 . . 3  |-  ( (
<. G ,  H >.  e.  DivRingOps  /\  H  e.  A
)  ->  ( G  e.  _V  /\  H  e.  A ) )
76ancoms 439 . 2  |-  ( ( H  e.  A  /\  <. G ,  H >.  e.  DivRingOps
)  ->  ( G  e.  _V  /\  H  e.  A ) )
8 rngoablo2 21105 . . . . 5  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
9 elex 2809 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
108, 9syl 15 . . . 4  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  _V )
1110ad2antrl 708 . . 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 443 . . 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 518 . 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 21089 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
1514eleq2i 2360 . . 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 3812 . . . . . 6  |-  ( g  =  G  ->  <. g ,  h >.  =  <. G ,  h >. )
1716eleq1d 2362 . . . . 5  |-  ( g  =  G  ->  ( <. g ,  h >.  e.  RingOps  <->  <. G ,  h >.  e.  RingOps ) )
18 rneq 4920 . . . . . . . . 9  |-  ( g  =  G  ->  ran  g  =  ran  G )
19 fveq2 5541 . . . . . . . . . 10  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
2019sneqd 3666 . . . . . . . . 9  |-  ( g  =  G  ->  { (GId
`  g ) }  =  { (GId `  G ) } )
2118, 20difeq12d 3308 . . . . . . . 8  |-  ( g  =  G  ->  ( ran  g  \  { (GId
`  g ) } )  =  ( ran 
G  \  { (GId `  G ) } ) )
2221, 21xpeq12d 4730 . . . . . . 7  |-  ( g  =  G  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )
2322reseq2d 4971 . . . . . 6  |-  ( g  =  G  ->  (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  =  ( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) ) )
2423eleq1d 2362 . . . . 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 691 . . . 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 3813 . . . . . 6  |-  ( h  =  H  ->  <. G ,  h >.  =  <. G ,  H >. )
2726eleq1d 2362 . . . . 5  |-  ( h  =  H  ->  ( <. G ,  h >.  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
28 reseq1 4965 . . . . . 6  |-  ( h  =  H  ->  (
h  |`  ( ( ran 
G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId `  G
) } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) ) )
2928eleq1d 2362 . . . . 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 691 . . . 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 4299 . . 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 248 . 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 868 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   {csn 3653   <.cop 3656   class class class wbr 4039   {copab 4092    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271   GrpOpcgr 20869  GIdcgi 20870   AbelOpcablo 20964   RingOpscrngo 21058   DivRingOpscdrng 21088
This theorem is referenced by:  zrdivrng  21115  isdrngo1  26690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059  df-drngo 21089
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