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Theorem isdivrngo 25631
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 4440 . . . . 5  |-  ( G DivRingOps H 
<-> 
<. G ,  H >.  e.  DivRingOps
)
2 df-drngo 25606 . . . . . . 7  |-  DivRingOps  =  { <. x ,  y >.  |  ( <. x ,  y >.  e.  RingOps  /\  ( y  |`  (
( ran  x  \  {
(GId `  x ) } )  X.  ( ran  x  \  { (GId
`  x ) } ) ) )  e. 
GrpOp ) }
32relopabi 5116 . . . . . 6  |-  Rel  DivRingOps
43brrelexi 5029 . . . . 5  |-  ( G DivRingOps H  ->  G  e.  _V )
51, 4sylbir 213 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  ->  G  e.  _V )
65anim1i 566 . . 3  |-  ( (
<. G ,  H >.  e.  DivRingOps  /\  H  e.  A
)  ->  ( G  e.  _V  /\  H  e.  A ) )
76ancoms 451 . 2  |-  ( ( H  e.  A  /\  <. G ,  H >.  e.  DivRingOps
)  ->  ( G  e.  _V  /\  H  e.  A ) )
8 rngoablo2 25622 . . . . 5  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
9 elex 3115 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
108, 9syl 16 . . . 4  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  _V )
1110ad2antrl 725 . . 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 455 . . 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 530 . 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 25606 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
1514eleq2i 2532 . . 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 4203 . . . . . 6  |-  ( g  =  G  ->  <. g ,  h >.  =  <. G ,  h >. )
1716eleq1d 2523 . . . . 5  |-  ( g  =  G  ->  ( <. g ,  h >.  e.  RingOps  <->  <. G ,  h >.  e.  RingOps ) )
18 rneq 5217 . . . . . . . . 9  |-  ( g  =  G  ->  ran  g  =  ran  G )
19 fveq2 5848 . . . . . . . . . 10  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
2019sneqd 4028 . . . . . . . . 9  |-  ( g  =  G  ->  { (GId
`  g ) }  =  { (GId `  G ) } )
2118, 20difeq12d 3609 . . . . . . . 8  |-  ( g  =  G  ->  ( ran  g  \  { (GId
`  g ) } )  =  ( ran 
G  \  { (GId `  G ) } ) )
2221sqxpeqd 5014 . . . . . . 7  |-  ( g  =  G  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )
2322reseq2d 5262 . . . . . 6  |-  ( g  =  G  ->  (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  =  ( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) ) )
2423eleq1d 2523 . . . . 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 708 . . . 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 4204 . . . . . 6  |-  ( h  =  H  ->  <. G ,  h >.  =  <. G ,  H >. )
2726eleq1d 2523 . . . . 5  |-  ( h  =  H  ->  ( <. G ,  h >.  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
28 reseq1 5256 . . . . . 6  |-  ( h  =  H  ->  (
h  |`  ( ( ran 
G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId `  G
) } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) ) )
2928eleq1d 2523 . . . . 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 708 . . . 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 4754 . . 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 257 . 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 898 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458   {csn 4016   <.cop 4022   class class class wbr 4439   {copab 4496    X. cxp 4986   ran crn 4989    |` cres 4990   ` cfv 5570   GrpOpcgr 25386  GIdcgi 25387   AbelOpcablo 25481   RingOpscrngo 25575   DivRingOpscdrng 25605
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-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774  df-rngo 25576  df-drngo 25606
This theorem is referenced by:  zrdivrng  25632  isdrngo1  30599
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