MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdivrngo Structured version   Unicode version

Theorem isdivrngo 24063
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 4394 . . . . 5  |-  ( G DivRingOps H 
<-> 
<. G ,  H >.  e.  DivRingOps
)
2 df-drngo 24038 . . . . . . 7  |-  DivRingOps  =  { <. x ,  y >.  |  ( <. x ,  y >.  e.  RingOps  /\  ( y  |`  (
( ran  x  \  {
(GId `  x ) } )  X.  ( ran  x  \  { (GId
`  x ) } ) ) )  e. 
GrpOp ) }
32relopabi 5066 . . . . . 6  |-  Rel  DivRingOps
43brrelexi 4980 . . . . 5  |-  ( G DivRingOps H  ->  G  e.  _V )
51, 4sylbir 213 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  ->  G  e.  _V )
65anim1i 568 . . 3  |-  ( (
<. G ,  H >.  e.  DivRingOps  /\  H  e.  A
)  ->  ( G  e.  _V  /\  H  e.  A ) )
76ancoms 453 . 2  |-  ( ( H  e.  A  /\  <. G ,  H >.  e.  DivRingOps
)  ->  ( G  e.  _V  /\  H  e.  A ) )
8 rngoablo2 24054 . . . . 5  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  AbelOp )
9 elex 3080 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
108, 9syl 16 . . . 4  |-  ( <. G ,  H >.  e.  RingOps 
->  G  e.  _V )
1110ad2antrl 727 . . 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 457 . . 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 532 . 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 24038 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
1514eleq2i 2529 . . 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 4160 . . . . . 6  |-  ( g  =  G  ->  <. g ,  h >.  =  <. G ,  h >. )
1716eleq1d 2520 . . . . 5  |-  ( g  =  G  ->  ( <. g ,  h >.  e.  RingOps  <->  <. G ,  h >.  e.  RingOps ) )
18 rneq 5166 . . . . . . . . 9  |-  ( g  =  G  ->  ran  g  =  ran  G )
19 fveq2 5792 . . . . . . . . . 10  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
2019sneqd 3990 . . . . . . . . 9  |-  ( g  =  G  ->  { (GId
`  g ) }  =  { (GId `  G ) } )
2118, 20difeq12d 3576 . . . . . . . 8  |-  ( g  =  G  ->  ( ran  g  \  { (GId
`  g ) } )  =  ( ran 
G  \  { (GId `  G ) } ) )
2221, 21xpeq12d 4966 . . . . . . 7  |-  ( g  =  G  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )
2322reseq2d 5211 . . . . . 6  |-  ( g  =  G  ->  (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  =  ( h  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) ) )
2423eleq1d 2520 . . . . 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 710 . . . 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 4161 . . . . . 6  |-  ( h  =  H  ->  <. G ,  h >.  =  <. G ,  H >. )
2726eleq1d 2520 . . . . 5  |-  ( h  =  H  ->  ( <. G ,  h >.  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
28 reseq1 5205 . . . . . 6  |-  ( h  =  H  ->  (
h  |`  ( ( ran 
G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId `  G
) } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) ) )
2928eleq1d 2520 . . . . 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 710 . . . 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 4708 . . 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 893 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3071    \ cdif 3426   {csn 3978   <.cop 3984   class class class wbr 4393   {copab 4450    X. cxp 4939   ran crn 4942    |` cres 4943   ` cfv 5519   GrpOpcgr 23818  GIdcgi 23819   AbelOpcablo 23913   RingOpscrngo 24007   DivRingOpscdrng 24037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-1st 6680  df-2nd 6681  df-rngo 24008  df-drngo 24038
This theorem is referenced by:  zrdivrng  24064  isdrngo1  28903
  Copyright terms: Public domain W3C validator